Expected Level of Geometric Brownian Motion

Solve the stochastic differential equation to find the explicit form of $S_t$.

Take expectations of the exponential of a normal random variable.

1. 1. Solve the SDE

   The solution to this geometric Brownian motion SDE is $S_t=S_0\exp \left((\mu -{\sigma }^2/2)t+\sigma W_t\right)$.

2. 2. Take expectations

   Since $W_t\sim N(0,t)$, using the moment generating function of a normal random variable gives the result.

   $$E[S_t]=S_0{e}^{(\mu -{\sigma }^2/2)t}E\left[e^{\sigma W_t}\right]=S_0{e}^{(\mu -{\sigma }^2/2)t}{e}^{{\sigma }^{2}t/2}=S_0{e}^{\mu t}$$

Answer: S_0 * e^(mu*t)