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

Practice Final Exam questions.

Given December 11, last revised December 12

Focus: Review

This should help you study for the final exam. There are more questions than could probably be on the actual final in order to review more material.

1. In each part, state whether the statement is true or false and give a short explination with a reason it is true (maybe something short of a full mathematical proof) or a counterexample.

a. Suppose dX_{t} = (X_{t}^{3} − 1)dt + X_{t}^{2}dW_{t} with X_{0} = 2. It is possible to compute an accurate approximation to E[X_{T}^{2}] with T = 3 without simulating a random process.

b. Suppose X(ω) and Y (ω) are functions of the random variable ω defined for ω ∈ Ω, a probability space. Let F_{X} and F_{Y} be the σ−algebras generated respectively by X and Y. Let F_{X,Y} be the σ−algebra generated by both X and Y. Then F_{X,Y} = F_{X} ∪ F_{Y}.

c. If (X_{t}^{(1)}, X_{t}^{(2)}) is a two component Markov process, then X_{t}^{(1)} separately is a one component Markov process.

d. If f(x, t) satisfies the PDE

∂_{t}f + (2 + sin(x))∂_{x}^{2}f = 0 ,

and f is bounded, then

f(x, t) ≤ max_{x′} f(x′, T) ,

whenever T > t.

e. If X_{t} and Y_{t} are independent Brownian motions, then X_{T}^{2}Y_{T} −∫_{0}^{T} Y_{t}dt is a martingale.

f. If σ_{t} ∈ F_{t} and dX_{t} = σ_{t}dW_{t} then X_{t} is a Markov process.

g. If there is a function b(x) so that σ_{t} = b(X_{t}) and dX_{t} = σ_{t}dW_{t}, then X_{t} is a Markov process.

h. If X is a two dimensional diffusion process that satisfies dX_{t} = a(X_{t})dt+σ(X_{t})dW_{t}, where W_{t} is a pair of independent Brownian motions, and dY_{t} = a(Y_{t})dt + σ(Y_{t})QdW_{t} where Q is a 2 × 2 orthogonal matrix, then the probability measure on the path space C([0, T] → R^{2}) defined by X_{t} and Y_{t} are the same even though X_{t} ≠ Y_{t} almost surely.

2. Suppose X_{t} is standard Brownian motion and that G is the σ− algebra generated by X_{1} and X_{2} (X evaluated at times t = 1 and t = 2). Let Y = ∫_{0}^{3} X_{t}dt. Write a formula for the conditional PDF u(y | G_{0}). This will be a function of the three variables y, x_{1}, and x_{2}. Hint: You know that the conditional probability densities multivariate normals is normal, so figure out something about the joint PDF of X_{1}, X_{2}, and Y. You need not write the joint PDF in complete detail.

3. Suppose (X_{1}, X_{2}, X_{3}) ∈ R^{3} is a three dimensional multivariate normal with E[X_{1}] = 1, E[X_{2}] = −1, and E[X_{3}] = 3 and covariance matrix

C = ( \begin{matrix}5 & 0 & −2 \\ 0 & 1 & 0 \\ −2 & 0 & 1 \end{matrix} ) .

Write a formula for the joint PDF, u(x_{1}, x_{2}, x_{3}). Hint: C^{−1} = ( \begin{matrix}1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 5 \end{matrix} ) .

4. Suppose dX_{t} = μX_{t}dt + σX_{t}dW_{t} where μ and σ are constants. Suppose V (x) = max(x − K, 0) and that we want to evaluate E[V (X_{T})M_{T}] where M_{T} = exp(−r ∫_{0}^{T} X_{t}dt). Write a PDE we could use.

5. We wish to solve the PDE ∂_{t}f + \frac{x^{2}+y^{2}}{2} ∂_{x}^{2}f + (x − y^{2})∂_{y}f + ryf = 0, where r is some constant, and f(x, y, T) = V (x, y) is given. Write an SDE and express f(x, y, 0) as the expectation of some function of the path X_{t}, Y_{t}.

