Is a continuous function of (integrable) Brownian motion always integrable?

Question

Let $Z_{t}=e^{-W(t)}$ with $\{W(t):t\geq0\}$ a Brownian motion. Is $Z_{t}$ integrable, since it is a continuous function of Brownian motion? Furthermore, are all continuous functions of Brownian motion integrable, since Bownian motion itself is integrable?

Answer 1

Continuous functions of integrable functions need not be intergable. In this case we can use the fact that if $X$ is normally distributed then $Ee^{cX}$ is integrable for any constant $c$. In fact $Ee^{cX}=e^{c\mu} e^{\sigma^{2}c^{2}/2}$ where $\mu$ is the mean and $\sigma^{2}$ the variance.

Answer 2

$e^{-W_t}$ is integrable: $Ee^{-W_t}=e^{\frac{1}{2}t}$.

Other continuous functions of $W_t$ need not be integrable. For example, $e^{W_1^2}$.