Preparation for Financial Mathematics

INSTRUCTOR: SCOT ADAMS

(topics summaries)

Topic 0001 (Foundational Material: Logic and Set Theory) Link to Topics website

- 0001(1): title slide
- 0001(2): notation: for all, there exists, such that, implies, iff
- 0001(3): transition slide
- 0001(4): limit definition using notation
- 0001(5): QED, box marks end of problem,
*e.g.*,*i.e.* - 0001(6): def'n: size, cardinality, the empty set, eg size, size of empty set,
- 0001(6): eg: union, intersection, complelement, notation (is an element of)
- 0001(6): notation: \Z:={integers}, \R:={reals}, \Q:={rationals}, \C:={complex numbers}
- 0001(7): eg: disjoint union
- 0001(8): notation: subset, superset
- 0001(9-21): Venn diagrams
- 0001(9-10): union
- 0001(11-12): intersection
- 0001(13-14): if (A \subseteq B), then (A \cap B = A)
- 0001(15): complement
- 0001(16): set-theoretic difference
- 0001(17-21): complement of the union = the intersection of the complements

- 0001(22): def'n: partition
- 0001(22-24): eg/non-eg: partition
- 0001(25): eg: partition
- 0001(26): def'n/notation: ordered pair, contrast with unordered pair
- 0001(26): def'n/notation: Cartesian product, A^n
- 0001(26-27): \R^2 and \R x {3} visualizations
- 0001(28): def'n: Euclidean n-space := \R^n

Topic 0002 (Numbers and sets of numbers) Link to Topics website

- 0002(1): title slide
- 0002(2): factorial notation and eg
- 0002(3-21): eg/visualization: intervals
- 0002(4-20): SKILL: graph interval
- 0002(5): eg: open, bounded
- 0002(7 and 9): eg: open, unbounded
- 0002(11): eg: compact
- 0002(11): def'n: compact := closed and bounded
- 0002(13 and 15): eg: closed, unbounded, noncompact
- 0002(17): eg: half-open, bounded
- 0002(19): eg: half-open, bounded
- 0002(20): def'n: degenerate
- 0002(20): eg: closed, bounded, compact, degenerate
- 0002(21): eg: neither open nor closed
- 0002(21): eg: both open and closed
- 0002(21): the empty set is not an interval
- 0002(21): SKILL: identify interval (open, closed, half-open, bounded, unbounded, degenerate, nondegenerate)

- 0002(22-24): visualization: \Z
- 0002(24): \Z meets some open intervals, but not others

- 0002(25-26): visualization: \Q
- 0002(26): \Q meets all open intervals
- 0002(26): def'n: dense := meets all open intervals

- 0002(27): def'n: extended real number
- 0002(27): notation: \bar\R := {extended real numbers}
- 0002(27): ordering the extended real numbers
- 0002(28): def'n/eg: extended interval
- 0002(28): def'n/eg: interval
- 0002(29): def'n/eg: lower bound, infimum/inf/greatest lower bound/glb
- 0002(30): def'n/eg: min/minimum
- 0002(31): some subsets of \bar\R have no min
- 0002(32): def'n/eg: upper bound, supremum/sup/least upper bound/lub, max
- 0002(32): some subsets of \bar\R have no max
- 0002(33): eg: sup/max/inf/min
- 0002(34): every subset of \bar\R has an inf and a sup
- 0002(35): def'n/eg: finite union of intervals/fUofI
- 0002(36): def'n/notation/eg: length of an interval
- 0002(37): bounded = finite length, for intervals
- 0002(38): def'n/notation/eg: measure of a fUofI
- 0002(39): Warning: must rewrite the fUofI as a

finite*disjoint*union of intervals - 0002(39): SKILL: express a fUofI as a finite
*disjoint*union of intervals - 0002(39): SKILL: measure of a fUofI
- 0002(40): def'n: partition by intervals
- 0002(41): eg/non-eg: partition by intervals
- 0002(42): def'n: fUofIs = finite unions of intervals
- 0002(42): def'n: partition by fUofIs
- 0002(43): eg: partition by fUofIs

Topic 0003 (Absolute value and distance) Link to Topics website

- 0003(1): title slide
- 0003(2): def'n/notation: |x| := the absolute value of x
- 0003(3-5): eg: absolute value
- 0003(3-5): SKILL: compute absolute value
- 0003(5): \sqrt{x^2} is |x|,
*not*x - 0003(6): eg: distance on \R
- 0003(6): distance from a to b is |b-a|,
*not*b-a - 0003(6): SKILL: compute distance on \R
- 0003(7): (|x-a| \le r) iff (a-r \le x \le a+r)
- 0003(8): (|x-a| < r) iff (a-r < x < a+r)
- 0003(9-12): SKILL: graph an absolute value inequality
- 0003(9-10): graph |x-a| < r
- 0003(11-12): graph |x-a| \le r
- 0003(13-18): SKILL: graph a neighborhood
- 0003(13): graph the 0.5-neighborhood of 1
- 0003(13): graph |x-1| < 0.5
- 0003(14): graph the punctured 0.5-neighborhood of 1
- 0003(14): graph 0 < |x-1| < 0.5
- 0003(15-18): more graphing of neighborhoods and punctured neighborhoods
- 0003(19-20): distance in \R, in \R^2 and in \R^3
- 0003(21): triangle inequality in \R^2
- 0003(22-27): triangle inequality in \R
- 0003(27-30): additivity of error
- 0003(27-30): [ (|x-s| < \sigma) and (|y-t| < \tau) ] implies [(x+y)-(s+t) < \sigma+\tau]
- 0003(31): def'n/notation/eg: x_+ := the positive part of x
- 0003(31): SKILL: compute x_+
- 0003(32): x_+ = (|x|+x)/2
- 0003(32): def'n/notation: x_- := the negative part of x
- 0003(32): x_- = (|x|-x)/2
- 0003(33-34): formula for x_-
- 0003(35): transition slide
- 0003(36): eg: negative part
- 0003(36): SKILL: compute x_-
- 0003(36): x_- \ge 0
- 0003(37): the reproducing equation (x_+ minus x_- is equal to x)
- 0003(37): the absolute value equation (x_+ plus x_- is equal to |x|)

Topic 0004 (Functions and expressions) Link to Topics website

- 0004(1): title slide
- 0004(2): def'n/eg: function; notation: f:A-->B, f(a)
- 0004(2): def'n: domain, target
- 0004(3): elements of the domain used exactly once
- 0004(4): elements of the target may not be used, or used more than once
- 0004(5-6): eg: the squaring function
- 0004(7-8): G(2,3,4) preferred to G((2,3,4))
- 0004(9): def'n: image, onto, surjective, one-to-one, 1-1, injective
- 0004(9): eg: onto, not 1-1
- 0004(10): eg: 1-1, not onto
- 0004(11): eg: neither 1-1 nor onto
- 0004(12): def'n/eg: bijective = 1-1 and onto
- 0004(13-14): eg: inverses
- 0004(15): def'n: composite
- 0004(16): def'n: identity; redefine inverses
- 0004(17): composition is associative
- 0004(18): composition is not commutative
- 0004(19): x^2, t^2 and the squaring function; notation: evaluation, change in output
- 0004(20): recall inf/infimum/glb, min, sup/supremum/lub, max
- 0004(21): notation: sup_A f = sup_{x\in A} f(x) = etc.
- 0004(22): eg: sup_A f = sup_{x\in A} f(x) = etc.
- 0004(23): notation: inf_A f = inf_{x\in A} f(x) = etc.
- 0004(23): eg: sup_A f = sup_{x\in A} f(x) = etc.
- 0004(24-25): notation: max_A f = etc.; max may not exist
- 0004(26): notation: min_A f = etc.
- 0004(27): The Extreme Value Theorem
- 0004(28-29): def'n: caglad, caglad in x, caglad in t
- 0004(30): def'n/visualization: increasing
- 0004(31): non-visualization: increasing
- 0004(32-33): maximal interval of increase
- 0004(34): def'n/visualization: decreasing, semi-increasing/nondecreasing, semi-decreasing/nonincreasing
- 0004(35): notation: rescaling function
- 0004(36): def'n: scalar := number
- 0004(37): constant = scalar
- 0004(38): notation: addition of functions
- 0004(39): def'n: linear operations, linear combination of functions/expressions, coefficient
- 0004(40): eg: linear combination of expressions of x; domains intersected
- 0004(41): eg: linear combination of expressions of t; domains intersected
- 0004(42): def'n: polynomial in x, degree
- 0004(43): def'n: polynomial in t, degree
- 0004(44): def'n: polynomial in r, degree
- 0004(45): def'n: polynomial, degree
- 0004(46): def'n: constant, linear, quadratic, cubic, quartic, quintic
- 0004(46): SKILL: find the degree of a polynomial
- 0004(47): def'n/eg: constant term, linear term, quadratic term, cubic term, quartic term, quintic term
- 0004(47): SKILL: identify the terms of a polynomial (by degree)
- 0004(48): def'n/eg: constant coefficient, linear coefficient, quadratic coefficient, cubic coefficient, quartic coefficient, quintic coefficient
- 0004(48): SKILL: identify the coefficients of a polynomial (by degree)
- 0004(49): def'n/eg: leading coefficient
- 0004(49): SKILL: identify the leading coefficient of a polynomial
- 0004(50): def'n/eg: leading term
- 0004(50): SKILL: identify the leading term of a polynomial
- 0004(51): def'n: exponentially bounded / exp-bdd
- 0004(52-53): visualization: exponentially bounded
- 0004(54): eg/non-eg: exp-bdd expressions in x

Topic 0005 (Completing the square) Link to Topics website

- 0005(1): title slide
- 0005(2): eg/SKILL: collecting like terms; decreasing degree; increasing degree
- 0005(3-6): eg: horizontal translations and replacing x by x-a
- 0005(7): eg: vertical translations and replacing y by y-a
- 0005(8): eg: horizontal dilations and replacing x by x/a
- 0005(9): eg: vertical dilations and replacing y by y/a
- 0005(10-18): eg: eliminate the linear term and graph a quadratic
- 0005(19): eg/SKILL: find the translation that elminates the linear term (in a quadratic)
- 0005(20): transition slide
- 0005(21): eg/SKILL: eliminate the linear term (in a quadratic)
- 0005(22-23): find the translation that elminates the linear term in ax^2 + bx + c
- 0005(24-25): elminate the linear term in ax^2 + bx + c
- 0005(26-30): special case: -(x^2/2) + bx + c
- 0005(26-28): eg: -(x^2/2) + 19x + 5
- 0005(26): replace x by x + (the linear coefficient)
- 0005(29): in general, -(x^2/2) + bx + c
- 0005(29): replace the linear term by (1/2)(linear coefficient)^2
- 0005(30): eg: -(x^2/2) + 8x - 3

Topic 0006 (Combinatorics) Link to Topics website

- 0006(1): title slide
- 0006(2-6): counting reorderings
- 0006(7-12): counting ways to choose k objects from a set of n
- 0006(7-8): motivation in finance
- 0006(9): notation/eg: "n choose k"
- 0006(12): (5 choose 2) = (5 choose 3); (n choose k) = (n choose n-k)

- 0006(13-14): reordering a partition respecting the comma; counting these reorderings
- 0006(15): each reordering of abcde is (up to removal of a comma) equal to

a reordering of one of the partitions of abcde into: two letters, three letters - 0006(16-20): listing all the reorderings of abcde by listing partitions, reordering, removing commas
- 0006(21): key point: 5! = (5 choose 2) x (2!) x (3!) = (5 choose 3) x (2!) x (3!)
- 0006(22): (5 choose 2) = [5!] / [(2!)(3!)]
- 0006(23): (n choose k) = [n!] / [(k!)((n-k)!)]
- 0006(24): eg: computation of (100 choose 9)
- 0006(24): SKILL: compute binomial coefficients
- 0006(25): (n choose 1) is a polynomial in n of degree 1
- 0006(26): (n choose 2) is a polynomial in n of degree 2
- 0006(27-28): (n choose 3) is a polynomial in n of degree 3
- 0006(28): SKILL: for fixed d, write (n choose d) as a degree d polynomial in n

Topic 0007 (The binomial formula) Link to Topics website

- 0007(1): title slide
- 0007(2-3): forumlas for (x+y)^n, with 2^n terms, with duplication
- 0007(4-12): formulas for (x+y)^n, with n+1 terms, without duplication
- 0007(6-8): following the coefficients from n=2 to n=3
- 0007(9): following the monomials in x and y (from x^4 to y^4)
- 0007(9): reorganization of coefficient computation to avoid duplicate writing
- 0007(10-12): rerorganization of coefficients into a triangle
- 0007(12): Pascal's triangle

- 0007(13-19): counting terms of the form x^2 y^3 in the 2^5-term expansion of (x+y)^n, with duplication
- 0007(18): answer is (5 choose 2)
- 0007(19): answer is (5 choose 3)

- 0007(20-22): formula for (x+y)^5 without duplication, using (5 choose k)'s as coefficients
- 0007(21): the binomial formula for (x+y)^5 with (5 choose k) coefficients and k decreasing from 5 to 0
- 0007(22): the binomial formula for (x+y)^5 with (5 choose k) coefficients and k increasing from 0 to 5
- 0007(22): def'n: binomial coefficients

Topic 0008 (Counting monomials) Link to Topics website

- 0008(1): title slide
- 0008(2): def'n/eg: monomial; eg: degree of a monomial
- 0008(3-9): counting monomials of degree \le d, in n variables
- 0008(3): eg: d=3,n=2; answer is 10; (3+2 choose 2) = (3+2 choose 3) = 10
- 0008(4): eg: d=3,n=4; answer is 35; (3+4 choose 3) = (3+3 choose 4) = 35
- 0008(5-9): eg: d=8,n=4
- 0008(5): answer is difficult; (8+4 choose 4) = 495
- 0008(6-9): counting without listing; 1-1 correspondence; answer is 495

- 0008(10-16): counting monomials of degree = d, in n variables
- 0008(11-15): d=6, n=4; counting without listing; 1-1 correspondence
- 0008(16): answer is (d+n-1 choose d) = (d+n-1 choose n-1)

- 0008(17): transition slide
- 0008(18): a relation between monomial counts
- 0008(19-22): explanation for the relation via Pascal's triangle

Topic 0009 (One variable diff calc) Link to Topics website

- 0009(1): title slide
- 0009(2): squaring and its derivative,
- as an expression in x, as an expression in t, as a function

- 0009(2): introducing d/dx, d/dt and prime
- 0009(3-4): allowable notation, not allowed, and usually avoided
- 0009(5): cosine and its derivative
- as an expression in x, as an expression in t, as a function

- 0009(6): logarithm and its derivative
- as an expression in x, as an expression in t, as a function

- 0009(7): (d/dx)[f(x)]=f'(x)
- 0009(8-11): the product rule
- 0009(12-13): the quotient rule
- 0009(14-17): the chain rule
- 0009(18-20): practice problem in differentiation
- 0009(21): def'n/eg: logarithmic derivative
- 0009(22): principle of logarithmic differentiation
- 0009(23-25): eg: logarithmic differentiation

Topic 0010 (The IVT and the MVT) Link to Topics website

- 0010(1): title slide
- 0010(2): def'n: value; Intermediate Value Theorem (IVT)
- 0010(3-7): illustration of the Mean Value Theorem (MVT) by a trip to Chicago
- 0010(8): statement and visualization of MVT
- 0010(9): non-uniqueness of the solution to the MVT equation
- 0010(10-17): applications of the MVT
- 0010(11): the increasing test
- 0010(12): the decreasing test
- 0010(13): the 1-1 test
- 0010(14): the constant test
- 0010(15): equality of derivatives
- 0010(16): the nonincreasing and nondecreasing tests

