As we learned in class, we define a diffusion process as a stochastic process with stochastic differential equation (SDE) given by:

dX = a(X,t) dt + b(X,t) dW X(0) = X₀ [**]

Its called diffusion because the randomness in the increments comes from Brownian motion only. (There are no jumps.)

Remember: Solving the SDE means describing X(T) as a random variable for all future times T.

In the case where a(X,t) = aX and b(X,t) = sX with constants a and s we have geometric Brownian motion (GBM).

For GBM how did we solve this SDE? The trick was to consider the function log(X). Using Itos formula we worked out d(log(X)), the stochastic increment of log(X).

Write down d(log(X)) in the case of GBM.