Question

Consider the stochastic process X(t) defined as X(t) = (W(t) - t)e^(W(t) - t/2), where W(t) is a Wiener process. Use Ito's formula to demonstrate that the process X(t) is a martingale.

Answer 1

To demonstrate that the process X(t) = (W(t) - t)e^(W(t) - t/2) is a martingale, we need to use Ito's formula, which allows us to compute the differential of X(t). By taking the expectation of this differential, we can show that it is equal to zero, proving that X(t) is a martingale.

Step-by-step explanation:

To demonstrate that the process X(t) = (W(t) - t)e^(W(t) - t/2) is a martingale, we need to show that it satisfies the martingale property.

According to Ito's formula, we can write the differential of X(t) as:

dX(t) = ((W(t) - t)e^(W(t) - t/2))dW(t) + (e^(W(t) - t/2) - (W(t) - t)e^(W(t) - t/2)/2)dt

Taking the expectation of dX(t) we get:

E(dX(t)) = E(((W(t) - t)e^(W(t) - t/2))dW(t)) + E((e^(W(t) - t/2) - (W(t) - t)e^(W(t) - t/2)/2)dt) = 0 + 0 = 0

Since the expectation of dX(t) is 0, the process X(t) satisfies the martingale property and is a martingale.