- 0010(17):stop slide

Topic 0011 (One variable integral calculus review) Link to Topics website

- 0011(1): title slide
- 0011(2-3): def'n/eg: antiderivative
- 0011(4): def'n/eg: antiderviative wrt (with respect to) x
- 0011(5): d/dx is not 1-1
- 0011(6): equality of derivatives controls the noninjectivity of d/dx
- 0011(6): the set of all antiderivatives of x^2 wrt x
- 0011(7): the (indefinite) integral of x^2 wrt x, notation: \int x^2 dx
- 0011(8-10): (1/3)x^3 + C = (1/3)x^3 + 6C
- 0011(11): indefinite integration and antidifferentiation wrt v
- 0011(12): indefinite integration and antidifferentiation wrt t
- 0011(13): indefinite integration and antidifferentiation wrt s
- 0011(14): indefinite integration and antidifferentiation
- 0011(15-32): area under a curve
- 0011(15): the goal
- 0011(16): three subintervals (3rd partition)
- 0011(17): left endpoint, midpoint, right endpoint for one subinterval
- 0011(18): left endpoints, midpoints, right endpoints
- 0011(19): transition from 3rd to 10th partition
- 0011(20): the 10th partition
- 0011(21): rectangles (left endpoints, 10th partition)
- 0011(22): 3rd partition, right endpoints
- 0011(23): rectangles (right endpoints, 3rd partition); total area
- 0011(23): right 3rd Riemann sum from a to b of f; R_n S_a^b f
- 0011(24): 3rd partition, midpoints
- 0011(25): rectangles (midpoints, 3rd partition); total area
- 0011(25): midpoint 3rd Riemann sum from a to b of f; M_n S_a^b f
- 0011(26): 3rd partition, left endpoints
- 0011(27): rectangles (left endpoints, 3rd partition); total area
- 0011(27): left 3rd Riemann sum from a to b of f; L_n S_a^b f
- 0011(28): rectangles (left endpoints, 10th partition)
- 0011(29): rectangles (left endpoints, 60th partition)
- 0011(30): In limit, as n-->\infty, all three Riemann sums converge to the same value
- 0011(31): def'n/notation: the definite integral from a to b of f(x) wrt x; \int_a^b f(x) dx
- 0011(32): the definite integral represents the area under the curve

- 0011(33): Riemann sums with varying points
- 0011(34-36): Riemann sums with intervals of varying lengths, mesh -->0
- 0011(34): def'n: mesh

Topic 0012 (Example of a definite integral) Link to Topics website

- 0012(1): title slide
- 0012(2): visualization: area under y=x^2 from x=0 to x=1; f(x):=x^2
- 0012(3): visualization: R_8 S_0^1 f
- 0012(4): computation of R_8 S_0^1 f as a sum
- 0012(5): computation of R_n S_0^1 f as a sum
- 0012(5): computation of \int_0^1 f as a limit of sums
- 0012(6-7): simplification of the sums (via an IOU); computation of the limit
- 0012(8-21): proof of the IOU
- 0012(8): transition slide; def'n: sequence
- 0012(9): def'n/notation/eg: difference operator on a sequence, \triangle a_n
- 0012(10): the difference of n^4; use of Pascal's triangle
- 0012(11): the coefficients come from Pascal's triangle
- 0012(12): the difference of the quartic n^4 is a cubic (in n)
- 0012(13): the difference of s_n := 1+...+n^2
- 0012(14): the difference of s_n, \triangle s_n, is a quadratic in n
- 0012(14): expect s_n to be a cubic in n
- 0012(15): the difference of n^3
- 0012(16): the difference of n^2 and of n
- 0012(17): matching the first two terms in \triangle s_n
- 0012(18): matching the last term in \triangle s_n
- 0012(19): a cubic in n with the same difference as s_n
- 0012(20): the cubic and s_n differ by a constant; the constant is 0
- 0012(21): bringing the cubic to a common denominator

Topic 0013 (The Fundamental Theorem of Calculus) Link to Topics website

- 0013(1): title slide
- 0013(2): connecting antidifferentiation to area:

The Fundamental Theorem of Calculus (FTC) - 0013(2): we'll connect change in position (antideriv. of velocity)

to area under the graph of velocity - 0013(3): problem of going from velocity to position, esp. change in position

specific problem: given velocity v(t)=t^2, compute [p(11)]-[p(5)] - 0013(3): start by splitting [5,11] into three subintervals
- 0013(4): transition slide
- 0013(5): estimate of [p(11)]-[p(5)] using midpoint velocities
- 0013(6): transition slide
- 0013(7-9): computation of M_3 S_5^{11} v
- 0013(10): [p(11)]-[p(5)] is approximately equal to M_3 S_5^{11} v
- 0013(11): [p(11)]-[p(5)] is approximately equal to M_n S_5^{11} v, and

error --> 0 as n --> \infty, so:

[p(11)]-[p(5)] is equal to \int_5^{11} t^2 dt - 0013(11): this connects change in position (antideriv. of velocity)

to area under the graph of velocity - 0013(12): \int_5^{11} t^2 dt is hard to calculate from the definition, but, as the change in an antiderivative, it's easy: [11^3/3] - [5^3/3]
- 0013(13): transition slide
- 0013(14): key idea: to compute a definite integral, find an antiderivative, then evaluate at the limits of integration, then subtract
- 0013(15): the FTC, definite integrals version
- 0013(16-17): (d/dx)\int_5^x t^2 dt = [t^2]_{t:\to x}
- 0013(18): the FTC, antiderivatives version
- 0013(19): change of notation; warning: \int_a^x f(x) dx is not acceptable
- 0013(20): recomputation of \int_0^1 x^2 dx via the FTC

Topic 0014 (Techniques of one variable integration) Link to Topics website

- 0014(1): title slide
- 0014(2-17): integration by substitution
- 0014(2-3): the general rule for indefinite integrals
- 0014(4): an example
- 0014(5): the general rule for definite integrals
- 0014(5-9): replacing x by x + 5 in a definite integral
- 0014(9): dx unchanged, limits of integration decreased by 5
- 0014(10-17): a definite integral of the exponential of a quadratic
- 0014(17): SKILL: integrate exp(quadratic)

- 0014(18-22):
- 0014(18): the general rule (up up minus the integral of new new)
- 0014(19-20): an example
- 0014(21): another example
- 0014(22): redoing the second example by tabular integration

Topic 0015 (Sequences and series) Link to Topics website

- 0015(1): title slide
- 0015(2): (recall def'n)/eg: (real) sequence
- 0015(3): tails of a sequence
- 0015(4): def'n/eg: liminf of a sequence
- 0015(5): def'n/eg: limsup of a sequence
- 0015(6): limit exists iff liminf = limsup
- 0015(7-8): eg: liminf and limsup
- 0015(8): limsup = sup of limits of subsequences

liminf = inf of limits of subsequences

- 0015(9): def'n/eg: (real) series
- 0015(10): (-1)+(-1)+(-1)+... often written -1-1-1-...
- 0015(11-12): eg: sequence of partial sums
- 0015(13): recall def'n: extended real number
- 0015(13): def'n: converges to an extended real number
- 0015(13): eg: series = extended real number means

the limit of the partial sums is the extended real number - 0015(14): eg: sums of series; a series may converge or diverge
- 0015(15): it may diverge to \infty or -\infty
- 0015(16): it may have no sum at all, eg: 1-1+1-1+1-1+... has no sum
- 0015(17): 1+(1/2)+(1/3)+... deferred to later
- 0015(18): recall: linear operations
- 0015(18): def'n: infinite linear combination of functions
- 0015(19): def'n: infinite linear combination of expressions of q
- 0015(20): def'n: infinite linear combination of expressions of s
- 0015(21): recall def'n: polynomial in x; def'n: power series in x
- 0015(22): transition slide
- 0015(23): eg: power series in x
- 0015(24): def'n/eg; power series in u

Topic 0016 (Polynomial approximation) Link to Topics website

- 0016(1): title slide
- 0016(2): def'n/notation/eg: n-jet of f(x) at a, (J^nf)(a) or J^n_af
- 0016(2): (J^4(sin))(\pi/6)
- 0016(3): \tilde f(x)=f(-x) implies (J^n\tilde f)(a)=...
- 0016(4): if f and g agree to order n at 0, then so do \tilde f and \tilde g
- 0016(5): def'n: second order Maclaurin approximation
- 0016(6): formula for second order Maclaurin approximation
- 0016(7): def'n: third order Maclaurin approximation
- 0016(8): formula for third order Maclaurin approximation
- 0016(9): def'n: nth order Maclaurin approximation
- 0016(10): formula for nth order Maclaurin approximation
- 0016(11): Maclaurin expansion
- 0016(12): def'n: Maclaurin expansion
- 0016(13): eg: Maclaurin expansion
- 0016(14): when is a function equal to its Maclaurin expansion?
- 0016(15): recall: decreasing test, nonincreasing test
- 0016(15-16): antidifferentiation of inequalities
- 0016(17-22): eg: antidifferentiation of inequalities
- 0016(20-22): traveling particle problems
- 0016(23-25): error estimate in third order Maclaurin approxmation at 5
- 0016(26): error estimate in nth order Maclaurin approxmation at a \ge 0
- 0016(27): condition under which a function is equal to the sum of its Maclaurin expansion
- 0016(28): proof of the result
- 0016(29): eg: e^9 is the sum of (the Maclaurin expansion of e^x) evaluated at x:-->9
- 0016(30): nothing special about 9
- 0016(31): transition slide
- 0016(32): various expressions are equal to the sum of their Maclaurin expansions
- 0016(e^x, sin x, cos x
- ln(1+x), for x \in (-1,1]

- 0016(33): the error in the 2nd-order Maclaurin approximation is o(x^2), i.e., tends to zero faster than x^2
- 0016(34): nothing special about 2
- 0016(35): the error in the nth-order Maclaurin approximation is o(x^n) i.e., tends to zero faster than x^n

Topic 0017 (Conditional convergence of series) Link to Topics website

- 0017(1): title slide
- 0017(2): recall some sums of series; 1+(1/2)+(1/3)+... still unknown
- 0017(3): the harmonic series is 1+(1/2)+(1/3)+...; goal: lower bound on 31st partial sum
- 0017(4): transition slide
- 0017(5): grouping of terms
- 0017(6): comparision with another sum
- 0017(7-8): the lower sum is 5/2
- 0017(9): 5/2 is a lower bound on the 31st partial sum of the harmonic series
- 0017(10): transition slide
- 0017(11): partial sums are unbounded; harmonic series diverges to \infty
- 0017(12): alternating harmonic series converges to ln 2
- 0017(13): see more terms
- 0017(14-17): a rearrangement (of the alternating harmonic series) converges to (3/2)(ln 2)
- 0017(18-19): the sum of the positive terms is \infty; the sum of the negative terms is -\infty
- 0017(20): transition slide
- 0017(21-24): there's a rearrangement (of the alternating harmonic series) converges to 1000
- 0017(24): there's a rearrangement that converges to any real number
- 0017(25): transition slide
- 0017(26): there's a rearrangement that sums to \infty; there's a rearrangement that sums to -\infty;

there's a rearrangement that has no sum - 0017(26): this happens for
*any*series whose positive terms sum to \infty and whose negative terms sum to -\infty - 0017(27): def'n: nonnegative series
- 0017(28): def'n/eg: generalized partial sum
- 0017(29): the sum of a nonnegative series is the sup of its generalized partial sums
- 0017(30): every nonnegative series has a sum (possibly \infty) and rearrangement doesn't affect it
- 0017(31): recall the positive/negative parts of x; recall the reproducing/absolute value eq'ns
- 0017(32): if the sums of positive and negative parts are both \infty, then rearrangements converge to any number
- 0017(32): if the sums of positive and negative parts are not both \infty, then the reproducing equation gives the sum of all rearrangements
- 0017(33): summary
- 0017(34-44): similar issues in integration
- 0017(34): \int_{-\infty}^\infty as a limit of \int_{-K}^K
- 0017(35-37): problem with change of variables
- 0017(38-43): problem persists even assuming smoothness
- 0017(44): problem persists even on a bounded interval

Topic 0018 (Some important indeterminate forms) Link to Topics website

- 0018(1): title slide
- 0018(2): limits of some rational expressions of n
- 0018(3): continuous compounding motivates limit of [(1+(r/n)]^n
- 0018(4-5): expansion of [(1+(r/n)]^n
- 0018(6-7): computation of the limit
- 0018(7): limit of [(1+(r/n)]^n is e^r
- 0018(7): e^r is the risk-free factor
- 0018(8): [1+(7/n)+o(1/n)]^n --> e^7
- 0018(9-15): proof
- 0018(15): [1+(x/n)+o(1/n)]^n --> e^x
- 0018(16): the renormalized powers of 1-x^2 converge to e^{-x^2}
- 0018(17-22): visualization of these renormalized powers,

and of e^{-x^2} - 0018(23): x:-->3
- 0018(24): if the 2nd order Macl (Maclaurin) approx of f is 1-7x^2,

then the renormalized powers of f(x) converge to a bell curve,

for x=3 - 0018(25-32): proof
- 0018(33-34): another proof using the "o" notation
- 0018(35): if the 2nd order Macl (Maclaurin) approx of f is 1-ax^2,

then the renormalized powers of f(x) converge to a bell curve,

- 0018(35): eg: the renormalized powers of cos x
- 0018(36-40): visualization of the renormalized powers of cos x,

and of their limit

Topic 0019 (Complex numbers) Link to Topics website

- 0019(1): title slide
- 0019(2): scalar typically means real number, but not in this topic
- 0019(2): def'n/eg: complex (cx) number

def'n/notation/eg: real part, imaginary part - 0019(2): SKILL: Finding real and imaginary parts of cx numbers
- 0019(3): SKILL: addition of cx numbers; eg: cx addition
- 0019(4-6): geometry of cx addition
- 0019(7): SKILL: absolute value of a cx number; modulus=absolute value
- 0019(7): notation: absolute value; eg: absolute value
- 0019(8): geometry of absolute value
- 0019(9-12): dist(u,v)=|u-v|
- 0019(12): eg: distance
- 0019(13): SKILL: distance between two cx numbers
- 0019(14): triangle inequality for cx numbers
- 0019(15): SKILL: multiplication of cx numbers; eg: cx multiplication
- 0019(16): addition/multiplication is commutative/associative
- 0019(16): multiplication distributes over addition
- 0019(17): def'n/eg/notation: complex conjugate
- 0019(17): SKILL: complex conjugate
- 0019(17): cx conjugation distributes over addition and multiplication
- 0019(18): proof that cx conjugation distributes over multiplication
- 0019(19): |z|=\sqrt{z \bar{z}}
- 0019(20): transition slide
- 0019(21): absolute value distributes over multiplication
- 0019(22): def'n: e^z; (e^z)(e^w)=e^{z+w}
- 0019(23-25): e^{ix} = (cos x) + i (sin x)
- 0019(26): transition slide
- 0019(27): visualization of re^{i\theta}; argument of a complex number
- 0019(28): geometry of cx multiplication
- 0019(29): SKILL: exponentiating cx numbers; eg: exp of a cx number
- 0019(30-34): exponentiation commutes with cx conjugation

Topic 0020 (Topology) Link to Topics website

- 0020(1): title slide
- 0020(2): def'n/eg: boundary point
- 0020(3-5): eg: boundary of a set
- 0020(5): boundary of product \ne product of boundaries
- 0020(6-7): eg: boundary
- 0020(8-11): def'n/eg/non-eg: open
- 0020(12-17):def'n/eg/non-eg: closed
- 0020(15): most sets are neither open nor closed

- 0020(18): def'n: clopen; the two clopen sets in \R^n;
- 0020(19-22): def'n/eg/non-eg: compact
- 0020(20): intution of compact sets: all homeomorphic images bounded

- 0020(23): SKILLs
- 0020(identify from picture if a set is open, closed, both or neither
- identify from picture if a set is compact
- identify from description if a set is open, closed, both or neither
- identify from description if a set is compact

- 23): are any sets both open and compact?
- 0020(24): def'n: f is supported on C
- 0020(24): def'n: f is compactly suported/has compact support
- 0020(24): visualization when the domain of f is \R

Topic 0021 (Basics of vector spaces) Link to Topics website

- 0021(1-2): title slides
- 0021(3): def'n/eg: dot product;def'n: vector; SKILL: vector dot product
- 0021(4-5): a game: determining a vector from its dot products
- 0021(6): def'n: linear operations; SKILLS: vector addition, scalar multiplication
- 0021(7): def'n: linear combination, coefficients: SKILL: linear combinations (l.c.)
- 0021(8): def'n/eg: subspace/vector subspace/linear subspace
- 0021(9): def'n/notation: span/linear span
- 0021(10): transition slide
- 0021(11): more notation for span of a finite set of vectors
- 0021(12): eg of a subspace of \R^4 described as a span of three vectors
- 0021(13): one of the vectors is a l.c. of the others
- 0021(14): transition slide
- 0021(15): that vector can be dropped without affecting the span
- 0021(16): def'n: linearly dependent (l.d.), linearly independent (l.i.)
- 0021(17): v not in the span of F implies

linear independence of F unaffected by adding v - 0021(18-19): l.i. iff (the only l.c. equal to 0 is the trivial one)
- 0021(20): def'n: spanning set, basis
- 0021(21): standard basis of \R^n

