STATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 4 DUE NOVEMBER 23, 26 Problem Normal Moments (A) Use the Itô formula and Brownian scaling to check that the even moments of the normal distribution are () E W 2n = (2n )(2n 3) 3 (This is not meant to be hard You should also obtain the formula by integrating against the standard normal density, if you haven t seen this before) (B) Show that the product on the right side of () is also the number of perfect matchings of the integers, 2,, 2n (A perfect matching is a partition of the set {, 2,, 2n} into n subsets of cardinality 2 These are sometimes called dimer coverings) (C) Can you give a direct combinatorial or probabilistic explanation of why these two things are the same? Problem 2 Concentration Inequality Let M t be an Itô integral process such that M = and [M ] (This means that M t = I t (V ) for some progressively measurable process V that satisfies V 2 s d s ) Prove that for every x, HINT: For every θ > the process is a positive supermartingale P {supm t x } e x 2 /2 t exp{θ M t θ 2 [M ] t /2} Problem 3 Local Time Recall from the lectures (see section 52 of the Notes) that the local time process L t = L t is the unique adapted, nondecreasing process such that for every t, W t = L t + (W s )d W s Furthermore, for any even, C probability density ψ(x ) with support [, ] the local time L t is given by L t = lim ψ(w s /ɛ)d s ɛ ɛ (A) (Easy) Check that L t increases only on the zero set of the Brownian motion W (B) Show that L t is not identically zero (C) Use this to show that η(a ) := inf{t : L t > a } < almost surely (D) Show that the process {η(a )} a is a stable 2 subordinator What are the jumps?
Problem 4 More Local Time For any ɛ > and any t >, define the number N t (ɛ) of upcrossings of the interval [,ɛ] by the process W up to time t as follows First, set α =, and for each n =,, 2, define β n = min{t > α n : W t = ɛ} α n+ = min{t > β n : W t = }; and then let N t (ɛ) = max{n : β n t } Prove that with probability one, lim ɛ ɛn t (ɛ) = L t (or maybe 2 ± L t ) HINTS: One way to approach this is to use the representation L t = lim ɛ ɛ ψ(w s /ɛ)d s If you follow this approach, you might want to first show that the random variable Y = α [,ɛ] ( W s ) d s has finite exponential moments Problem 5 Feynman-Kac Formula I Let ϕ : be a twice-continuously differentiable function that satisfies the second order ODE (2) 2 ϕ (x ) = k(x )ϕ(x ) for some continuous function k :, called the potential function Assume that k (Thus, any positive solution ϕ of the differential equation will be convex) Standard existence/uniqueness theorems for ordinary differential equations imply that if k is piecewise C, or more generally Lipschitz continuous, then (2) will have a unique solution for any specification of an initial value ϕ(x ) and initial first derivative ϕ (x ) You may assume this fact as known Assume that ϕ is a solution of (2), and for any two real numbers a < < b set τ(a ) = min{t : W t = a }; τ(b ) = min{t : W t = b }; T = T a,b = min(τ(a ),τ(b )); Z t = ϕ(w t ) exp k(w s )d s (A) Show that {Z t T } t is an L 2 bounded martingale relative to the filtration, under any of the measures P x, where x [a, b ] NOTE: P x denotes the probability measure under which the process W t is a Brownian motion started at W = x (B) Show that for every x [a, b ], (3) ϕ(x ) = E x ϕ(w T ) exp and k(w s )d s Note: This is a time-independent version of the Feynman-Kac formula
