Stochastic Calculus Final Exam December 19, 2002

• Answer all questions in the blue answer books.

• On each book, write your name and the number of books used. Number each book, e.g. “book 2 of 3”.

• Cross out or erase all work you know to be wrong. If you have a right and a wrong answer, I will take off for the wrong answer as well as giving credit for the correct one.

• Explain and show all work, as time allows. I will take off points for formulas that are not explained.

1. In each case, indicate whether the statement is true or false and give a short explination of your answer, with either a reason for it to be true or a counterexample.

a. The random variables X and Y are jointly normal with mean zero, var(X) = 1, var(Y) = 1, and cov(X, Y) = \frac{1}{2}. Define U = X − 2Y. This U is uncorrelated with X, and independent of X.

b. With X and Y as above, define V = X^{2} + X − 2Y. This V is uncor-

related with X and independent of X. c. Let X_{n} be a discrete time martingale. Let G_{n} be the σ−algebra gener-

ated by X_{n} only (i.e. not X_{n−1}, etc.). Then E[X_{n+1} | G_{n}] = X_{n}. d. If Y_{n} has E[Y_{n+1} | G_{n}] = Y_{n}, then Y_{n} is a martingale. e. If F and G are two algebras of sets in the probability space Ω, then the

intersection F ∩ G is also an algebra of sets. f. If Q is a probability measure on the space of continuous paths C([0, T] →

R) that is absolutely continuous with respect to Brownian motion measure (Wiener measure), then the total variation of a path X_{t} for 0 ≤ t ≤ T is infinite with Q probability one. More formally, if A ⊂ C([0, T] → R) is the set of paths with infinite total variation, then Q(A) = 0 if Q is absolutely continuous with respect to Wiener measure.

2. a. For a normal random variable X ∼ N (0, σ^{2}), write the PDF for X and

use it to calculate E[X^{2}e^{X}]. b. For X and Y as in question 1a, calculate E[Y^{2}e^{X}]. Hint: express X

and Y in terms of independent gaussians.

3. An n state Markov chain has transition matrix P. The state after t transitions is X_{t}, which is one of the values (1, …, n). We also have a Markov stopping time, τ ≥ 0, independent of the Markov chain. A Markov stopping time depends on the parameter, q = Pr(τ > t | τ ≥ t). a. Write an expression for Pr(τ = t).

b. Find a matrix equation involving P for the τ step transition probabilities v_{jk} = Pr(X_{τ} = k | X_{0} = j). These v_{jk} form the τ step transition probability matrix V. Write a matrix equation involving V and P. Hint: Starting from state j, break the event A_{k} = {X_{τ} = k} into two sets, one with τ = 0 (Z = {τ = 0}) and one with τ > 0 (B = {τ > 0}). It is easy to calculate Pr(X_{τ} = k | Z, X_{0} = j). For the B part, we can calculate

Pr(X_{τ} = k | B, X_{0} = j) = ∑_{l} Pr(X_{τ} = k | X_{1} = l)·Pr(X_{1} = l | B, X_{0} = j) .

Because τ and X_{t} are Markov, the first factors in the sum on the right are entries of V.

4. Suppose x = (x_{1}, x_{2})^{∗} ∈ R^{2} and f = (f_{1}, f_{2})^{∗} ∈ R^{2}. The ^{∗} means transpose, so that the column vector x has x^{∗} = (x_{1}, x_{2}), which is a row vector. The component of f perpendicular to x is g = f −(x^{∗}f/x^{∗}x)x = f −(x^{∗}f/r^{2})f, where r^{2} = ∥x∥^{2} = x_{1}^{2} + x_{2}^{2}. The quantity in parentheses is a number, not a vector.

a. Show that x^{∗}g = 0.

b. Show that for any f(x), the system of two ordinary differential equa-

tions

\frac{dx}{dt} = g(x) = f(x) − \frac{x^{∗}f(x)}{r^{2}}x

has r^{2} = const by calculating

\frac{d}{dt}x^{∗}(t)x(t) = 2x^{∗} \frac{dx}{dt} or d(r^{2}) = d(x^{∗}x) = 2x^{∗}dx = 2x^{∗}gdt .

c. Suppose we have the two dimensional Ito SDE

dX(t) = dW (t)− \frac{X(t)^{∗}dW (t)}{R(t)^{2}}X(t) with R(t)^{2} = X_{1}(t)^{2}+X_{2}(t)^{2} .

Here X_{1}(t) and X_{2}(t) refer to the first and second components of X(t) ∈ R^{2}. Show that R(t) is not constant in time.

d. Find a formula for R(t) in terms of R(0) and t. Show that if R(0) is known, then R(t) is not random.

5. We have a system of two SDEs:

dS_{t} = rS_{t}dt + \sqrt{V_{t}}S_{t}dW_{t}^{(1)} ,
dV_{t} = (a − bV_{t})dt + μdW_{t}^{(2)} ,

where W^{(1)} and W^{(2)} are independent Brownian motions. The initial conditions for the SDE are S_{0} = 1, V_{0} = .4. We want to calculate

f = E [∫_{0}^{T} S_{t}^{2}dt] .

Explain how to do this using a PDE. Give the PDE and whatever other information (possibly including initial data, final data boundary conditions, etc.) is needed. Make sure to give a probabilistic definition of the quantity that satisfies the PDE.