Topic 0022 (Basics of linear transformations) Link to Topics website

- 0022(1): title slide
- 0022(2): linear relationships between variables
- 0022(3): transition slide
- 0022(4): exact vs. approximate linear relationships
- 0022(5): a linear relationship as a function L_0 : \R^4 --> \R^2
- 0022(6): transition slide
- 0022(7): def'n/eg: matrix; dimensions of a matrix; def'n/eg: L_M
- 0022(8-10): dotting the rows of the matrix by the vector input
- 0022(10): SKILL: compute L_M(p)
- 0022(11): def'n: linear map between subspaces
- 0022(12): def'n/notation: linear function/linear transformation corresponding to M
- 0022(12-13): a game: determining the matrix M from outputs of L_M
- 0022(14): the jth column of M is L_M(e_j)
- 0022(14): notation: the matrix [A] of a linear transformation A
- 0022(14): L_{[A]}=A and [L_M]=M
- 0022(15): every linear transformation comes from a matix
- 0022(15): no two matrices give the same linear transformation
- 0022(15): def'n: isomorphism/vector space isomorphism
- 0022(15): isomorphic means "the same to a linear algebraist"
- 0022(16): def'n/notation: kernel and image
- 0022(16): kernels and images are subspaces
- 0022(16): onto iff (image = target)
- 0022(17): transition slide
- 0022(18): 1-1 iff (ker = {0})
- 0022(19): def'n/eg: ordered basis
- 0022(20-24): ordered bases of a subspace V in 1-1 correspondence with

isomorphisms between V and Euclidean space

Topic 0023 (Matrix operations) Link to Topics website

- 0023(1): title slide
- 0023(2-10): eg: given A and B find C s.t. L_C = L_A \circ L_B
- 0023(11): def'n/notation: product of matrices
- 0023(12): to define AB, need that

the number of columns in A is equal to

the number of rows in B - 0023(12): matrix multiplication is not commutative
- 0023(13): matrix multiplication is associative
- 0023(14): eg: matrix multiplication
- 0023(15): SKILL: matrix multiplication; eg for discussion: matrix multiplication
- 0023(16): associations between vectors, column vectors and row vectors
- 0023(17): L_M(v) "is" Mv; more precisely:
- 0023(17): the column vector of L_M(v) is

M times the column vector of v - 0023(18): def'n/eg/notation/SKILL: matrix addition
- 0023(19): def'n/eg/notation/SKILL: direct sum of matrices
- 0023(20): def'n/notation: tensor product of matrices
- 0023(21-23): eg: tensor product of matrices
- 0023(24): the standard basis of R^{2x2}
- 0023(24-30): the reproducing equation
- 0023(31): def'n: the multiplication map \scrM
- 0023(31-37): \scrM( A \otimes B ) = AB
- 0023(38): def'n/eg: zero matrix
- 0023(39): zero matrix is an additive identity
- 0023(40): def'n: the diagonal of a square matrix
- 0023(41): the diagonal of a non-square matrix
- 0023(42): def'n: identity matrix
- 0023(43): identity matrix is a left identity
- 0023(44): identity matrix is a right identity
- 0023(45): def'n: inverse; invertible matricies are square
- 0023(45): a matrix cannot have two inverses
- 0023(46): def'n: left conjugate, right conjugate, conjugate
- 0023(46): SKILL: Matrix conjugation
- 0023(47): def'n: transpose
- 0023(47-50): as you pull M across the dot product, it gets transposed

Topic 0024 (Matrix types) Link to Topics website

- 0024(1): title slide
- 0024(2): def'n: diagonal matrix
- 0024(3): def'ns:
- upper triangular matrix
- strictly upper triangular matrix
- lower triangular matrix
- strictly lower triangular matrix

- 0024(4): def'n: symmetric matrix
- 0024(5): def'n: anti-symmetric matrix
- 0024(6): def'n: nilpotent matrix
- 0024(6-8): any strictly upper triangular matrix is nilpotent
- 0024(8): any conjugate of a strictly upper triangular matrix is nilpotent
- 0024(9): def'n: scalar matrix
- 0024(10): def'n: standard nilpotent
- 0024(11): def'n: Jordan block
- 0024(11): every 1x1 is a Jordan block
- 0024(12): def'n/notation: exponential (exp) of a matrix
- 0024(12): AB=BA implies e^{A+B} = e^A e^B
- 0024(13): multiplication of diagonal matrices
- 0024(14-17): eg: exp of a diagonal matrix
- 0024(17): diagonal matrices are easier to studay than generic matrices
- 0024(18): reflection in the y-axis in \R^2
- 0024(19): counterclockwise rotation in \R^2
- 0024(20): def'n: orthogonal = distance-preserving
- 0024(21): orthogonal = length-preserving
- 0024(22): orthogonal = dot-product-preserving
- 0024(23): M orthogonal iff M^t M = I

Topic 0025 (Intro to row and col ops) Link to Topics website

- 0025(1): title slide
- 0025(2-7): solving a system of linear equations
- 0025(8-10): solving the same system using row operations in matrices
- 0025(11): the primary elementary row operations, with examples
- 0025(12): elementary row operations are left multiplications
- 0025(13-15): the secondary elementary row operations, with examples
- 0025(16-20): interchanging two rows via primary row operations
- 0025(21): the primary elementary column operations
- 0025(22): the secondary elementary column operations
- 0025(23-24): elementary column operations are right multiplications
- 0025(25-26): elementary matrices are invertible
- 0025(27): def'n/eg: column-padded identity
- 0025(28): SKILL: row magic
- 0025(29): eg of row magic
- 0025(30): SKILL: row canonical form
- 0025(31-33): eg of row canonical form
- 0025(34): column-padded identity, column magic and column canonical form
- 0025(35): how to go from row canonical form to fully canonical form
- 0025(36): SKILL: fully canonical form
- 0025(37): eg of fully canonical form
- 0025(38): decomposition theorem: any matrix = (prod of elem)(fully canonical)(prod of elem)
- 0025(39-43): eg decomposition theorem
- 0025(44): SKILL: decomposition theorem
- 0025(44): SKILL: list n by k fully canonical matrices; eg: list 4 x 3 fully canonical matrices

Topic 0026 (Row and col ops and linear algebra) Link to Topics website

- 0026(1): title slide
- 0026(2): onto map {1,2,3} --> {1,2,3,4}? No
- 0026(2): onto map \R^3 --> \R^4? Yes, even continuous
- 0026(2): onto linear map \R^3 --> \R^4? No; start of proof
- 0026(2-4): proof that there's no onto linear map \R^3 --> \R^4
- 0026(5): if there's an onto linear \R^n --> \R^k, then n\ge k
- 0026(6): 1-1 map {1,2,3,4} --> {1,2,3}? No
- 0026(6): 1-1 map \R^4 --> \R^3? Yes, but not continuous
- 0026(6): 1-1 linear map \R^4 --> \R^3? No; similar proof
- 0026(7): if there's an 1-1 linear \R^n --> \R^k, then n\le k
- 0026(7): recall: if there's an onto linear \R^n --> \R^k, then n\ge k
- 0026(7): if there's an isomorphism \R^n --> \R^k, then n = k
- 0026(8): transition slide
- 0026(9): any two bases of a subspace have the same size
- 0026(10): any set of k+1 vectors in \R^k is linearly dependent
- 0026(11): transition slide
- 0026(12): any subspace of \R^k has a basis; start of proof
- 0026(13): end of proof
- 0026(14): def'n/eg: dimension of a subspace of Euclidean space
- 0026(15): SKILL: determine if a vector is in the span of some others
- 0026(15): elementary row operations don't change the row span
- 0026(15-17): eg: determine if a vector is in the span of some others
- 0026(18): SKILL: extract a basis from a spanning set
- 0026(18): algorithm for that skill
- 0026(19): SKILL: determine if a set of vectors is linearly independent, and, if not, express one of them as a linear combination of the others
- 0026(19): algorithm for that skill
- 0026(20): SKILL: find the kernel and image of a linear transformation
- 0026(20): elementary row operations don't change the row span
- 0026(20): elementary column operations don't change the column span
- 0026(21): eg: find the kernel and image of linear transformations
- 0026(22-23): eg: find the kernel of a linear transformation
- 0026(24): eg: find the image of a linear transformation
- 0026(25): elementary row and column operations change neither the dimension of the kernel nor the dimension of the image
- 0026(26): dimensions of kernel and image of a fully canonical matrix
- 0026(27): theorem: dim kernel + dim image = dim domain; intuition for this theorem
- 0026(28-30): proof of this theorem
- 0026(31): for square matrices, 11 properties that are equvalent to invertible
- 0026(32): if a matrix has a left and right inverse, then they're equal
- 0026(32): for square matrices, XY = I implies YX = I
- 0026(33): for non-square matrices:
- a short wide matrix times a tall thin matrix can sometimes equal the identity
- a tall thin matrix times a short wide matrix can never equal the identity
- a non-square matrix may have a left inverse
- a non-square matrix may have a right inverse
- a non-square matrix cannot have both at once

- 0026(34-40): Inversion of (square) matrices
- 0026(35): recall that elementary matrices are invertible
- 0026(36-37): eg: finding a left inverse (of a square matrix)
- 0026(38): algorithm for finding left inverses of square matrices
- 0026(39): transition slide
- 0026(40): a left inverse of a square matrix is a two-sided inverse

- 0026(41): SKILL: determine if a matrix is invertible
- 0026(41): SKILL: given an invertible matrix, find its inverse
- 0026(41): SKILL: given a collection of linear systems of equations with constant left side, but varying right side,

find an efficient algorithm for solving them - 0026(42-44): convert the collection of systems to a collection of matrix equations
- 0026(45): efficient algorithm for solving them

Topic 0027 (Determinants exist) Link to Topics website

- 0027(1): title slide
- 0027(2): def'n/eg: oriented parallelogram
- 0027(3): transition slide
- 0027(4): visualization of oriented parallelogram
- 0027(5): def'n;eg: oriented 3-parallelpiped
- 0027(5): visualization discussion
- 0027(6): def'n: standard oriented parallelogram
- 0027(6): def'n: standard oriented 3-parallelpiped
- 0027(7): def'n: oriented n-parallelpiped
- 0027(7): def'n: standard oriented n-parallelpiped
- 0027(8): def'n: degenerate oriented n-parallelpiped
- 0027(8): def'n/eg: degenerate oriented parallelograms
- 0027(9): def'n/eg: degenerate oriented 3-parallelpiped
- 0027(10): def'n/non-eg: positive oriented n-parallelpiped
- 0027(11): transition slide
- 0027(12): explanation, via visualization, of why ((1,3),(2,4)) is not positive
- 0027(13): def'n: negative oriented n-parallelpiped
- 0027(13): def'n: signed n-volume of an oriented n-parallelpiped
- 0027(13): def'n/eg: signed area of an oriented parallelogram
- 0027(14): eg: signed area of an oriented parallelogram
- 0027(15): transition slide
- 0027(16): IOU: signed area of ((1,3),(4,2))
- 0027(17): def'n/eg: AP, where A is an nxn matrix and P is an oriented n-parallelpiped
- 0027(18): def'n/eg/notation: "has a determinant"
- 0027(19): eg: determinant; det(AB)=[det(A)][det(B)]
- 0027(19): det(diagonal) = product of diagonal entries
- 0027(20): geometry of shearing
- 0027(21): shearing preserves area
- 0027(22): shearing preserves signed area
- 0027(23): det(shear) = 1; any elementary matrix has a determinant
- 0027(24): any matrix has a determinant
- 0027(25): sv(AP) = [det(A)][sv(P)], for *all* A and P
- 0027(25): vol(AP) = [|det(A)|][vol(P)], for all A and P
- 0027(26-28): calculating the volume of L_A ( [-a,b] x [-c,d] )
- 0027(29): eg: deteterminant of the matrix [ (1,3) (2,4) ] is -10
- 0027(30-31): sv ( (1,3) , (2,4) ) = -10

Topic 0028 (Properties of determinants) Link to Topics website

- 0028(1): title slide
- 0028(2): transpose does not affect determinant
- 0028(3-8): analysis of the determinant of the direct sum
- 0028(8): the determinant of the direct sum is the product of the determinants
- 0028(9-11): determinant is alternating in columns
- 0028(12-13): determinant is alternating in rows
- 0028(13): if two columns are equal, determinant = 0
- 0028(13): if two rows are equal, determinant = 0
- 0028(14): determinant is additive in the second column (for 2x2 matrices)
- 0028(15-16): determinant is additive in columns
- 0028(17-21): determinant is multilinear in columns
- 0028(22): determinant is additive in rows
- 0028(22): determinant is multilinear in rows
- 0028(23-25): if a row is zero, then the determinant is zero
- 0028(26-28): if a column is zero, then the determinant is zero
- 0028(29-32): formula for the determinant in the 2x2 case
- 0028(33-39): expanding along the first column in the 3x3 case
- 0028(40): formula for the determinant in the 3x3 case
- 0028(41): expanding along the first column in the 4x4 case
- 0028(42): expanding along the first row in the 4x4 case
- 0028(42): description of the formula for the determinant in the 4x4 case
- 0028(43): determiant(upper triangular) = product of diagonal entries
- 0028(43): determiant(lower triangular) = product of diagonal entries
- 0028(44): invertible iff (determinant is nonzero)
- 0028(45-51): a square matrix multiplied by (its transposed cofactor matrix)

is equal to (its determinant times the identity) - 0028(51): matrix of minors, cofactor matrix, transposed cofactor matrix
- 0028(52): the inverse of a square matrix is

its transposed cofactor matrix divided by its determinant - 0028(53-54): Cramer's rule for finding one unknown in a system of linear equations

Topic 0029 (Bilinear forms and quadratic forms) Link to Topics website

- 0029(1): title slide
- 0029(2): def'n/eg: polynomial in x
- 0029(2): def'n: polynomial in x,y
- 0029(3): def'n: polynomial in x,y,z
- 0029(4): def'n: homogeneous polynomial in x,y,z
- 0029(5): eg: homogeneous polynomial in x,y,z
- 0029(5): def'n: homogeneous polynomial \R^3 --> \R^5
- 0029(6): names of variables don't matter
- 0029(7): single-variable polynomial approximation by Maclaurin approximations
- 0029(7): Black-Scholes as a function \R^4 --> \R^2
- 0029(8): describing the Black-Scholes function F : \R^4 --> \R^2
- 0029(9): Approximating G(w,x,y,z) = F(100+w,97+x,0.01+y,0.2+z), for small w,x,y,z
- 0029(10): description of second-order approximation by polynomial in w,x,y,z
- 0029(11): this approximation is called the second order Maclaurin approximation of G
- 0029(11-12): some synonyms:
- constant = degree 0
- linear = degree 1
- quadratic = deggree 2
- cubic = deggree 3
- quartic = deggree 4

- 0029(12): do we need "quadratic algebra", "cubic algebra", "quartic algebra", etc
- 0029(13): no because tensor algebra reduces all of these to linear algebra
- 0029(13): we only need this reduction for quadratic approximations, where it's relatively easy
- 0029(14): start of naïve formulation of quadratic tensor algebra
- 0029(14): reduction to scalar-valued functions
- 0029(14): def'n/eg: quadratic form
- 0029(15): def'n: bilinear form
- 0029(16-17): def'n/notation: the matrix [B] of a bilinear form B
- 0029(18-19): a bilinear form is determined by its matrix
- 0029(20): def'n/eg: symmetric bilinar form
- 0029(20): a bilinear form is symmetric iff its matrix is symmetric
- 0029(21): from bilinear forms to quadratic forms: "restrict to the diagonal"
- 0029(22-25): more than one bilinear form gives rise to the same quadratic form
- 0029(26): square matrices are not in 1-1 correspondence with quadratic forms
- 0029(27): symmetric matrices are in bijective correspondence with quadratic forms
- 0029(27): description of the map in one direction (matrices to forms)
- 0029(28): start of search for the inverse (forms to matrices)
- 0029(29): goal: a procedure for going from quadratic forms to symmetric bilinear forms
- 0029(30): eg: going from a quadratic form to a symmetric matrix
- 0029(31-33): going from that symmetric matrix to a symmetric bilinear form
- 0029(34): recall going directly from a symmetric bilinear form to a quadratic form (restricting to the diagonal
- 0029(34-36): going directly from a quadratic form to a symmetric bilinear form
- 0029(36): def'n: polarization; the polarization formula
- 0029(36): the polarization map (from quadratic forms to symmetric biliear forms)
- 0029(37): def'n/notation: the matrix, Q_M, of a quadratic form
- 0029(37): recall notation for the matrix, [B], of a symmetric bilinear form
- 0029(37): recall notation for the bilinear form, B_M, of a matrix
- 0029(37): recall formula for B_M in terms of L_M
- 0029(37): formula for Q_M in terms of L_M
- 0029(38): multiplication is the polarization of squaring
- 0029(39): dot product is the polarization of length squared
- 0029(40-42): BIG IDEAs:
- polynomial algebra reduces to linear algebra
- everything reduces to polynomial algebra
*ergo*: everyting boils down to linear algebra

Topic 0030 (Cauchy-Schwarz) Link to Topics website

- 0030(1): title slide
- 0030(2-7): how big can v \cdot w be when |v| \le 3 and |w| \le 5
- 0030(4): upper bounds of 17, 377/9, 25
- 0030(4): general upper bounds of [9t+(25/t)]/2, for all t
- 0030(4): want to minimize [9t+(25/t)]/2
- 0030(6): the geometric mean is the minimum of the modified arithmetic means
- 0030(7): minimum upper bound of 15

- 0030(7): Cauchy-Schwarz for dot product
- 0030(8): transition slide
- 0030(9-10): Cauchy-Schwarz for a general positive semidefinite quadratic form
- 0030(10): def'n: positive semidefinite; def'n: positive definite
- 0030(11-12): buzz phrase: the absolute polarization at v,w of a positive semidefinite quadratic form

is less than or equal to

the geometric mean of its values at v and w.