(C) Use the result of (B) to solve for the Laplace transform E x e λt NOTE: This Laplace transform was obtained by other martingale methods in the lecture notes on Brownian motion The problem here is to get the same result by solving a second-order differential equation NOTE: The results of Problem 5 can be extended to higher dimensions to give Brownian path integral representations of the solutions of the PDE ϕ = kϕ 2 This PDE is the time-independent Schrödinger equation, which is of basic importance in quantum mechanics The Feynman-Kac formula arose out of Feynman s attempt to give a path integral formulation of quantum mechanics Problem 6 Feynman-Kac Formula II Let k(x ) = θ (,] (x ), where θ > is a positive constant This function has a discontinuity at x =, so the results of Problem 5 do not apply directly Nevertheless, one can use Problem 5 indirectly, as follows Let k k 2 k 3 be a non-decreasing sequence of C functions such that limk n (x ) = k(x ), and for each n let ϕ n be the solution of the equation 2 ϕ n (x ) = k n(x )ϕ n (x ) that satisfies the boundary conditions ϕ n (a ) = ϕ n (b ) =, for some fixed constants a < < b (A) Show that for each x [a, b ] the sequence ϕ n (x ) is non-increasing in n, and converges to a finite limit ϕ(x ) HINT: Rewrite the differential equation as an integral equation (B) Show that the limit function ϕ(x ) is continuous and twice differentiable (except at the argument x = ), and satisfies the boundary value problem (C) Show that for all x [a, b ] 2 ϕ (x ) = k(x )ϕ(x ) and ϕ(a ) = ϕ(b ) = ϕ(x ) = E x ϕ(w T ) exp θ (,] (W s )d s (D) Solve the boundary value problem in part (B) to obtain an explicit formula for the Laplace transform of the total occupation time random variable Y = (,] (W s )d s
Bonus Problems: The Hermite functions and Itô calculus (These are optional, not to be handed in) The Hermite functions H n (x, t ) are polynomials in the variables x and t that satisfy the backward heat equation H t + H x x /2 = The first few Hermite functions are (4) H (x, t ) =, H (x, t ) = x, H 2 (x, t ) = x 2 t, H 3 (x, t ) = x 3 3x t, H 4 (x, t ) = x 4 6x 2 t + 3t 2 The easiest way to define them is by specifying their exponential generating function: (5) H n (x, t ) θ n n! = exp{θ x θ 2 t /2} n= Problem 7 Use the generating function to show that the Hermite functions satisfy the two-term recursion relation H n+ = x H n nt H n Conclude that every term of H 2n is a constant times x 2m t n m for some m n, and that the lead term is the monomial x 2n Conclude also that each H n solves the backward heat equation Problem 8 Show that for each n, H n+ (W t, t ) = (n + )H n (W s, s )d W s = = (n + )! n d W tn+ d W tn d W t and use this to show that H n (W t, t ) is a martingale HINT: Use the Itô formula on exp{θ W t θ 2 t /2} The standard Hermite polynomials are the one-variable polynomials H n (x ) = H n (2x, /2) Equivalently, (6) H n (x ) θ n n! = exp{2θ x θ 2 } n= Problem 9 (A) Show that the Hermite polynomials satisfy the second-order differential equations H n (x ) 2x H n (x ) = 2nH n(x ) Thus, the Hermite polynomials are the eigenfunctions of the generator of the Ornstein-Uhlenbeck process (see next problem) HINT: First differentiate the generating function with respect to x to obtain a first-order ODE for H n Then combine this with the two-term recurrence relation H n+ = 2x H n 2nH n gotten by specialization from Problem 7 above (B) Show that H n (x )H m (x ) e x 2 π d x = C 2 n δ n,m and evaluate the normalizing constants C n Here δ n,m is the Kronecker delta, that is, is m n and if n = m Conclude that the polynomials H n /C n are an orthonormal basis for L 2 (Φ), where Φ is the normal distribution with mean and variance /2 HINT: Take the product of the generating
function (6) with itself, but evaluated at two different values of θ (say, θ and ξ), and integrate this against the normal density with mean zero and variance /2 Problem The Ornstein-Uhlenbeck process is the diffusion process that satisfies (A) Show that for each n the process is a martingale d Y t = Y t d t + d W t M n (t ) := e nt H n (Y t ) (B) Deduce that for each function f (y ) in the linear span of the Hermite polynomials, and for each t > and x, E x f (Y t ) = p t (x, y )f (y )d y where p t (x, y ) = e y 2 /2 e nt H n (x )H n (y )/C 2 π n This is the eigenfunction expansion of the transition probability kernel (C) Use the representation Y t = 2 /2 e t W e 2t of the stationary Ornstein-Uhlenbeck process to obtain another formula for the transition probabilities: p t (x, y ) = ( e 2t ) /2 exp (x e t y ) 2 π e 2t n=