6. Suppose dX_{t} = σX_{t}dW_{t} with constant σ and X_{0} = 1. Express

Y_{T} = ∫_{0}^{T} X_{t}dX_{t}

as a function of X_{T} and the random variable

U_{T} = ∫_{0}^{T} X_{t}^{2}dt .

Verify that E[Y_{T}] = 0 (why is Y_{T} a martingale?) by calculating the expected values of the two terms f(X_{t}) and U_{T}.

7. The state of a discrete time finite state space Markov chain is X_{t} at time t = 0, 1, 2, . . .. The state space is S = {0, 1, …, n}. For x ≠ 0, n we the transition probabilities are \frac{1}{3} to increase by one X → X + 1, decrease by one, or stay the same. When X = 0, the transition probabilities are \frac{1}{2} to go to one X_{t} = 0 → Xt + 1 = 1 or stay the same. For X = n the transition X_{t} = n → X_{t+1} = n has probability \frac{2}{3} and the transition X_{t} = n → X_{t+1} = n − 1 has probability \frac{1}{3}.

a. For n = 4, write the 4 × 4 transition matrix. b. For n = 4, use two matrix multiplications to calculate all the time 4 transition

probabilities a_{jk} = P (X_{t+4} = k | X_{t} = j). (This might be too much arithmetic for the actual exam.)

c. We construct a sequence of stopping times τ_{1} = min(t such that X_{t} = 0 or n), and τ_{k+1} = min(t > τ_{k} such that X_{t} = 0 or n). The sequence Y_{k} = X_{τ_{k}} consists of the 0 and n values of X_{t} with all other values taken out. Show that the sequence Y_{k} is a two state space Markov chain. This fact is a special case of what is called the “strong Markov property”.

d. Calculate the 2 × 2 transition matrix for the Y_{k} chain. To make it easier, replace the X transition probabilities at 0 by P (X_{t+1} = 0 | X_{t} = 0) = \frac{2}{3} and P (X_{t+1} = 1 | X_{t} = 0) = \frac{1}{3}. This makes the X transition matrix (and therefore the Y transition matrix) symmetric. Do not assume n = 4.

8. Let V_{t} satisfy the familiar SDE dV_{t} = −γV_{t}dt + αdW_{t}. Let be the first hitting time for 0, τ = min(t such that V_{t} = 0. Let s = min(τ, 5). Let F_{s} be the σ−algebra generated by all V_{t} for t ≤ s. Calculate G = E[V_{5}^{2} | F_{s}] by showing that it is given as a simple function of one random variable.

9. Let X_{t} be Brownian motion starting at a point X_{0} > 0 and let A_{T} be the event X_{t} > 0 for all t ∈ (0, T). Let u_{0}(x, t) be the probability density for paths in A_{t} to land at x at time t: u_{0}(x, t)dx = P (X_{T} ∈ A_{t} and x < X_{t} < x + dx).

a. Write the partial differential equation we can solve to determine u(x, t) including initial conditions and boundary conditions.

b. Write a formula for the solution of this PDE as a sum (difference) of two gaussian functions.

c. Suppose instead that dX_{t} = vdt + dW_{t} (Brownian motion with constant drift velocity, v). Write the PDE (with initial and boundary conditions) that determines u_{v}(x, t)dx = P (X_{T} ∈ A_{t} and x < X_{t} < x + dx).

d. Use Girsanov’s theorem to express u_{v}(x, t) in terms of u_{0}(x, t).

10. Suppose W_{t}^{(1)} and W_{t}^{(2)} are Brownian motions with correlation coefficient ρ. For any two random variables X and Y, ρ(X, Y) = cov(X, Y)/\sqrt{var(X)var(Y)}. Here, we suppose that ρ(W_{t}^{(1)}, W_{t}^{(2)} is a constant independent of t. We have the pair of SDE’s

dX_{t} = r_{t}X_{t}dt + σX_{t}dW_{t}^{(1)}
dr_{t} = μ(\overline{r} − r_{t})dt + α\sqrt{r_{t}}dW_{t}^{(2)} .

We want to compute

F = E[V (X_{T})e^{− ∫_{0}^{T} r_{t}dt}] .

Define a quantity f(x, r, t) as a conditional expectation value and give a backward equation satisfied by f so that f(x, r, 0) is F.