Topic 0031 (Rotations, reflections, orthogonal transformations) Link to Topics website

- 0031(1): title slide
- 0031(2): recall definition/notation of the length of a vector
- 0031(2): v \cdot v is the length squared of v
- 0031(2): def'n: normal; def'n/notation: orthogonal
- 0031(2): def'n: orthonormal
- 0031(3): def'n/notation: Kronecker delta
- 0031(3): rephrase definition of orthonormal, using the Kronecker delta
- 0031(4): fact: a matrix is orthogonal iff its columns "are" orthonormal
- 0031(5): transition slide
- 0031(6): fact: a matrix is orthogonal iff its rows "are" orthonormal
- 0031(7-13): Gram-Schmidt Orthonormalization
- 0031(7): description of the algorithm
- 0031(8-13): a worked example with three vectors in \R^4

- 0031(14): SKILLS:
- find the length of a vector
- determine if two vectors are orthogonal
- Gram-Schmidt orthonormalization

- 0031(15): fact: [R orthogonal] implies [(det R = 1) or (det R = -1)]
- 0031(15): def'n: rotation; def'n: reflection
- 0031(15): def'n: orthogonal linear transformation
- 0031(15): def'n: rotation linear transformation
- 0031(15): def'n: reflection linear transformation
- 0031(16): any normal vector "is" the first column of some rotation matrix
- 0031(16-17): eg: a certain normal vector in \R^4 "is" the first column of some rotation matrix
- 0031(18): proof that: any normal vector "is" the first column of some rotation matrix
- 0031(19): def'n: unit vector (same as normal vector)
- 0031(19): SKILL: Given a unit vector v, make a rotation matrix whose first column "is" v
- 0031(20): fact: for any two unit vectors v and w, there is a rotation R s.t. L_R(v)=w
- 0031(21): fact: for any two vectors v and w of the same length, there is a rotation R s.t. L_R(v)=w
- 0031(22): def'n: angle between two vectors
- 0031(23): if a = e_1 and b is in the span of e_1,e_2,

then a \cdot b is the cosine of the angle between a and b - 0031(24): for any two unit vectors a and b, a \cdot b is the cosine of the angle between a and b
- 0031(25): for any two vectors a and b, a \cdot b is the product of three quantities:

- the length of a
- the length of b
- the cosine of the angle between a and b

- 0031(26): summary of all the equivalent definitions of orthogonal
- 0031(26): orthogonal = rotation or reflection; formula for a \cdot b
- 0031(27): restate Cauchy-Schwarz
- 0031(27): def'n/eg: perfectly correlated
- 0031(28): def'n/eg: perfectly anti-correlated
- 0031(29): def'n/eg: positively correlated
- 0031(30): def'n/eg: negatively correlated
- 0031(31): def'n/eg: uncorrelated
- 0031(32-33): motivation for Spectral Theory

Topic 0032 (Eigenvalues and eigenvectors) Link to Topics website

- 0032(1): title slide
- 0032(2): description of Ramagrog, with Ramatins and Grogali
- 0032(3): transition slide
- 0032(4-6): population after one year
- 0032(6): population development as two scalar equations
- 0032(7-8): population development as one matrix equation, matrix B
- 0032(9): transition slide
- 0032(10): a way of finding the population after 10 years
- 0032(11): big herds and little herds; initial population in herds
- 0032(12): population development in herds
- 0032(13): transition slide
- 0032(14): population after 10 years in herds
- 0032(15): transition slide
- 0032(16): eigenvector/eigenvalue terminology
- 0032(16): initial population as a linear combination of eigenvectors
- 0032(17): the big idea of eigenvectors and eigenvalues: simplifying computations
- 0032(18): transition from herd count to Ramatin Grogali count
- 0032(19): transition from Ramatin Grogali count to herd count
- 0032(20): ease of computation of diagonal matrices
- 0032(21): the population development matrix (called B) is a conjugate of a diagonal matrix D; that is, B = CDC^{-1}
- 0032(22-24): going from eigenvectors and eigenvalues to B = CDC^{-1}
- 0032(24-26): double check B = CDC^{-1}
- 0032(27-28): computation of B^2 and B^{10}, in terms of C and D
- 0032(29): some questions
- given B, how to find C and D,
*i.e.*, how to diagonalize B,*i.e.*, how to find eigenvectors and eigenvalues - are all square matrices diagonalizable?
- if some are not, how to ideentify and diagonalize those that are?
- if some are not, how can we find a computationally good form for them?

- given B, how to find C and D,
- 0032(30-31): nonzero nilpotent matrices are not diagonalizable
- 0032(32): strictly upper triangular matrices are nilpotent
- 0032(33): how to find eigenvalues
- 0032(34): transition slide
- 0032(35-44): how to find eigenvectors for each eigenvalue
- 0032(45): transition slide
- 0032(46-49): how to diagonalize, given eigenvectors and eigenvalues
- 0032(49): how to take 10th power, given diagonalization
- 0032(50): def'n: characteristic polynomial
- 0032(51): def'n: eigenvalue = characteristic root
- 0032(52): transition slide
- 0032(53): def'n: eigenvector; def'n: eigenspace
- 0032(54): transition slide
- 0032(55): 0 is never and eigenvalue, but is in every eigenspace
- 0032(56): SKILL: given a square matrix, find its eigenvalues and,

for each eigenvalue, find a basis for its eigenspace

Topic 0033 (Diagonalization of matrices) Link to Topics website

- 0033(1): title slide
- 0033(2-19): how to identify and diagonalize matrices that are diagonalizable?
- 0033(2-10): real diagonalizable and complex diagonalizable are different
- 0033(11-12): the diagonalization algorithm
- 0033(13-17): where the diagonalization algorithm can fail
- 0033(18): criterion for diagonalizability:

the multiplicity of each eigenvalue is the dimension of its eigenspace - 0033(19): SKILL: Given a square matrix, determine if it's diagonalizable.

If it is, diagonalize it.

- 0033(20): Jordan blocks are not diagonalizable
- 0033(21-25): recall: scalar, standard nilpotent, Jordan block, direct sum
- 0033(26): direct sum is diagonalizable iff the summands are
- 0033(26): Jordan canonical form:

every matrix is conjugate to a direct sum of Jordan blocks - 0033(26): a direct sum of Jordan blocks is diagonalizable iff all the blocks are 1 x 1
- 0033(27-49): Is Jordan form computationally good?
- 0033(28-36): tenth power of a Jordan block
- 0033(37-48): exponential of a Jordan block
- 0033(49): powers and exponential distribute over direct sum
- 0033(49): Jordan form is computationally good.

- 0033(50-55): the exponential of an antisymmetric 2 x 2 matrix is a rotation matrix

Topic 0034 (The Spectral Theorem) Link to Topics website

- 0034(1): title slide
- 0034(2): recall orthogonal matrix
- 0034(3): recall rotation matrix, reflection matrix
- 0034(3): recall orthogonal linear transformation, rotation linear transformation, reflection linear transformation
- 0034(4): recall the quadratic form of a symmetric matrix; recall polarization
- 0034(4): def'n: equivalent quadratic forms
- 0034(5): formula for Q_S \circ L_X
- 0034(6): def'n: t-equivalent
- 0034(6): two quadratic forms are equivalent iff their matrices are t-equivalent
- 0034(7): proof of formula for Q_S \circ L_X
- 0034(8): SKILL: Given S and X, produce a matrix N such that Q_S \circ L_X = Q_N
- 0034(9): def'n: diagonal quadratic form
- 0034(10): eg/non-eg: diagonal quadratic form
- 0034(11): SKILL: recognize whether a given quadratic form is diagonal
- 0034(12): def'n: rotationally diagonalizable
- 0034(12): goal: rotationally diagonalizable = symmetric
- 0034(13-17): preliminaries
- 0034(13): fact: the eigenvalues of a symmetric (real) matrix are real
- 0034(14): fact: if two matrices are conjugate, then they have the same characteristic polynomial, and the same eigenvalues
- 0034(15): fact: the eigenvalues of an upper triangular matrix are its diagonal entries
- 0034(16): fact: conjugation of a symmetric matrix by an orthogonal matrix yields a symmetric matrix
- 0034(17): recall the definition of rotationally diagonalizable
- 0034(17): fact: any rotationally diagonalizable matrix is symmetric

- 0034(17): statement of the Spectral Theorem
- 0034(18): transition slide
- 0034(19): statement of the Spectral Corollary
- 0034(19): proof of the Spectral Corollary, given the Spectral Theorem
- 0034(20): the idea of the Spectral Corollary: any quadratic form can be easily studied by diagonalizing it
- 0034(22): transition slide
- 0034(22-36): application of the Spectral Corollary to graphing the level set of a quadratic form, Q
- 0034(22): goal is to get rid of the mixed term
- 0034(23-25): how to proceed if we have a rotation R such that Q \circ R is diagonal (and we know the coeffients of that diagonal form)
- 0034(25): the graph ends up being an hyperbola
- 0034(26): how to find the rotation and the coefficients of the diagonalized form?
- 0034(27-28): setup
- 0034(29): the eigenvalues, a and b, of the matrix, P, of the quadratic form
- 0034(30): recall that these eigenvalues must be real
- 0034(31): transition slide
- 0034(32): exercise to find an a-eigenvalue, normalize it, and make it the first column of a rotation matrix R_0
- 0034(32): the first column of R_0^{-1} P R_0 has entries a_0 and 0
- 0034(33-34): full computation of R_0^{-1} P R_0
- 0034(35): transition slide
- 0034(36): proof that, if R is the linear transformation corresponding to R_0, then Q \circ R is diagonal

- 0034(37): a direct sum of rotationally diagonalizable matrices is again rotationally diagonalizable
- 0034(38): recall that a rotationally diagonalizable matrix is symmetric
- 0034(38): recall the statement of the Spectral Theorem (now proved in the 2x2 case, as part of diagonliazing Q)
- 0034(39): transition slide
- 0034(40-42): proof of the Spectral Theorem in the 3x3 case, ginen the 2x2 case

Topic 0035 (Principal component analysis) Link to Topics website

- 0035(1): title slide
- 0035(2): a pairwise-orthogonal collection of nonzero vectors must be linearly independent
- 0035(3): start of "Principal Component Analysis" (PCA) or "Singular Value Decomposition" (SVD)
- 0035(4): the PCA/SVD theorem
- 0035(5-6): motivation: any matrix, after some post-processing (multiplication on the left by an orthogonal matrix),
has pairwise-orthogonal (
*i.e.*, uncorrelated) rows, and so is easier to study - 0035(7): motivation in terms of tracking measurements from general measuring devices
- 0035(8): motivation in terms of tracking financial assets
- 0035(9): a 2x5 example with the two rows almost equal, general description
- 0035(10): transition slide
- 0035(11): the 2x5 example, worked out in detail
- 0035(12-14): show that the data in the original matrix is driven by one factor, distorted by noise
- 0035(15): easier version of PCA/SVD; proof of this easier version
- 0035(16-22): preliminaries to proving the PCA/SVD theorem
- 0035(23): proof of the PCA/SVD theorem
- 0035(24): transition slide
- 0035(25-26): motiviation in terms of approximating a matrix by a nearby matrix of whose rows span a smaller subspace
- 0035(27-34): application to estimating movie preferences
- 0035(27): setup and notation
- 0035(27): panel of composite peope, a collection of composite movies
- 0035(28): change to the composites
- 0035(29): use of PCA to get simple ratings
- 0035(30): formula for the rating person 0 gives to composite movies 1,...,q
- 0035(31): proof of that formula
- 0035(32): recall the formula, and that it works for composite movies 1,...,q
- 0035(33): we approximate that the formula will work for movie 0
- 0035(34): the approximation for person 0's rating of movie 0
- 0035(34): difficulty if the singular values are 0, or close to 0

Topic 0036 (Cayley's Theorem) Link to Topics website

- 0036(1): title slide
- 0036(2): def'n: matrix extension
- 0036(2): statement of Cayley's Theorem
- 0036(3-4): matrix extension for polynomials of two or more variables
- 0036(5): notation: matrix is T, E^{ij} is the standard basis of \R^{nxn}
- 0036(5): notation: I:=identity, P is {polynomials in T},
- 0036(5): notation: det : \R^{nxn} --> \R is determinant
- 0036(5): notation: DET : P^{nxn} --> P is its matrix extension
- 0036(6): DET in the 3x3 case
- 0036(7): recall A\otimes B
- 0036(7): (A \otimes B)(C \otimes D) = (AC) \otimes (BD)
- 0036(8-9): the reproducing equation
- 0036(10): T \otimes I and I \otimes \Lambda
- 0036(11): transition slide
- 0036(12): the matrix extension formula of the characteristic polynomial of T
- 0036(13-15): proof of that formula in the 3x3 case
- 0036(16): recall the statement of Cayley's Theorem
- 0036(17): U := (T \otimes I) - (I \otimes T); Want: DET(U)=0
- 0036(18): T \otimes I , I \otimes T both in P^{nxn}
- 0036(19): transition slide
- 0036(20): recall transposed cofactor matrix, notation: tr-cof
- 0036(21): TR-COF, it's matrix extension
- 0036(22): [tr-cof(M)]M = M[tr-cof(M)] = (det M)I
- 0036(22): [TR-COF(M)]M = M[TR-COF(M)] = I \otimes (DET M)
- 0036(23): mult is linear and mult ( A\otimes B) = AB
- 0036(24): C := TR-COF(U);
- 0036(24): Want: mult ( C (T \otimes I) ) = mult ( C (I \otimes T) )
- 0036(25): entries of C all are in P, and so commute with T
- 0036(26): computation of mult ( C (T \otimes I) ) and mult ( C (I \otimes T) )
- 0036(27): conclusion of proof

Topic 0037 (Multivariable polynomial approximation) Link to Topics website

- 0037(1): title slide
- 0037(2): single variable linear approximation
- 0037(3-4): visualization of single variable linear approximation
- 0037(5-15): a two variable linear approximation problem
- 0037(16): the general formula for two variable linear approximation
- 0037(17-18): def'n: partial derivatives for two variables
- 0037(19-22): the general formula for two variable linear approximation in the language of partial derivatives
- 0037(23): transition slide
- 0037(24-25): simplifying the notation, using p=(x,y), \triangle p = (h,k)
- 0037(26): def'n: gradient for two variables; notation: \nabla g, g'
- 0037(27): simplifying notation using gradient
- 0037(28): comparison with single variable linear approximation
- 0037(29): def'n: graph; a the mountain climber's problem
- 0037(30-32): solving the mountain climbers problem
- 0037(33): practice problems in partial derivatives
- 0037(34): SKILL: compute partial derivatives, compute the gradient
- 0037(35): SKILL: compute second partials; partials commute
- 0037(36-37): def'n: n variable partial derivatives
- 0037(38-42): alternate notations for partial deriviatives
- 0037(43): def'n: gradient for n variables; notation: \nabla g, g'
- 0037(44): SKILL: comptute partial derivatives, compute the gradient
- 0037(45): notations for higher order partial derivatives
- 0037(46): SKILL: compute the second order partial derivatives
- 0037(47): recall single variable second order Maclaurin approximation
- 0037(47): def'n: two variable second order Maclaurin approximation
- 0037(48): transition slide
- 0037(49-50): compute coefficients for the two variable second order Maclaurin approximation
- 0037(51-52): the formula for the two variable second order Maclaurin approximation
- 0037(53): transition slide
- 0037(54): exercise: third order Maclaurin approximation to f(x,y)
- 0037(54): exercise: second order Maclaurin approximation to f(x,y,z)
- 0037(55): transition slide
- 0037(56-58): rewriting the second order Maclaurin approximation to f(x,y) in terms of the gradient and Hessian
- 0037(56): def'n: gradient; def'n: Hessian
- 0037(59): gradient and Hessian approximations for vector-valued functions
- 0037(60): notations for the gradient, gradient matrix and Hessian
- 0037(61): def'n: k-jet at 0 of a function of n-variables with values in R^q
- 0037(62): def'n: Maclaurin approximation for a function of n-variables with values in R^q
- 0037(63): SKILLS:
- gradient and Hessian of a function of n variables
- k-jet of a function of n variables
- k'th order Maclaurin approximation of a function of n variables
- number of terms in the k'th order Maclaurin approximation of a function of n variables
- number of entries in the k-jet at 0 of a function of n variables

