Symmetrical function of Brownian Motion

Question

Let $W_t$ be Brownian motion. Using software, I can compute $E[e^{\beta t} \sin{(\gamma W_t})] = 0$. Could one void this computation with a clever symmetrical argument. That is: Since $sin(t)$ is an odd function (symmetric about the x-axis), we must have that $sin(W_t)$ is positive w.p. 1/2 and negative w.p. 1/2. Hence, the expected value must vanish. How can I make this argument more formal?

Answer 1

There are lots of random variables that are positive with probability $1/2$ and negative with probability $1/2$, but their expected value is not $0$. But the point about $\sin$ being odd is a good one. What it means is that the distribution of your random variable is symmetric about $0$, and that does imply that the expected value (if it exists) is $0$.