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

Assignment 4.

Given September 26, due October 10 (note: two weeks) Last revised, October 7.

Objective: Continuous conditioning, Gaussian random variables.

1. Suppose X ∼ N (μ, σ^{2}). Find a formula for E[e^{X}]. Hint: write the integral and complete the square in the exponent.

2. In finance, people often use N(x) for the CDF (cumulative distribution function) for the standard normal. That is, if Z ∼ N (0, 1) then N(x) = P (Z ≤ x). Suppose S = e^{X} for X ∼ N (μ, σ^{2}). Find a formula for E[max(S, K)] in terms of the N function. Hint: as in problem 1. This calculation is part of the Black–Scholes theory of the value of a vanilla European style call option. K is the known strike price and S is the unknown stock price.

3. Suppose X = (X_{1}, X_{2}, X_{3}) is a 3 dimensional Gaussian random variable with mean zero and covariance

E[XX^{∗}] = ( \begin{matrix}2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{matrix} ) .

Set Y = X_{1} + X_{2} − X_{3} and Z = 2X_{1} − X_{2}.

a. Write a formula for the probability density of Y.

b. Write a formula for the joint probability density for (Y, Z).

c. Find a linear combination W = aY + bZ that is independent of X_{1}.

d. Find a (modern style) formula for E[Y | Z].

e. Interpret your answer to part d in classical style as E[Y | 2X_{1} − X_{2} = z].

4. In some applications, linear filtering may be thought of as trying to find the best approximation to an unknown signal that has been corrupted by noise. Much of the theory of linear filtering rests on models using linear recurrence relations and Gaussian noise. Here is a simple example of that theory. We model the random process representing the signal as a sequence, X_{n}, which satisfies the recurrence relation

X_{n+1} = αX_{n} + Z_{n} ,

where α is a “decay” parameter, |α| < 1, and the Z_{n} are independent standard normal random variables. Assume that X_{0} = 0.

a. For any T, show that X = (X_{1}, …, X_{T}) is jointly Gaussian. You need not calculate the T × T covariance marix.

The noisy observations Y_{n} are modeled as

Y_{n} = X_{n} + βW_{n} ,

where the W_{n} are standard normals independent of the X_{m} and each other, β is a noise parameter. Let F_{n} be the σ−algebra generated by the first n observations Y_{1}, …, Y_{n}.

b. Compute X̂_{n} = E[X_{n} | F_{n}]. This is a best (least squares) estimate of X_{n} from the noisy observations Y_{k} for 1 ≤ k ≤ n. Hint: first find X̂_{1}, then find a formula for X̂_{n+1} in terms of X̂_{n} and Y_{n+1}.

Remark (for statisticians): The expected value of a gaussian is the same as the maximum likelihood estimate because the maximum of the probability density as the mean value. This might not be true for other random variables.

5. Suppose we have independent random variables Z_{k} ∼ N (0, 1) and Y_{k} with density u(y) so that u(y) = 3y^{2} if 0 ≤ y ≤ 1 and u(y) = 0 otherwise. Consider the sums S_{n} = ∑_{k=1}^{n} Z_{k}, T_{n} = ∑_{k=1}^{n} Z_{k}^{2}, V_{n} = ∑_{k=1}^{n} Y_{k}, and W_{n} = ∑_{k=1}^{n} Y_{k}^{2}. Note that Z_{k} is hardly independent of Z_{k}^{2}. Use the central limit theorem to decide which is more true:

a. For large n, S_{n} is nearly independent of T_{n}.

b. For large n, V_{n} is nearly independent of W_{n}.