- 0037(64): the second order Maclaurin approximation of the Black-Scholes function
- 0037(65): transition slide
- 0037(66-75): explaining the meaning of "agrees to order two"

Topic 0038 (Vector fields and ODEs) Link to Topics website

- 0038(1): title slide
- 0038(2): def'n: vector field; def'n: linear vector field
- 0038(3): e.g. of vector field: the tornado
- 0038(3): def'n: flowline
- 0038(4-5): visualization of a a flowline of the tornado
- 0038(5): guess ( cos t , sin t ) based on the visualization
- 0038(6): checking that ( cos t , sin t ) is right
- 0038(7): existence of short-time flowlines
- 0038(7): a specific example of a short-time flowline in one dimension that traverses \R in finite time
- 0038(8): def'n: footed (footpoint), extends, maximal
- 0038(9): transition slide
- 0038(10): existence and uniques of maximal flowlines
- 0038(10): explanation of why the specific example above does not have a long-time flowline
- 0038(11): an ODE (Ordinary Differential Equation) and an equivalent flowline problem
- 0038(11): the continuous compounding ODE
- 0038(12): an ODE system with initial value conditions and an equivalent flowline problem
- 0038(13): the Euler method
- 0038(14): the Euler method on the tornado flowline
- 0038(15-24): the Euler method for a continuous compounding ODE
- 0038(25): transition slide
- 0038(26): checking that the Euler method gave the correct solution
- 0038(27): an alternate solution to the continuous compounding ODE, via logarithmic change of variables
- 0038(28-45): a third approach via fixpoint methods in infinite dimensional function spaces
- 0038(28): reformulation of the ODE problem as a fixpoint problem for an integral mapping
- 0038(29): transition slide
- 0038(30): def'n: contraction; def'n: contraction factor
- 0038(30): fact: in a certain distance, the integral mapping is a contraction with contraction factor 1/2
- 0038(31-41): proof of the fact
- 0038(31): the distance used in the uniform distance
- 0038(42): why a contraction mapping leads to a fixpoint in complete metric spaces
- 0038(43): the uniform distance is complete
- 0038(44-45): finding the short-time flowline for the continuous compounding ODE

- 0038(46): a coupled linear system of ODES and the corresponding flowline problem
- 0038(47): rewriting the ODE system as a matrix ODE
- 0038(48): transition slide
- 0038(49): solving the matrix ODE using the matrix exponential map
- 0038(50): comparision of the matrix ODE solution with the continuous compounding ODE solution
- 0038(51): finding the constant from the initial condition
- 0038(52-54): writing the solution with the correct constant
- 0038(54): exercise: diagonalize the matrix to compute the exponentials that solve the matrix ODE
- 0038(55): SKILLS:
- NOTE: "integrate" means find the flowline at an arbitrary footpoint, i.e., find all the flowlines
- SKILL: integrate a constant vector field
- SKILL: integrate a homogeneous linear vector field
- SKILL: integrate an inhomogeneous linear vector field

- 0038(56): finding flowlines for a constant vector field
- 0038(56): finding flowlines for an inhomogeneous linear vector field
- 0038(57): def'n: functional
- 0038(57): def'n: local max; def'n: local min
- 0038(57): goal: maximize/minimize a smooth functional
- 0038(57): related goal: find local max/min for a smooth functional
- 0038(58): def'n: critical point; local max at a implies critical point at a
- 0038(58): second derivative test
- 0038(59): reverse gradient flow to minimize a functional
- 0038(60): gradient flow to maximize a functional

Topic 0039 (The multivariable chain rule) Link to Topics website

- 0039(1): title slide
- 0039(2): the chain rule for composites \R^k --> \R^m --> \R^n
- 0039(2): discussion of a formula for the Hessian of a composite \R^k --> \R^m --> \R
- 0039(3-9): the chain rule for composites \R --> \R^m --> \R
- 0039(9): discussion of a formula for the Hessian of a composite \R --> \R^m --> \R
- 0039(10-20): using the chain rule to show:

If q(x,y) is the 2nd order Maclaurin approximation to g(x,y),

then g(x,y) = q(x,y) + o(x^2+y^2) for (x,y) --> (0,0)

Topic 0040 (Lagrange mult, constrained optimization) Link to Topics website

- 0040(1): title slide
- 0040(2): problem: find the major and minor axes of 3x^2+2xy+3y^2=8
- 0040(2-11): solution of the problem via Spectral Theory, i.e., via diagonalization of the matrix of 3x^2+2xy+3y^2
- 0040(12-30): solution of the problem via constrained optimization
- 0040(12): reprhasing the minor axis problem as a constrainted minimization problem
- 0040(13): transition slide
- 0040(14): level sets of f(x,y) = x^2+y^2 compared to level sets of Q(x,y) = 3x^2+2xy+3y^2
- 0040(15): comparing the gradients of f and Q at a point
- 0040(15): key point: the gradient is perpendicular to the level set
- 0040(16): a starting point one E = Q^{-1}(8)
- 0040(17-18): which way to go on E to make f decrease
- 0040(19-20): why the decrease ceases at the endpoint of a minor axis
- 0040(21): def'n: critical point for a constrained optimization problem
- 0040(22-30): finding the critical points for minimizing f constrained to E

- 0040(31): the general format of a constrained minimization problem, with one scalar constraint
- 0040(31): def'n: objective; def'n: constraint; def'n: Lagrange multiplier
- 0040(31): counting equations and unknowns
- 0040(32): def'n: non-smooth points for one constraint
- 0040(33-35): example that shows we must check non-smooth points as well as critical points
- 0040(36): the general format of a constrained minimization problem, with k scalar constraints
- 0040(36): counting equations and unknowns
- 0040(37): def'n: non-smooth points for k constraints

Topic 0041 (Multivariable change of variables) Link to Topics website

- 0041(1): title slide
- 0041(2): start of multivariable integral calculus
- 0041(3): single variable change of variables formula
- 0041(4-5): some special cases, as they're presented in a calc course
- 0041(6): goal is to find an analog for two variable functions
- 0041(7): setting up the definition of the Riemann integral for functions of two variables
- 0041(8-20): review of the definition of the Riemann integral for functions of one variable
- 0041(8): the setup
- 0041(9): a partition
- 0041(10): picking points
- 0041(11): focus on one subinterval
- 0041(12): name it
- 0041(13): look at the graph
- 0041(14): form the rectangle
- 0041(15): estimate the integral over that subinterval as the area of the rectangle
- 0041(16): add over all the boxes to estimate the integral over the full interval as a Riemann sum
- 0041(17): make a sequence of partitions with mesh --> 0
- 0041(18): transition slide
- 0041(19-20): define the integral as a limit of Riemann sums

- 0041(21-28): repeate this for functions of two variables
- 0041(21): the setup
- 0041(22): a partition
- 0041(23): picking points
- 0041(24): focus on one subset
- 0041(25): estimate the integral over a subset as the volume of a box
- 0041(26): sum to estimate the integral
- 0041(27): make a sequence of partitions with mesh --> 0
- 0041(28): define the integral as a limit of Riemann sums

- 0041(29): Fubini's Theorem on rectangles
- 0041(30-43): the two-variable change of variables formula
- 0041(30): the setup and goal
- 0041(31): the answer
- 0041(32): estimating the LHS and RHS of the answer via Riemann sums
- 0041(33): reducing the problem to an estimation of area of the image of a subset
- 0041(34): zoom in on a subset, and parametrize the box by h and k
- 0041(34): estimate the change of variables by linear approximation
- 0041(34): estimate the image of the subset by linear approximation
- 0041(35): estimate the area of the image of the subset by linear approximation
- 0041(36): the absolute value of the determinant relates area before a linear map to area afterward
- 0041(37): the area of the box is the base time the height
- 0041(38-39): complete the required estimation of the area of the image of a subset
- 0041(40): complete the proof that the LHS and RHS are close
- 0041(41): transition slide
- 0041(42-43): complete the proof that the LHS and RHS are equal

- 0041(44-49): polar coordinates
- 0041(49): replacements:
- x by r \cos \theta
- y by r \sin \theta
- x^y + y^2 by r^2
- dx dy by r dr d\theta

- 0041(50-53): proof that \int_{-\infty}^\infty e^{-x^2/2} dx is equal to \sqrt{2\pi}
- 0041(54): spherical coordinates

Topic 0042 (Variations on Stokes' Theorem) Link to Topics website

- 0042(1): title slide
- 0042(2): start of Green's Theorem and Cauchy's Theorem
- 0042(3): def'n: directed line segment; def'n: starting point; def'n: ending point
- 0042(4): def'n: standard parametrization; def'n: constant velocity
- 0042(5): def'n: simple chain
- 0042(6): def'n: rectangle
- 0042(7):
*e.g.*: open rectangle - 0042(8): descriptive def'n and notation: counterclockwise boundary
- 0042(9):
*e.g.*: counterclockwise boundary - 0042(10): formal def'n and notation: counterclockwise boundary
- 0042(11): def'n and notation: line integral in the plane over a directed line segment
- 0042(12): def'n and notation: line integral in the plane over a chain
- 0042(13): statement of Green's Theorem
- 0042(14): def'n: zero-form in two variables; def'n: one-form in two variables
- 0042(14): def'n: exterior derivative of a zero-form in two variables
- 0042(15): SKILL: compute exterior derivative of a zero-form
- 0042(16): the conventions for computing wedge products
- 0042(17): zero-forms move through wedge products
- 0042(18): an example computation of a wedge product in two variables
- 0042(19): an example computation of an exterior derivative of a one-form in two variables
- 0042(20): def'n: two-form in two variables
- 0042(20): def'n: exterior derivative of a one-form in two variables
- 0042(21): transition slide
- 0042(22): an example computation of an exterior product in two variables
- 0042(22): def'n: integral of a two-form in $x$ and $y$ over a rectangle in $\R^2$.
- 0042(23): restatement of Green's Theorem in the language of forms
- 0042(24-26): example computations of wedge products of three one-forms in three variables
- 0042(27-29): example computations of exterior derivatives of one-forms in three variables
- 0042(30): an example computation of a line integral of a one-form in two variables
- 0042(31-32): an example computation of a line integral of a one-form in three variables
- 0042(33): recall the definition of line integral over a directed line segment in \R^2
- 0042(34): def'n: line integral over a directed line segment in \C
- 0042(35): recall the definition of line integral over a chain in \R^2
- 0042(36): def'n: line integral over a chain in \C
- 0042(37): recall Green's Theorem on rectangles in \R^2
- 0042(38): Green's Theorem on rectangles in \C
- 0042(39-41): exterior derivative of 1-form in one variable is always zero (over \R)
- 0042(42): expecting 0 in Green's Theorem on rectangles in \C
- 0042(43): def'n: complex differentiable
- 0042(43): exercise: show that f(z)=e^{3z} is complex differentiable
- 0042(44): transition slide
- 0042(45): example of a function (namely, f(z)=|z|^2) that is smooth, but not complex differentiable
- 0042(46-57): proof that f(z)=|z|^2 is not complex differentiable
- 0042(57): necessary condition for complex differentiability
- 0042(58-60): going from that necessary condition to the Cauch-Riemann equations
- 0042(60): the Cauchy-Riemann equations
- 0042(61): finding the real and imaginary parts of e^{z^2/2}
- 0042(61): exercise to verify the Cauchy-Riemann equations for the real and imaginary parts of e^{z^2/2}
- 0042(62): recall Green's Theorem for rectangles in \C
- 0042(63): in the analytic case, the exterior derivative vanishes
- 0042(64): Cauchy's Theorem for rectangles in \C
- 0042(65-75): proof of Green's Theorem for rectangles in \R^2

Topic 0043 (Planimeters) Link to Topics website

- 0043(1): title slide
- 0043(2-3): description of a simple design of a planimeter; the curve C
- 0043(4-7): modeling the rate of turning of the wheel
- 0043(8): goal is to show that the total turning of the wheel is proportional to the area enclosed inside C
- 0043(9): the vector fields R and W and V
- 0043(9): computation of R
- 0043(10): computation of W
- 0043(11): the scalar quantities c and s
- 0043(11-12): computation of V in terms of c and s
- 0043(13): transition slide
- 0043(14-15): computation of c and s
- 0043(16): transition slide
- 0043(17): recall the goal (proportionalty of area enclosed to total turning of the wheel)
- 0043(18): \gamma parametrizes the curve C
- 0043(19): \gamma = ( \alpha , \beta )
- 0043(19): p and q are the components of V
- 0043(20): \omega = p(x,y) dx + q(x,y) dy
- 0043(20-25): the total turning of the wheel as a line integral of \omega over C via the parametrization \gamma
- 0043(26): \omega in terms of s, x, c and y
- 0043(27): Stokes' Theorem for the curve C in \R^2
- 0043(28): transition slide
- 0043(29): R := the region enclosed inside C, A := the area enclosed as an integral over R of dx \wedge dy
- 0043(30-37): computation of d\omega, the exterior derivative of \omega
- 0043(37): d\omega is proportional to dx \wedge dy
- 0043(38): the total turning of the wheel is propotonal to the area enclosed inside C

Topic 0044 (Problems in integration) Link to Topics website

- 0044(1): title slide
- 0044(2): def'n: \Phi as an antiderivative of H := e^{-x^2/2} / \sqrt{2\pi}
- 0044(2): the definite integral of H from p to \infty
- 0044(3): the definite integral of e^{ax} H dx from p to \infty
- 0044(4): the definite integral of ( e^{ax} - b ) H dx from p to \infty
- 0044(5-6): the definite integral of ( e^{ax} - b )_+ H dx from -\infty to \infty
- 0044(7-16): indefinite integral of x^k H dx
- 0044(7): k=0 case
- 0044(8): k=1 case
- 0044(9): simplest antiderivative in the k=1 case
- 0044(10-11): from k=2 to k=0
- 0044(12): conclusion of the k=2 case
- 0044(13): from k=3 to k=1
- 0044(14): conclusion of the k=3 case
- 0044(15): from k to k-2
- 0044(16): exercise: the k=4 case

- 0044(17-21): the definite integral of x^k H dx from -\infty to \infty
- 0044(17): setting up the problem
- 0044(18): solution when k is even
- 0044(19): solution when k is odd
- 0044(20): transition slide
- 0044(21): the two cases (k even and k odd) on one slide

Topic 0045 (Basics of PCRVs) Link to Topics website

- 0045(1): title slide
- 0045(2): def'n: piecewise constant random variable (PCRV)
- 0045(2): finitely many pieces
- 0045(2): \Omega := [0,1]
- 0045(2):
*e.g.*of a PCRV (called X) - 0045(3): transition slide
- 0045(4): intuition behind PCRVs (Tyche chooses elements of \Omega)
- 0045(5): some probability computations
- 0045(6): def'n and
*e.g.*s: deterministic - 0045(6): pieces can have zero length
- 0045(7): another
*e.g.*of a PCRV (called Y) - 0045(7): use of "almost surely" and "surely"
- 0045(8): modeling a coin-flip: C_1
- 0045(8): modeling two coin-flips: C_1 and C_2 (bad choice of C_2)
- 0045(9): good choice of C_2
- 0045(10): def'n: distribution of a PCRV
- 0045(11):
*e.g.*: distribution of X - 0045(12):
*e.g.*: distribution of Y - 0045(13):
*e.g.*: distribution of C_1 and C_2 - 0045(13): C_1 and C_2 are identically distributed, but are not equal
- 0045(14): def'n: joint distribution
- 0045(14): to get distribution of X + Y, it is enough to know the joint distribution of (X,Y)
- 0045(15):
*e.g.*to show that the distributions of X and Y do not determine the distribution of X + Y - 0045(16): def'n: f(A), where f is a function and A is a PCRV
- 0045(17): def'n: mean (same as expectation) of a PCRV; notation: E[T] = mean (or expectation) of T
- 0045(17): mean is linear
- 0045(17): def'n: T^\circ, where T is a PCRV
- 0045(17): variance of a PCRV; notation: Var[T] = variance of T
- 0045(17): variance is always nonnegative
- 0045(17): a PCRV is deterministic iff its variance is zero
- 0045(17): variance = second raw moment minus the square of the first, i.e., Var[T] = (E[T^2]) - (E[T])^2
- 0045(18-22): proof that variance = (second raw moment) minus (the square of the first)
- 0045(23): recall X, compute E[X]
- 0045(23): intuition: mean is a measure of (average) size
- 0045(24): transition slide
- 0045(25): compute Var[X]
- 0045(25): intuition: variance is a measure of risk
- 0045(26): return-loving vs. risk-averse
- 0045(27): variance is not linear; it's quadratic
- 0045(28): def'n: covariance is the polarization of variance; notation: Cov[S,T] is the covariance of S and T
- 0045(28): Cauchy-Schwarz; def'n: correlation; notation: Corr[S,T] is the correlation of S and T
- 0045(28): correlation is between -1 and 1; correlation is not defined if either PCRV is deterministic
- 0045(28): def'n: uncorrelated
- 0045(29): transition slide
- 0045(30): covariance is (the expectation of the product) minus (the product of the expectations)
- 0045(30): uncorrelated iff (covariance zero) iff (variance of sum = sum of variances) iff (expectation product = product of expectations)
- 0045(31-32): S-T deterministic implies Var[S]=Var[T]
- 0045(33-34): calculation of variance of a binary PCRV
- 0045(35): transition slide
- 0045(36): def'n: standard PCRV
- 0045(36): coin-flipping variables are standard
- 0045(37): def'n: standard deviation; notation: SD[S] is the standard deviation of S
- 0045(37): def'n: U_\circ, for a non-deterministic PCRV U; the "standardization of U"
- 0045(38): any non-deterministic PCRV is "almost" standard
- 0045(39): SD[cS] = |c| SD[S]
- 0045(39-40): variance measures risk, but standard deviation measures it better
- 0045(41): identically distributed implies same mean, variance, standard deviation
- 0045(41): need identical joint distribution for same covariance or for same standard deviation
- 0045(42): standard deviation of a binary PCRV; def'n: arithmetic mean and geometric mean
- 0045(43): def'n: indicator function
- 0045(43-45): problems of calculating probabilities can be converted into problems of calculating expected values
- 0045(46): applications to finance: reduction of risk
- 0045(47): comparing additivity and subadditivity of standard deviation, for perfectly correlated and uncorrelated portfolios, respectively
- 0045(48): description of modern portfolio theory, as a constrained optimization problem
- 0045(49-52): given standard deviations and correlations for two assets, and a requirement to hold a certain amount of one of them,

find the amount of the other to short so as to minimize the standard deviation of the portfolio -- example problem

