Expected value of exponential integrate Brownian motion

Question

I would like to show that \begin{equation} \mathbb{E}\left[\exp\left\{ \int_{0}^{T} B_t^4 dt\right\}\right]=+\infty, \end{equation} as I know that the process $X_{t}:=\exp\{B_{t}^{4}\}$ is not integrable as well. I tried to use recursively Ito's lemma but I cannot show it. Any suggestions?

Answer 1

It looks like Jensen's inequality takes care of it: \begin{align*} \mathbb{E}\left[\exp\left(\int_0^T B_t^4dt\right)\right] \ge \mathbb{E}\left[\frac 1T\int_0^T\exp\left( TB_t^4\right)dt\right] = \frac 1T \int_0^T \mathbb{E}[e^{TB_t^4}]dt = \infty. \end{align*}