MATH375: Tutorial 3

1. Let (F W (t),t ≥ 0) by a filtration for Brownian motion (W (t),t ≥ 0). Show that the following processes are martingales relative to this filtration:

2. Using Itˆo’s formula, show that the following processes are martingales with respect to the natural filtration (F W (t),t ≥ 0) of Brownian motion, and find their representation:

3. Consider the stochastic differential equation:

for some given constants a, b, σ. Find its solution (X(t),t ≥ 0).

[Hint: Consider the ordinary differential equation

i.e. y(t) = e −at is a known function. Apply Itˆo’s formula to X(t)y(t) so that the unknown X(t) on the right-hand side of the stochastic differential equation is eliminated.]