Topic 0046 (Cholesky decomposition) Link to Topics website

- 0046(1): title slide
- 0046(2): given Z_1,...,Z_n uncorrelated, with variance 1, goal is to

find linear combinations of the Z_j that have prescribed variance-covariance matrix - 0046(2): necessary conditions on a variance-covariance matrix: symmetric, positive semidefinite
- 0046(3): careful mathematical phrasing of the problem
- 0046(4-9): converting the problem into a linear algebra problem:

given C, solve C=AA^t, for A - 0046(10-11): another way to convert the problem to C=AA^t
- 0046(12): using the Spectral Theorem to solve C=AA^t for A
- 0046(12): Spectral Theorem is hard to implement because it requires finding zeroes of a polynomial
- 0046(13-24): an easier way to solve C=AA^t, using row and column operations -- an example
- 0046(25): first statement of the Cholesky decomposition (positive definite C, lower triangular A s.t. C=AA^t)
- 0046(26-29): example of using row and column operations to get a lower triangular A
- 0046(30-34): example with a positive semidefinite C
- 0046(35): improved Cholesky decomposition (positive semidefinite C, lower triangular A s.t. C=AA^t)
- 0046(36-37): why the row/column algorithm works for positive semidefinite C:

a zero on the diagonal implies zeroes all through that row and column - 0046(38): a worked example, done by solving equations (as opposed to row/column operations)
- 0046(39-41): another worked example, done by solving equations
- 0046(42): for semideifnite, a zero on the diagonal implies zeroes all through that row and column
- 0046(43): second improved Cholesky decomposition (positive semidefinite C, upper triangular A s.t. C=AA^t)
- 0046(44): third improved Cholesky decomposition (positive semidefinite C, lower triangular B s.t. C=BB^t)
- 0046(45): fourth improved Cholesky decomposition (positive semidefinite C, upper triangular B s.t. C=BB^t)
- 0046(46): SKILL: given a positive semidefinite, symmetric matrix, find all four Cholesky decompositions of it
- 0046(47): end of linear algebra

Topic 0047 (Cond prob, indep and the CLT) Link to Topics website

- 0047(1): title slide
- 0047(2-6): conditional probability of one event given another -- motivating example
- 0047(7): def'n: conditional probability of one event given another
- 0047(8): coin-flip example (C_1 and C_2) to motivate independence of events
- 0047(9): def'n: two indpendent events when second one has positive probability
- 0047(10): toward a better definition
- 0047(11): def'n: two independent events (the probability of both is the product of the probabilities)
- 0047(12): def'n: two independent PCRVs
- 0047(13):
*e.g.*: C_1 and C_2 are independent PCRVs - 0047(14): def'n: independence of three or more events
- 0047(14): def'n: independence of three or more PCRVs
- 0047(15):
*e.g.*: an independent sequence of PCRVs -- C_1,C_2,C_3,... - 0047(16): for indpendent PCRVs, the joint distribution is determined by the individual distributions
- 0047(17): independent implies uncorrelated
- 0047(18): independence is preserved under postcomposition with functions
- 0047(19): proof that independence is preserved under postcomposition with functions
- 0047(20): A,B independent implies that, for all function f,g,

f(A),g(B) are uncorrelated - 0047(20): converse also true, but not proved
- 0047(21): D_n = C_1 + ... + C_n
- 0047(21): graph and distribution of D_2
- 0047(22): standardization of D_n is D_n/\sqrt{n}
- 0047(23): start of preview of Central Limit Theorem (CLT)
- 0047(24): statement of the CLT in terms of convergence in distribution to a standard normal Z
- 0047(24): explanation of convergence in distribution to Z via "test function"s
- 0047(25): transition slide
- 0047(26): two possible meanings of "test function"
- 0047(27): transition slide
- 0047(28): example problem using the CLT
- 0047(29): def'n: augmented expectation; notation: AE[X]
- 0047(29-31): expectation almost asymptotically commutes with exponentiation
- 0047(32-37): from the mean and standard deviation of an iid sum to the mean and standard deviation of the summands
- 0047(37): an uncorrelated portfolio is better because the standard deviations don't add; they subadd
- 0047(38): the formula for Var[A+B] shows that it's good to have A and B uncorrelated if they represent future asset values
- 0047(38): even better is if A and B have negative correlation
- 0047(39): remember Cauchy-Schwarz
- 0047(39): def'n: perfectly correlated; def'n: perfectly anti-correlated;
- 0047(40): remember the definition of correlation, for non-deterministic PCRVs
- 0047(40): perfectly correlated, uncorrelated and perfectly anti-correlated, in terms of correlation
- 0047(41): def'n: positively correlated; def'n: negatively correlated; recall the def'n of uncorrelated
- 0047(41): positively correlated, uncorrelated and negatively correlated, in terms of correlation
- 0047(42): recall the definition of standard deviation; motivation behind standard deviation; formula for SD[cX]
- 0047(43-45): for perfectly correlated PCRVs, standard deviations add
- 0047(46): for uncorrelated PCRVs, variances add

Topic 0048 (Bayes' Law) Link to Topics website

- 0048(1): title slide
- 0048(2): recall the definition of conditional probability
- 0048(3): an incorrect cancellation formula for Pr[C|A]
- 0048(3): a correct cancellation formula for Pr[(B and C)|A]
- 0048(4): long form vs. short form
- 0048(5): suggestion: move to the long form before doing cancellation
- 0048(6-7): s cancellation formula for Pr[B]
- 0048(8): def'n/notation: {E}_A^x; def'n/notation: {E}_A^o; examples
- 0048(9): more examples of {E}_A^o; factors without A disappear
- 0048(10): def'n/notation: Odds[A]
- 0048(11): def'n/notation: Odds[A|B], conditional odds of A given B
- 0048(11): example of odds in horse-racing
- 0048(12): [probability of (sickness) given (positive medical test)] vs. [probability of (positive medical test) given (sickness)]
- 0048(13): Bayes' Law
- 0048(14-15): application of Bayes' Law to computing [probability of (sickness) given (positive medical test)]
- 0048(16-18): computing [probability of (positive medical test) given (not sick)]
- 0048(19): applying {\bullet}_A^o to Bayes' Law for Pr[A|B]
- 0048(19): result is the odds form of Bayes' Law
- 0048(20): transition slide
- 0048(21): def'n/notation: LQ^A[B], the likelihood quotient of B with respect to A
- 0048(21): general philosophy of the odds form of Bayes' Law: to update the odds based on new information, multiply by a carefully chosen "likelihood quotient"
- 0048(22): checking the odds form of Bayes' Law for sickness and positive medical test
- 0048(23-26): second level of Bayes' Law
- 0048(23): second level of Bayes' Law in terms of probabilities
- 0048(24): apply {\bullet}_S^o
- 0048(25): def'n/notation: LQ^A[C|B], the conditional likelihood quotient of C given B with respect to A
- 0048(26): the odds form of the second level of Bayes' Law

- 0048(27): application of the second level of Bayes' Law, via high-risk group
- 0048(28): explicit formula for LQ^A[C|B]
- 0048(29): general philosophy of the odds form of Bayes' Law: to update the odds based on new information, multiply by a carefully chosen "likelihood quotient"
- 0048(30): the third level of Bayes' Law
- 0048(30): general philosophy of the odds form of Bayes' Law: to update the odds based on new information, multiply by a carefully chosen "likelihood quotient"
- 0048(31-33): cancellation formulas for likelihood quotients
- 0048(34-36): proving the second and third level of Bayes' Law using the first level,

together with cancellation formulas from likelihood quotients - 0048(36): general philosophy of the odds form of Bayes' Law: to update the odds based on new information, multiply by a carefully chosen "likelihood quotient"

Topic 0049 (Conditional expectation) Link to Topics website

- 0049(1): title slide
- 0049(2-3): an example of calculating the conditional expectation of a PCRV given an event
- 0049(4): notation: E[X|W], the conditional expectation of X, given W
- 0049(4): a formula for E[X|W] in terms of integration
- 0049(5): def'n: conditional expectation of a PCRV given an event
- 0049(6-14): the conditional expectation of one PCRV given another
- 0049(7-10): a worked example based on two PCRVs, X and Y
- 0049(7): calculation of E[Y|X=a] for two values of a
- 0049(8): transition slide
- 0049(9): def'n of the PCRV E[Y|X] and the graph of of E[Y|X]
- 0049(10): general philosophy: average Y over the level sets of X

- 0049(11-14): a worked example based on two PCRVs, X' and Y
- 0049(11): calculation of E[Y|X'=a] for two values of a
- 0049(12): transition slide
- 0049(13): def'n of the PCRV E[Y|X'] and the graph of of E[Y|X']
- 0049(14): general philosophy: average Y over the level sets of X'

- 0049(7-10): a worked example based on two PCRVs, X and Y
- 0049(15-23): conditional expectation of a PCRV given a partition
- 0049(16): calculation of the level sets of X, where X is defined above
- 0049(16): def'n: partition of a PCRV; notation: \scrP_V
- 0049(16): calculation \scrP := \scrP_X
- 0049(17): calculation of the level sets of X', where X' is defined above
- 0049(17-18): calculation \scrP := \scrP_X = \scrP_{X'}
- 0049(19): calculation of E[Y|\scrP], where Y is defined above
- 0049(20): def'n of E[V|I], where I is an interval
- 0049(20): def'n of E[V|\scrQ], where \scrQ is a partition of \Omega by intervals; \Omega:=[0,1]
- 0049(21): def'n: E[V|W], where V and W are PCRVs
- 0049(21): difficulty: sets in the partition of W may not be intervals, but, rather, fUofIs (finite unions of intervals)
- 0049(22): def'n of E[V|I], where I is a fUofI
- 0049(22): def'n of E[V|\scrQ], where \scrQ is a partition of \Omega by fUofIs; \Omega:=[0,1]
- 0049(23): new difficulty
- 0049(23): fix that difficulty by only considering PCRVs W whose level sets all have positive size
- 0049(23): more on that difficulty later

- 0049(24-73): more on conditional expectations given PCRVs and partitions
- 0049(24): def'n: a.e. constant
- 0049(24): def'n: measurability of a PCRV w.r.t. a partition
- 0049(24): philosophy: a partition is a measurement of information
- 0049(24): philosophy: a PCRV is measurable w.r.t. a partition if the information of the partition is sufficient to calculate the PCRV a.s.
- 0049(25): the idea of starting with Y, which is *not* \calP-measurable and perturbing as little as possible into a PCRV that is
- 0049(26): one way of interpreting "as little as possible": keep the same conditional expectations on sets in \calP
- 0049(27-28): another interpretation: develop a distance between PCRVs and minimize the distance
- 0049(27): recall distance in the plane
- 0049(28): L^2 distance between PCRVs and L^2 norm of a PCRV

- 0049(29): a worked example with two PCRVs W and V
- 0049(29): averaging V over the level sets of W gives U, i.e., U = E[V|W]
- 0049(29): goal: interpret U as a the L^2-closest PCRV to V that is (\scrP_W)-measurable
- 0049(29): let X be a generic (\scrP_W)-measurable PCRV (with values x and y)
- 0049(30): write out X and V
- 0049(31): compute X-V
- 0049(32): compute (X-V)^2 and then E[(X-V)^2] which is the squared L^2 norm of X-V
- 0049(33): finish off the minimization problem
- 0049(34): difficulty: some level sets may have size 0, and so you can't average over them
- 0049(35): make a change to W so that one level set has size 0
- 0049(35): average over those level sets that do have positive size
- 0049(36): make arbitrary choices elsewhere
- 0049(37): then E[V|W] is not well-defined, and, instead of (U=E[V|W]), we write (U=E[V|W] a.s.)
- 0049(38): another possibility for E[V|W]
- 0049(39): the Tower Law
- 0049(39): def'n: finer
- 0049(39): P finer than Q implies any Q-measurable PCRV is P-measurable
- 0049(40): transition slide
- 0049(41): the Tower Law (not proved)

- 0049(42-47): recall the coin-flipping variables C_1,C_2,...
- 0049(48): recall the Central Limit Theorem
- 0049(48): def'n: PCRV approximation
- 0049(49-51): key goal of probability theory:

define random variables and expectation so that every PCRV approximation has a limit in distribution - 0049(52-53): for example: we seek a random variable Z such that (C_1 + ... + C_n) / \sqrt{n} ---> Z in distribution
- 0049(54-70): SKIPPED
- 0049(71): independent implies (deterministic conditional expectation) implies uncorrelated
- 0049(72): uncorrelated does not imply (deterministic conditional expectation) does not imply independent
- 0049(73): SKIPPED

- 0049(74-77): linearity of conditional expectation (or: taking out what you know)
- 0049(74): linearity of E[\bullet] over \R
- 0049(74): linearity of E[\bullet|P] over P-measurable PCRVs
- 0049(74): special case when P is as coarse as possible
- 0049(74): special case: E[\bullet|P] commutes with multiplication by P-measurable PCRVs
- 0049(75): why this commutativity is called "taking out what you know"
- 0049(76-77): proof of linearity of E[\bullet|P] over P-measurable PCRVs

Topic 0050 (Stirling's Formula) Link to Topics website

- 0050(1): title slide
- 0050(2): def'n of asymptotics, notation: a_n \tilde b_n
- 0050(2): some illustrations of asymptotics
- 0050(2): (x_n \tilde 5/n) implies (1+x_n)^n ---> e^5
- 0050(3): the graph of \ln
- 0050(4-5): trapezoidal approximation of \int_1^4 \ln x dx, and notation for the error, as a sum of "slivers" c_1+c_2+c_3
- 0050(6): transition slide
- 0050(7): the slivers, on translation form disjint subsets in a rectangle of area \ln 2
- 0050(8): c_1+c_2+c_3 \le \ln 2
- 0050(9): generalizing to n trapezoids, with n slivers, to estimate \int_1^n \ln x dx, with error
- 0050(10-12): computing the exact value of \int_1^n \ln x dx, by integration by parts
- 0050(13): exponentiation of estimate with error gives a formula for n!
- 0050(14): the error factor K_n is asymptotic to some constant K
- 0050(14-15): from the aysmptotics of K_n to the asymptotics of n!
- 0050(16): IOU: K = \sqrt{2\pi}; this yields Stirling's Formula
- 0050(17): Know that n! is asymptotic to a certain formula involving K
- 0050(18): transition slide
- 0050(19): I_n := \int_0^{\pi/2} \sin^n(x) dx
- 0050(19-24): a formula relating I_n to I_{n-2} via integration by parts
- 0050(24): a recursive formula for I_n in terms of I_{n-2}
- 0050(25): transition slide
- 0050(26): recall the definition of asymptotics, some asymptotics involving I_n-2
- 0050(27): I_n is asymptotic to I_{n-2}
- 0050(28): I_{n-1} is between I_n and I_{n-2}, so all three are asymptotic
- 0050(28): I_n is asymptotic to I_{n-1}
- 0050(28): relpacing n by n+1, I_{n+1} is asymptotic to I_n
- 0050(28): replacing n by 2n, I_{2n+1} is asymptotic to I_{2n}
- 0050(29-37): calculation of I_0 through I_9
- 0050(38): two formulas, one for I_{2n}, and another for I_{2n+1}
- 0050(39): the two formulas are asymptotic
- 0050(40-45): rewriting this asymptotic relation in terms of factorials
- 0050(46): transition slide
- 0050(47): cancellation of terms from this rewritten asymptoic formula yields that 2\pi is asymptotic to K^2
- 0050(47): two constants are asymptotic iff they are equal, so 2\pi = K^2
- 0050(47): K is positive, completing the IOU
- 0050(48): Stirling's formula is proved

