Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I

4.2.3 Risk-Neutral Measure

1. 1. Geometric Brownian Motion. Consider an economy consisting of a risk-free asset and a stock price (risky asset). At time , the risk-free asset and the stock price have the following diffusion processes
   

   where is the risk-free rate, is the stock price drift rate, is the stock price volatility(which are all time dependent) and is a -standard Wiener process on the probability space .

   From the following discounted stock price process

   show, using Girsanov's theorem, ...