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

Assignment 7.

Given October 31, due November 7 Last revised November 5.

Objective: The Ito integral

1. We want to calculate from the definition

Y_{T} = ∫_{0}^{T} X_{t}^{2}dX_{t} ,

where X_{t} is standard Brownian motion. Were X_{t} a differentiable function of t, the answer would be Y_{T} = \frac{1}{3}X_{T}^{3}. There will be an Ito correction to this. To find the correction, use the notations X_{k} = X_{t_{k}}, t_{k} = kΔt, and ΔX_{k} = X_{k+1} − X_{k}, then calculate

X_{k+1}^{3} − X_{k}^{3} = 3X_{k}^{2}ΔX_{k} + 3X_{k}ΔX_{k}^{2} + ΔX_{k}^{3} .

Turning this around gives

X_{k}^{2}ΔX_{k} = \frac{1}{3}(X_{k+1}^{3} − X_{k}^{3}) − X_{k}ΔX_{k}^{2} − \frac{1}{3}ΔX_{k}^{3} .

Thus

Y_{T}^{(n)} = ∑_{k=0}^{n−1} X_{k}^{2}ΔX_{k} = \frac{1}{3}X_{T}^{3} − A_{n} − B_{n} − \frac{1}{3}C_{n} ,

where

A_{n} = ∑_{k=0}^{n−1} X_{k}Δt ,

B_{n} = ∑_{k=0}^{n−1} X_{k}(ΔX_{k}^{2} − Δt) ,

C_{n} = ∑_{k=0}^{n−1} ΔX_{k}^{3} .

Find the limit as Δt → 0 of A_{n} as an integral involving X_{t}. Show that B_{n} → 0 and C_{n} → 0 as n → ∞ with n = 2^{L} by calculating E[B_{n}^{2}] and E[C_{n}^{2}]. Note that if ∑_{n} E[B_{n}^{2}] < ∞ then ∑_{n} B_{n}^{2} < ∞ almost surely (why?), which implies that B_{n}^{2} → 0 as n → ∞ almost surely.

2. Use the Ito isometry to calculate E[Y_{T}^{2}], where

Y_{T} = ∫_{t=0}^{T} (∫_{s=0}^{t}(t − s)X_{s}dX_{s}) dX_{t} .

It may help to find a forumla for E[Z^{4}] and E[Z^{6}] where Z ∼ N (0, σ^{2}).

3. We want simple approximations to the integral

Y_{T} = ∫_{0}^{T} V (X_{t})dX_{t}

when T is small. For any random variable, W_{T}, we say that W_{T} = O(t^{p}) if E[|W_{T} |] = O(Δt^{p}). Note that this is not the same as saying that there is a constant C so that |W_{T} | ≤ Ct^{p}. For example, if X_{T} is standard Brownian motion, we already used the fact that X_{T} = O(\sqrt{T}).

a. Use the Cauchy Schwartz inequality to show that if W_{T}^{2} = O(Δt^{2p}) then W_{T} = O(Δt^{p}).

b. Use the Ito isometry formula to show that if F_{t} ∈ F_{t}, F_{t}^{2} = O(t^{p}), and W_{T} = ∫_{0}^{T} F_{t}dX_{t}, then W_{t} = O(t^{(p+1)/2}).

c. Suppose V (x) is a smooth function of x and take enough Taylor series terms of V to show that

Y_{T} = V (X_{0})ΔX_{T} + \frac{1}{2}V′(X_{0})(ΔX_{T}^{2} − T) + O(T^{3/2}) .

We suppose the Brownian motion path X_{t} starts at X_{0}, which is not necessarily zero. We use the notation ΔX_{T} = X_{T} − X_{0}. Hint: integrate the Taylor series for V (X_{t}) term by term, and use parts a-c to get a bound for what we get by integrating the remainder term.

4. Find a backwards equation for

f(x, t) = E_{x,t} [exp (∫_{s=t}^{T} V (X_{s})dX_{s})] .

Hint: First derive the formula using the tower property for conditional expectation,

f(x, t) = E_{x,t} [exp (∫_{s=t}^{t+Δt} V (X_{s})dX_{s}) · f(x + ΔX, t + Δt)] ,

then use Taylor series, including the answer to question 3.