# For what $p$ do we have $Z_{T} \in \mathcal{L}^{p}$, where $Z_{T}$ is an exponential martingale?

Let me define $Z_{t}$ to be the stochastic exponential $\mathcal{E}(-\dfrac{B_{s}}{\sigma},B_{s})$, where $B_{s}$ is a standard 1-d Brownian motion and $\sigma$ is a positive constant, i.e. $Z_{t}$ is the following exponential martingale: $$Z_{t}=exp\{-\int_{0}^{t}\dfrac{B_{s}}{\sigma}dB_{s}-\dfrac{1}{2}\int_{0}^{t}\dfrac{B_{s}^{2}}{\sigma^{2}}ds\}.$$

Now I fix $T>0$ and $p >1$, my question is: for what p do we have $Z_{T} \in \mathcal{L}^{p}$? i.e. for what $p>1$ do we have $\mathbb{E}[exp\{-p\int_{0}^{t}\dfrac{B_{s}}{\sigma}dB_{s}-\dfrac{p}{2}\int_{0}^{t}\dfrac{B_{s}^{2}}{\sigma^{2}}ds\}] < \infty \$?