Stochastic Calculus, Fall 2002 (http://www.math.nyu.edu/faculty/goodman/teaching/StochCalc/)

Assignment 5.

Given October 19 due October 24 Last revised, October 23.

Objective: Simple martingales and PDE’s for Brownian motion

1. Suppose h(x) has h′(x) > 0 for all x so that there is a unique x for each y so that y = h(x). Consider the process Y_{t} = h(X_{t}), where X_{t} is standard Brownian motion. Suppose the function h(x) is smooth. The answers to the questions below depend at least on second derivatives of h.

a. With the notation ΔY_{t} = Y_{t+Δt} − Y_{t}, for a positive Δt, calculate a(y) and b(y) so

that E[ΔY_{t} | F_{t}] = a(Y_{t})Δt + O(Δt^{2}) and E[ΔY_{t}^{2} | F_{t}] = b(Y_{t})Δt + O(Δt^{2}). b. With the notation f(Y_{t}, t) = E[V (Y_{T}) | F_{t}], find the backward equation satisfied

by f. c. Writing u(y, t) for the probability density of Y_{t}, use the duality argument to find the

forward equation satisfied by u. d. Write the forward and backward equations for the special case Y_{t} = e^{cX_{t}}. Note (for

those who know) the similarity of the backward equation to the Black Scholes partial differential equation. We will get it exactly right soon.

2. Use a calculation similar to the one we used in class to show that Y_{T} = X_{T}^{4} − 6 ∫_{0}^{T} X_{t}^{2}dt is a martingale. Here X_{t} is Brownian motion.

3. Show that Y_{t} = cos(kX_{t})e^{k^{2}t/2} is a martingale. Hint 1: It suffices to show that E[Y_{t+Δt} | F_{t}] = Y_{t} + O(Δt^{2}), which is a calculation in this case. Now, to show E[Y_{t′} | F_{t}] = Y_{t} for t′ > t, divide the interval (t, t′) into n intervals of size Δt = (t′ − t)/n and use the tower property for the expanding family F_{t_{k}} ⊂ F_{t_{k+1}}, with t_{k} = t + kΔt. This will show that E[Y_{t′} | F_{t}] = Y_{t} + O(nΔt^{2}). The error term goes to zero as n → ∞. Hint 2: Use the fact that a certain function satisfies the backward equation.

4. Fourier solution of the heat equation

a. Show that

∫_{−π/2}^{π/2} cos(kx) cos(jx)dx = \frac{π}{2} δ_{jk}

if j and k are positive integers. This is the “orthogonality” relation between cosines.

b. Suppose f(x) = ∑_{k=1}^{∞} a_{k} cos(kx) (a “Fourier series”). Show that

a_{j} = \frac{2}{π} ∫_{−π/2}^{π/2} cos(jx)f(x)dx ,

assuming the sum converges rapidly enough. The numbers a_{j} are the “Fourier coefficients” of f. c. Find a formula for ∫_{−π/2}^{π/2} cos(jx)u(x, t)dx, where u(x, t) is the density for Brown-

ian motion X_{t} with X_{0} = 0 and τ > t with τ being the hitting time of ±\frac{π}{2}. This can be done by using the result of question 3 and the observation that ∫ cos(jx)u(x, t)dx = E[cos(jY_{t})] = E[cos(jX_{t})] (Y_{t} being the stopped Brwonian motion).

d. This gives a representation u(x, t) = ∑_{k=1}^{∞} a_{k}(t) cos(kx). Show by direct calculation that each of the terms satisfies the u forward equation and boundary conditions.

e. Conclude that for large t, u(x, t) ≈ e^{−t/2} cos(x) and P (τ > t) ≈ e^{−t/2}.