Topic 0051 (From Stirling to the CLT) Link to Topics website

- 0051(1): title slide
- 0051(2): recall Stirling's formula, giving asymptotics of n!
- 0051(2-4): recall asymptotics and some examples of asymptotics
- 0051(4): if x_n is asymptotic to y_n and q_n is a sequence of integers that tends to infinity,

then x_{q_n} is asymptotic to y_{q_n} - 0051(4): from Stirling's formula to asymptotics of, say, (2n)!
- 0051(5-11): applied coin-flipping
- 0051(5): modeling male height
- 0051(6-9): modeling gravitational acceleration as an extremely mildly probabilistic phenomenon, with N in the denominator instead of \sqrt{N}
- 0051(9): maybe all physics should be probabilistics, but sometimes we can't see it because, e.g., N is in the denominator
- 0051(10): applying coin-flipping to finance, an expected payout problem
- 0051(11): a (simplistic) probabilistic model of stock price evolution, with N=10^{10^{100}} time subperiods
- 0051(11): computation of that expected payout, assuming a 68% probability that (H-T)/\sqrt{N} is between -1 and 1
- 0051(11): our goal is now to prove that that probability really is 68%

- 0051(12): changing from [the constant N=10^{10^{100}}] to [2n, with n a variable integer]; we will eventually look at the n=N/2 case
- 0051(13): an example path when n=9, i.e., with 2n=18 time subperiods, based on 18 coin-flips
- 0051(14): probability of a random path (with n=9) ending at 4/\sqrt{18}
- 0051(15): transition slide
- 0051(16-22): for n=9, building a histogram (a.k.a. bar graph) illustrating probabilities of ending at various numbers
- 0051(22): the span of the n=9 histogram
- 0051(23): generalize to arbitrary n, and compute, in the nth histogram:
- the heights of all the bars
- the span of the histogram

- 0051(24): to prove the 68% percent goal from above, we study the n=N/2 histogram
- 0051(25): f_n(x) is defined as the function whose graph is the top of the nth histogram
- 0051(25): goal: f_n(x) ---> e^{-x^2/2}/\sqrt{2\pi}, as n ---> \infty
- 0051(26-28): when n > x^2/2, x is in the span of the nth histogram
- 0051(29): statement of Central Limit Theorem (CLT) in this notation
- 0051(29): for n > x^2/2, x is in the span of the nth histogram, k_n is the number of the bar under which it lies and h_n is the height of that bar
- 0051(29): we wish to show that h_n ---> e^{-x^2/2}/\sqrt{2\pi}
- 0051(30): we focus on x=7
- 0051(31-34): aysmptotics of k_n
- 0051(35): transition slide
- 0051(36-38): asymptotics of n + k_n and of n - k_n
- 0051(39): transition slide
- 0051(40-42): develop the formula for h_n
- 0051(42): recall Stirling's formula
- 0051(43): transition slide
- 0051(44-52): asymptotics of h_n
- 0051(53-54): h_n ---> e^{-x^2/2}/\sqrt{2\pi}, for x=7, except for two IOUs
- 0051(55): a new IOU: general result about 1^\infty indeterminate forms
- 0051(55-60): using that general result to prove the first two IOUs
- 0051(61-62): proof of the general result

Topic 0052 (Piecewise constant processes) Link to Topics website

- 0052(1): title slide
- 0052(2): def'n: (\triangle t)-piecewise constant process ((\triangle t)-PCP)
- 0052(2): def'n: X_t
- 0052(2): idea of def'n: an evolving PCRV, but can only change at multiples of \triangle t
- 0052(3): a 2-PCP is a 1-PCP
- 0052(4): a (\triangle t)-PCP is a ((1/2)(\triangle t))-PCP
- 0052(5): def'n: piecewise constant process ((\triangle t)-PCP)
- 0052(6): def'n: \triangle X, when X is a PCP
- 0052(7-8): recall coin-flipping sequence C_1,C_2,...
- 0052(9): a PCP X that approximates Brownian motion (BM)
- 0052(10): transition slide
- 0052(11): for that X (that approximates BM), the CLT gives good approximation for E[g(X_1)]
- 0052(12): recall definition of bounded function \R ---> \R
- 0052(12): recall definition of PCRV approximation
- 0052(13): def'n: bounded fuction X ---> R, where X is any set, (
*e.g.*, X=\R^k) - 0052(14): def'n: PCP approximation
- 0052(15): example of a PCP approximation: the standard Brownian motion approximation
- 0052(16): with X^{(N)} = that standard BM approximation, do the following:
- problem: for all t\ge0, compute the limit, as n--->\infty, of E[( e^{X_t^{(N)} - 5 )_+]
- subproblem: compute the limit, as n--->\infty, of: E[( e^{X_3^{(N)} - 5 )_+]

- 0052(16-23): solution to the subproblem, using the CLT
- 0052(24): guess solution to the problem, based on solution to the subproblem
- 0052(24): a second subproblem: compute the limit, as n--->\infty, of: E[( e^{X_\pi^{(N)} - 5 )_+]
- 0052(25-26): solution to the second subproblem, except for an IOU
- 0052(27): transition slide
- 0052(28-32): proof of the IOU
- 0052(33-36): motivation of Stochastic Differential Equations via "the random bank"

Topic 0053 (Functional analysis) Link to Topics website

- SKIPPED

Topic 0054 (The heat equation) Link to Topics website

- SKIPPED

Topic 0055 (One period pricing and hedging) Link to Topics website

- 0055(1): title slide
- 0055(2-8): a pricing problem involving Dan and Alice
- 0055(3): description of a foward (or futures) contract
- 0055(3): description of an option contract
- 0055(4): transition slide
- 0055(5): the payoff or (contingent) claim
- 0055(6-8): rewriting the payoff using (\bullet)_+ notation

- 0055(9): transition slide
- 0055(10-11): pricing step 1: model the underlying
- 0055(11): explanation of the 1-subperiod 70-30 CRR model
- 0055(12): transition slide
- 0055(13-17): pricing step 2: calibrate the model
- 0055(13): explanation of drift and volatility
- 0055(14): transition slide
- 0055(15-16): calculation of u and d from the drift and volatility
- 0055(17): calculation of the uptick and downtick factors

- 0055(18-25): pricing step 2: find a perfect hedging strategy
- 0055(19): transitio slide
- 0055(20): description of the hedging portfolio (a.k.a. "the hedge")
- 0055(21): using the payoff to find the ending value of the hedge
- 0055(22): transition slide
- 0055(23): setting up two equations in two unknowns to find the hedging parameters
- 0055(24): a third equation in a third unknown
- 0055(25): solving the equations

Topic 0056 (Risk-neutrality and Delta-hedging) Link to Topics website

- 0056(1): title slide
- 0056(2): recall the three equations in three unknowns from the preceding topic
- 0056(2): probability is used in the calibration step (step 2), but continues to be useful in the hedging step (step 3)
- 0056(2): trick: imagine another universe with uptick and downtick probabilities that help us in the calculation of the price
- 0056(3): in *our* universe (modeled with 70-30 probabilities) two expected value problems
- 0056(4): transition slide
- 0056(5): the expected value and return of 1 US dollar, invested in the bank (following our 70-30 model)
- 0056(6-7): the expected value and return of 1 US dollar, invested in Euros (following our 70-30 model)
- 0056(8): transition slide
- 0056(9): investors in our universe are "risk-averse"
- 0056(9): imagine a universe where they are "risk-neutral"
- 0056(10): computation of new probabilities (60-40)
- 0056(11): recall Euro return under real world probabilities
- 0056(12): $1 Euro expected value under the new (60-40) probabilities
- 0056(13): $1 bank expected value under the new (60-40) probabilities
- 0056(14): $1 Euro and bank expected return under the new (60-40) probabilities
- 0056(14): the 60-40 world is the risk-neutral world
- 0056(15): $3 bank expected return under the risk-neutral (60-40) probabilities
- 0056(16): $2 Euro expected return under the risk-neutral (60-40) probabilities
- 0056(17): ($3 bank plus $2 Euro) expected return under the risk-neutral (60-40) probabilities
- 0056(17): in the risk-neutral world all portfolios have the same expected return
- 0056(18): using this constancy of expected returns to compute the price of the option that Alice sells to Dan
- 0056(19): coin-flippers got price
- 0056(20): recall the diagram yielding the price and hedging parameters
- 0056(20): difference in the hedging parameters (x-y) is the price (called "?")
- 0056(20): as pricers, we know ?, so, if we can compute x, we can solve x-y=? for y
- 0056(21): naïve volatility of Euro, bank and the hedging portfolio
- 0056(22): 40 Euros would not have enough naïve volatility
- 0056(23): we need 50 Euros; number of Euros (called "x") is the naïve volatility of the option divided by that of the Euro
- 0056(24): pricers got hedge

Topic 0057 (Pricing/hedging in 3 subperiods) Link to Topics website

- 0057(1): title slide
- 0057(2): a pricing problem involving Harry and Gail involving a three monnth call option on stock
- 0057(3): Gail's model (three subperiod 90-10 CRR)
- 0057(4-8): calibration to get the one month uptick and downtick factors
- 0057(9): computing the one month logarithmic risk-free factor
- 0057(10): transition slide
- 0057(11): computing the one monnth risk-free factor
- 0057(11): comparing the uptick and downtick factors to the risk-free factor
- 0057(12): dropping the 90-10 real-world probabilities
- 0057(13): computing the risk-neutral probabilities
- 0057(14): all portfolios have the same expected return each month
- 0057(15): setting up the template of prices and hedging parameters
- 0057(16-17): putting the share prices in the template
- 0057(18-19): using the payoff function to work out the contingent claim = the ending hedge values
- 0057(20): working out the hedge values after two months, using risk-neutrality
- 0057(21): working out the hedge values after one month, using risk-neutrality
- 0057(22): working out the starting hedge value = the price of the option, using risk-neutrality
- 0057(23-24): going directly from the contingent claim to the option price, using risk-neutrality
- 0057(25): summary
- work out the underlying in forward time
- use the payoff function to find the contingent claim
- use risk-neutrality to work back to the price (either in stages or in a single step)

- 0057(26): transition slide
- 0057(27): plan to use \Delta-hedging, a.k.a. "pricers got hedge"
- 0057(28): working out one hedging parameter (number of shares) as the needed naïve volatility of the option divided by the naïve volatility of a single share
- 0057(29): a comment on rounding error
- 0057(30): filling that hedging parameter into the template
- 0057(31): too few shares gives too little naïve volatility; too many gives too much
- 0057(32-33): working out the other hedging parameter (bank loan) at that same node via the equation of the node
- 0057(34): filling in all the other hedging parameters into the template
- 0057(34): underlying in forward time, the derivative in backward time, hedging parameters in any order
- 0057(35): answers to more decimals

Topic 0058 (Pricing/hedging many subperiods, 1) Link to Topics website

- 0058(1): title slide
- 0058(2): a pricing problem involving Harry and Gail involving a 30 day call option on stock
- 0058(2): Gail's model (N subperiod 50.001-49.999 CRR, with N = 30 x 24 x 60 x 60 = number of seconds in 30 days)
- 0058(3-7): calibration to get the one second uptick and downtick factors
- 0058(8-10): computing the one second logarithmic risk-free factor
- 0058(10): computing r := the one second risk-free factor
- 0058(10-11): the one-second risk-free factor is the average of the uptick and downtick factors
- 0058(12): transition slide
- 0058(13): recall the one-second uptick and downtick factors and the one-second risk-free factor
- 0058(14): goal: price via hedging; difficulty: many subperiods; savlation: CLT
- 0058(15): the payoff function
- 0058(16): computing the risk-neutral probabilities
- 0058(17): returns are the same in the risk-netural world
- 0058(18): the evolution of the share price in our model, organized by a large recombinant tree
- 0058(18): the ending share prices
- 0058(19): the contingent claim, via the payoff function
- 0058(20): V = price of option = initial value of hedging portfolio
- 0058(21): e^{rN}V = expected contingent claim
- 0058(21): the probability of each outcome in the contingent claim
- 0058(22): an equivalent coin-flipping game
- 0058(23): hard to find the expected value in the coin-flipping game
- 0058(24): first do a probability problem, then move to an expected value problem
- 0058(24): the probability problem: compute Pr[-\sqrt{N}

Topic 0059 (Central Limit Theorem) Link to Topics website- 0059(1): title slide
- 0059(2): recall the probability problem: compute Pr[-\sqrt{N}
0059(3): rephrasing the problem in terms of the standardization, X := (H-T)/\sqrt{N}, of H-T - 0059(4-5): notation H_j, T_j, D_j
- 0059(5): the program: D_1, D_1/7, D_2, D_N, X = D_N/\sqrt{N}
- 0059(6-9): understanding D_1
- 0059(6-7): the distribution of D_1
- 0059(8): its generating function and Fourier transform (FT)
- 0059(9): inverse Fourier transform

- 0059(10-11): understanding D_1/7
- 0059(11): (the FT of the distribution of D_1/7) is obtained by replacing t by t/7 in (the FT of the distribution of D_1)
- 0059(12-13): understanding D_2
- 0059(13): (the FT of the distribution) of D_2 is the square of (the FT of the distribution of D_1)
- 0059(14): understanding D_N
- 0059(15): understanding X
- 0059(16): transition slide
- 0059(17): themes (generating functions, Fourier transforms, Fourier analysis, infinite-dimensional spectral theory)
- 0059(17): recall the probability problem: compute Pr[-1
0059(17): (the FT of the distribution) of X is approximately equal to e^{-t^2/2} - 0059(18): transition slide
- 0059(19): ( a random variable whose distribtion has FT equal to e^{-t^2/2} ) should be close to X
- 0059(19): approximate answer to the probability problem
- 0059(20): description of the distribution of a certain random variable Z
- 0059(21): Z has infinite support
- 0059(22): correcting a small mistake
- 0059(23): Pr[Z=7] = 0
- 0059(24): Pr[2
0059(25): generating function and Fourier transform of Z - 0059(26): Z is close to X
- 0059(27): approximate answer to the probability problem: : Pr[-1
0059(27): Berry-Esseen Theorem bounds the error - 0059(28): next: expected value in the coin-flipping game

Topic 0060 (Pricing/hedging many subperiods, 2) Link to Topics website- 0060(1): title slide
- 0060(2): recall the coin-flipping game and the expected value problem
- 0060(2): goal: the approximate expected value, E, of f(e^{Hu+Td})
- 0060(3): the expected value of f(D_2)
- 0060(4): the expected value of g(D_2), for any g
- 0060(5-7): the expected value of f(Z)
- 0060(8): the expected value of g(Z), for any exp-bdd g
- 0060(9-10): highlighting the change from Z to x, and then integrating against e^{-x^2/2}/\sqrt{2\pi} =: h(x)
- 0060(11): recall X = D_N / \sqrt{N}, with N = 30 x 24 x 60 x 60, and D_N = H_N - T_N = H - T = C_1 + ... + C_N
- 0060(12): the approximate expected value of g(X), for any exp-bdd g
- 0060(13): subgoal: choose g so that g(X) = f(e^{Hu+Td})
- 0060(14): make sure that g is exp-bdd
- 0060(15-16): writing H and T in terms of X
- 0060(17-19): writing e^{Hu+Td} as Ce^{kX}
- 0060(20): figuring out g and checking that g is exp-bdd
- 0060(21): figuring out C and k
- 0060(22): setting up an integral that is approximately equal to E, our goal
- 0060(23): transition slide
- 0060(24-30): evaluating the integral
- 0060(30): E is approximately 121.07
- 0060(31): recall the coin-flipping game, recall V = e^{-rN} E is the price of the option
- 0060(31): compute V approximately
- 0060(31): V is approximately 120.76
- 0060(32-36): computing E exactly as a sum involving binomial coefficients
- 0060(36): to two decimals, E is 121.11; the approximation we found was accurate to 4 cents
- 0060(37): to two decimals, V is 120.80; the approximation we found was accurate to 4 cents
- 0060(37-41): computing the (real-world) expected return of the stock after one year, from its drift and volatility
- 0060(41): let S_1 denote the stock price after 1 year
- 0060(41): augmented drift := [drift] + [1/2][(volatility)^2] = [the augmented expectation of (ln S_1]
- 0060(41): key point: [the expected return on the stock] is [the expectation of (the exponential of (ln S_1))], which is:
- the exponential of (the augmented expectation of (ln S_1)), or
- the exponential of the augmented drift

