16.5 Examples of SDEs
I strongly believe that the only way one learns stochastic calculus is by practicing exercises. There aren't many tools one needs to use; in fact, the most important one to know is the Itô lemma.
Note that the coefficients are and which satisfy the linear growth condition and the Lipschitz condition. Therefore, the solution exists and it is weakly unique.
Note that this is an example we have seen before of the geometric Brownian motion. We again take and . Applying the Itô formula, we obtain
So , and therefore ...