Conditional expectation of geometric brownian motion at q = 0.5, given t = 1

Question

Let $(B_t, t>0)$ be standard Brownian Motion. What is $E(e^{aB_{0.5}}|B_1)$?

For $E(e^{aB_1}|B_{0.5})$, that is easy, but I am a little confused when the expectation is about an in-between point, and a future point is given.

Answer 1

You can use the following strategy: find a $c$ such that $B_{1/2}-c B_1 $ is independent of $B_1$ (since $(B_{1/2},B_1)$ is Gaussian, we just need to look at uncorrelatedness). Then write $$ e^{aB_{1/2}}=\exp\left(a\left(B_{1/2}-c B_1\right)\right)\exp\left( c B_1 \right).$$