- 0060(41): the expected return is NOT the exponential of the drift

Topic 0061 (Intro to the Black-Scholes Formula) Link to Topics website- 0061(1): title slide
- 0061(2): set up a T-year European call option on one share of a stock struck at K
- 0061(2): S_0 := current price of the stock
- 0061(2): more notation for payoff function, drift, volatility, logarithmic risk-free factor
- 0061(2): asterisk subscript is used for annualized figures, unsubscripted figures are over T years
- 0061(2): choose (modeled) real-world uptick probability p, and let q := 1-p
- 0061(2): our sequence of models: the n-subperiod (p,q) CRR model, with n = 1,2,3,...
- 0061(2): the models "tend to Black-Scholes"
- 0061(2): goal: limit of (the option price under the nth model), as n ---> \infty
- 0061(3): the evoluation (over one subperiod) of the stock price, and its logarithm
- 0061(4-7): setting up the equations that must be solved to do the calibration
- 0061(8-10): finding the risk-neutral probabilities
- 0061(11): V_n := inital value of the hedge in the nth model = option price in the nth model
- 0061(11): goal is limit of V_n, as n ---> \infty
- 0061(11-14): finding a summation formula for the price of the option in the nth model
- 0061(14): the CRR Option Pricing Formula
- 0061(15): K' := present value of the strike price
- 0061(15): def'n: bogus at the money quotient, at the money quotient, logarithmic at the money quotient
- 0061(16): def'n: Black-Scholes center
- 0061(17): def'n: Black-Scholes interval with endpoints d_- and d_+
- 0061(18): transition slide
- 0061(19): theorem: \lim V_n = [S_0][\Phi(d_+)] - [K'][\Phi(d_-)]
- 0061(20): version 0 of the Black-Scholes Option Pricing Formula (BSOPF) is [S_0][\Phi(d_+)] - [K'][\Phi(d_-)]
- 0061(21): simplicity and centrality of the BSOPF
- 0061(22): inputs, outputs, asymptotic output
- 0061(23-27): change to annualized figures
- 0061(28-29): centrality and simplicity of BSOPF using annualized figures
- 0061(30-32): writing out the BSOPF using annualized figures
- 0061(32): first version of the BSOPF
- 0061(33-34): rewriting K' in terms of K, and adjusting the BSOPF to involve K not K'
- 0061(34): second version of the BSOPF
- 0061(35-36): rewriting the BSOPF in terms of forward prices
- 0061(36): third version of the BSOPF
- 0061(37): transition slide
- 0061(38): first, second and third version of BSOPF are present, neutral and forward formulas
- 0061(39): version zero is a time-normalized present formula

Topic 0062 (Testing the B-S Formula) Link to Topics website- 0062(1): title slide
- 0062(2): reall versions zero to four of the BSOPF (Black-Scholes Option Pricing Formula)
- 0062(2): do these formulas approximate the exact CRR price, with a large number of subperiods?
- 0062(3): recall: Kyle and Gail, with N = 30 x 24 x 60 x 60 subperiods
- 0062(4): recall: the exact CRR price
- 0062(5): transition slide
- 0062(6): recall the logarithmic risk-free factor
- 0062(7): transition slide
- 0062(8): recall the volatility and strike price
- 0062(8): recall the initial stock price
- 0062(8): recall the formulas for K', for d_+ and for d_-
- 0062(9): compute K'
- 0062(10): compute the logarithmic at the money quotient
- 0062(11): compute d_+ and d_-
- 0062(12): transition slide
- 0062(13): compute \Phi(d_+) and \Phi(d_-)
- 0062(14): transition slide
- 0062(15): compute the BSOPF price and compare with the exact CRR price
- 0062(16): the approximation is good
- 0062(17): transition slide
- 0062(18): BSOPF as a function, BlSch, of five inputs:
- tenor T
- annual volatility \sigma_*
- annual logarithmic risk-free factor r_*
- initial price of the stock S_0
- strike price K

- 0062(18): for fixed T,r_*,S_0 and K, this function of \sigma_* is increasing
- 0062(19): def'n: implied volatility
- 0062(20): Why teach BS (Black-Scholes)? Comparison with home mortgage interest rates.
- 0062(21): def'n: volatility smile and skew
- 0062(22): def'n: volatility surface

Topic 0063 (Prelim to the TCLT and B-S) Link to Topics website- 0063(1): title slide
- 0063(2): recall decay rates on Maclaurin error estimates
- 0063(3): decay rate on error estimate for e^{x_n} \approx 1 + x_n + (x_n)^2/2, with x_n ---> 0 as n ---> \infty
- 0063(3): decay rate on error estimate for e^{3x_n} \approx 1 + 3x_n + 9(x_n)^2/2, with x_n ---> 0 as n ---> \infty
- 0063(4): def'n: h(x) = e^{-x^2}/\sqrt{2\pi}
- 0063(4): def'n: Z_n ---> Z in distribution
- 0063(5): remembering the formula: change every Z_n to x and then integrate against h
- 0063(6): equivalent definition for Z_n ---> Z in distribution
- 0063(6): fact: adding a sequence of scalars tending to zero
- 0063(6): fact: multiplying by a sequence of scalars tending to one
- 0063(7): def'n: Z_n ---> Z in distribution against continuous exp-bdd
- 0063(7): same two facts for convergence in distribution against continuous exp-bdd
- 0063(8-9): recall: dividing the PCRV by 7 causes the Fourier transform to have t replaced by t/7
- 0063(10): recall: the Fourier transform for D_2 is the square of the one for D_1
- 0063(11-12): recasting D_2 as a sum of iid PCRVs whose distributions have Fourier transform cos t
- 0063(12): if you add two independent PCRVs, the Fourier transform is the product of the two Fourier transforms
- 0063(13): the idea of convolution of distributions
- 0063(14): Fourier transform simplifies convolution to multiplication
- 0063(15): if two PCRVs have distributions with nearly equal Fourier transforms, then their distributions are nearly equal
- 0063(16): if a sequence of PCRVs have distributions whose Fourier transforms approach e^{-t^2/2}, then the PCRVs approach Z in distriution
- 0063(17): notation for the Fourier transform of the distribution
- 0063(18): facts about Fourier transforms
- 0063(18): def'n: identically distributed (i.d.) set of PCRVs
- 0063(19): def'n and notation: [mean/expectation, variance, standard deviation (SD), Fourier transform of distribution (FTD)] of an i.d. set, A, of PCRVs
- 0063(19): def'n: c+A, cA
- 0063(19): (A is i.d.) implies (cA is i.d.)
- 0063(19): formulas for mean, variance and SD of cA
- 0063(20): formulas for mean, variance and SD of c+A
- 0063(21): def'n: S := \{ standard PCRVs \}
- 0063(21): S is not i.d.
- 0063(22): an i.d. subset of S has mean 0 and variance 1
- 0063(22): an i.d. set meeting S must be contained in S
- 0063(23): def'n: iid n-fold sum, notation \sum_n
- 0063(24): (A is i.d.) implies (\sum_n A is i.d.)
- 0063(24): formulas for mean, variance, SD and FTD of \sum_n A
- 0063(24): formula for [ \sum_n on ((c/n)+A) ]
- 0063(24): formula for [ \sum_n on (cA) ]
- 0063(24-25): renormalized iid sum preserves and reflects standardness
- 0063(26): def'n: B^{p,u}_{q,d}, specific set of binary variables
- 0063(26): B^{p,u}_{q,d} is i.d.
- 0063(26): multplying a constant by B_{q,d}^{p,u}
- 0063(26): adding a constant to B_{q,d}^{p,u}
- 0063(27): def'n: B^p_q, general set of binary variables
- 0063(27): analysis of \sum^n B^p_q
- 0063(28): the iid sum of general binaries is the union of the iid sums of specific binaries
- 0063(29-30): \sum^n B^p_q is invariant under adding a constant or multiplying by a nonzero constant
- 0063(31): recall the definition of X_\circ, standardization
- 0063(31): \sum^n B^p_q is invariant under standardization
- 0063(32): iid products
- 0063(32): exp of an iid sum is an iid product
- 0063(32): formula for the mean of the an iid product

Topic 0064 (the Triangular CLT) Link to Topics website- 0064(1): title slide
- 0064(2): statement of the TCLT (Triangular Central Limit Theorem)
- 0064(2): the sequence p_n of probabilities is bounded away from 0 and 1
- 0064(2): Y_n is an standard iid sum of n binary variables, using probabilities p_n and 1 - p_n
- 0064(2): recall the definition of convergence to Z in distribution
- 0064(2): recall the definition of B^p_q
- 0064(3): the triangle of variables
- 0064(4): recall that the iid sum of general binaries is the union of the iid sums of specific binaries
- 0064(5): q_n := 1 - p_n
- 0064(6): start of proof of the TCLT
- 0064(7): identify Y_n as an iid sum of specific binary variables, using u_n and d_n
- 0064(8): transition slide
- 0064(9-11): find the equations to solve to find u_n and d_n
- 0064(11): solve the equations to find u_n and d_n
- 0064(12): show that u_n and d_n are bounded
- 0064(13): transition slide
- 0064(14-16): compute the Fourier transform [\phi(t/\sqrt{n})]^n of the distribution of Y_n
- 0064(17): goal is to show that [\phi(t/\sqrt{n})]^n convergest to e^{-t^2/2}
- 0064(18): transition slide
- 0064(19): focus on t=5
- 0064(19-21): second order approximation of \phi(5/\sqrt{n}), with error term
- 0064(22): transition slide
- 0064(23): [ 1 + (x/n) + o(1/n) ]^n ---> e^x finishes the proof of the TCLT
- 0064(24): TCLT, version 2
- 0064(25): def'n of convergence in distribution to \sigma Z + \mu
- 0064(26-28): equivalent definitions for convergence in distribution to \sigma Z + \mu
- 0064(29-40): TCLT, version 3
- 0064(29): start of statement; note that PCRVs may be nonstandard
- 0064(30): conclusion of statement
- 0064(31): transition slide
- 0064(32): start of proof: def'n of \mu_n and \sigma_n, standardization of the PCRVs, and use of first TCLT
- 0064(33): transition slide
- 0064(34-36): replacing \sigma_n by its limit
- 0064(37-39): replacing \mu_n by its limit
- 0064(40): recall the definion of convergence in distribution to \sigma Z + \mu
- 0064(40): conclude the proof

Topic 0065 (1st proof of Black-Scholes) Link to Topics website- 0065(1): title slide
- 0065(2-3): notation
- f(x) is the payoff function (for a call option)
- T is the tenor
- S_t is the stock price at time T
- X = \ln ( S_T / S_0 )
- r is the logarithmic risk-free rate over time T
- \mu is the drift of the stock over time T
- \sigma is the drift of the stock over time T

- 0065(3): we select as sequence of models (the n-subperiod (p,q) CRR models, with n=1,2,...)
- 0065(3): this models X as a sequence of PCRVs, X_n, in \Sum^n B^p_q
- 0065(4): change X to X_n for the modeled real world
- 0065(5): transition slide
- 0065(6): let u_n and d_n be the (unknown) uptick and downtick amounts
- 0065(6): then X_n \in \Sum^n B^{p,u_n}_{q,d_n}
- 0065(6): start calibration to find u_n, d_n from \mu,\sigma
- 0065(7): find the risk-neutral (R-N) probabilities p_n and q_n
- 0065(7): IOU: for n sufficiently large, the risk-free rate is between the downtick and uptick factors
- 0065(8): transition slide
- 0065(9): in the R-N world, change X_n to \tidle X_n \in \Sum^n B^{p_n,u_n}_{q_n,d_n}
- 0065(9-10): verify that the expeted value of e^{\tilde X_n} is e^r,

*i.e.*, an investment in bank or stock yields the same expected return - 0065(11-12): comparison of the real world (about which little is known for sure), the modeled real world, and the R-N world
- 0065(12): key point: in the R-N world, the nth forward option price is equal to the expected payout
- 0065(13): transition slide
- 0065(14): \mu_n := the R-N drift; \sigma_n := the R-N volatility
- 0065(15): compare calibration formulas that allow us to go from \mu,\sigma to u_n,d_n

to formulas that take us from u_n,d_n to \mu_n,\sigma_n - 0065(16): transition slide
- 0065(17): more IOUs: as n ---> \infty, we have
- Girsanov: \sigma_n ---> \sigma
- p_n ---> p, q_n ---> q
- \mu_n ---> r - ( \sigma^2 / 2 )
- key point: the limit of \mu_n does not involve \mu
- as a result, \mu does not appear in the Black-Scholes Option Pricing Formula

- 0065(17): standardize \tilde X_n, getting ( \tilde X_n - \mu_n ) / \sigma_n
- 0065(18-19): apply TCLT version 2 to see that ( \tilde X_n - \mu_n ) / \sigma_n ---> Z in distribution against continuous exp-bdd
- 0065(20): replacing \sigma_n by its limit \sigma
- 0065(21): replacing \mu_n by its limit \nu := r - ( \sigma^2 / 2 )
- 0065(21): then Z_n := ( \tilde X_n - \nu ) / \sigma ---> Z in distribution againts continuous exp-bdd
- 0065(22): finding a test function g so that the forward option price is E[g(Z_n)]
- 0065(23-25): using Z_n ---> Z to reduce calculation of E[g(Z_n)] to an integration problem
- 0065(25): IOU: the solution to the integration problem
- 0065(26): conclude that the nth option price is given by version 0 of the Black-Scholes formula

Topic 0066 (IOUs in proof of B-S) Link to Topics website- 0066(1): title slide
- 0066(2-8): IOU: the solution to the integration problem
- 0066(2): recall the setup
- 0066(3): computing g(x) and recalling that h(x) = e^{-x^2/2} / \sqrt{2\pi}
- 0066(3): writing out the integral to be computed
- 0066(4): finding the lower limit of integration that allows us to drop the positive part
- 0066(5): splitting the resulting integral into two integrals, and computing one of them
- 0066(6): changing x to (x plus the linear coefficient) to handle the second integral
- 0066(6): some computations and simplificiations in the second integral
- 0066(7): transition slide
- 0066(8): computing the second integral

- 0066(9-12): IOU: for n sufficiently large, the risk-free rate is between the downtick and uptick factors
- 0066(9): completion of calibration
- 0066(10): limit (as n ---> \infty) of u_n\sqrt{n} is positive
- 0066(10): limit (as n ---> \infty) of d_n\sqrt{n} is negative
- 0066(11): limit (as n ---> \infty) of (r/n)\sqrt{n} is 0
- 0066(12): conclusion of proof

- 0066(13-23): IOU: p_n ---> p, q_n ---> q
- 0066(13): recall the calibration formulas for p_n, q_n
- 0066(13): get at asymptotics of p_n and q_n by studying the asymptotics of the formula

p_n e^{u_n} + q_n e^{d_n} = e^{r/n} - 0066(13): asymptotics of e^{(c/\sqrt{n})+(k/n)}
- 0066(14): asymptotics of e^{u_n}
- 0066(15): asymptotics of e^{d_n}
- 0066(16): asymptotics of e^{r/n}
- 0066(17): asymptotics of p_n e^{u_n} + q_n e^{d_n}
- 0066(18-19): asymptotics of p_n e^{u_n} + q_n e^{d_n} = e^{r/n}
- 0066(20): asymptotics of p_n \alpha + q_n \beta
- 0066(21): transition slide
- 0066(22-23): asymptotics of p_n and q_n

- 0066(24-25): IOU: \sigma_n ---> \sigma
- 0066(24): (u_n - d_n) \sqrt{n} is constant
- 0066(25): (formula \sigma_n = ...) minus (formula \sigma = ...)
- 0066(25): show that \sigma_n - \sigma tends to 0

- 0066(26-30): IOU: \mu_n ---> r - ( \sigma^2 / 2 )
- 0066(26): recall that the expeted value of e^{\tilde X_n} is e^r,

*i.e.*, an investment in bank or stock yields the same expected return - 0066(26-27): recall that W_n := (\tilde X_n - \mu_n) / \sigma ---> Z in distribution against exp-bdd
- 0066(27): multiply E[e^{\tilde X_n}] = e^r by e^{-\mu_n} and write left hand side in terms of W_n
- 0066(27-29): use the fact that W_n ---> Z to express the limit of e^{r-\mu_n} as an integral
- 0066(30): computing that integral, show that e^{r-\mu_n} ---> e^{\sigma^2/2}
- 0066(30): conclude that r-\mu_n ---> \sigma^2/2,
*i.e.*, that \mu_n ---> r - ( \sigma^2 / 2 )

- 0066(26): recall that the expeted value of e^{\tilde X_n} is e^r,