geometric Brownian motion
Let S(t) be a positive stochastic process that satisfies the generalized geometric Brownian motion differential equation (see Example 4.4.8)
where a(t) and a(t) are processes adapted to the filtration ℱ(t) , t ≥ 0, associated with the Brownian motion W(t) , t ≥0. In this exercise, we show that S(t) must be given by formula (4.4.26) (i.e., that formula provides the only solution to the stochastic differential equation (4.10.2)) . In the process, we provide a method for solving this equation.
(i) Using (4.10.2) and the lt-Doeblin formula, compute d log S(t) . Simplify so that you have a formula for d log S(t) that does not involve S(t) .
(ii) Integrate the formula you obtained in (i) , and then exponentiate the answer to obtain (4.4.26) .
geometric Brownian motion