I have a question about Novikov condition and martingale.
$T>0$: fix. Let $(\Omega, \mathcal{F}, \left(\mathcal{F}_{t}\right)_{t \in [0,T]}, P)$ be a filtered probability space and $(B_{t})_{t \in [0,T]}$ be a $(\mathcal{F}_{t})_{t \in [0,T]}$ adapted Wiener process on $(\Omega, \mathcal{F}, \left(\mathcal{F}_{t}\right)_{ t \in [0,T]}, P)$.
Novikov condition satets:
Let $(X_{t})_{ t \in [0,T]}$ be a $(\mathcal{F}_{t})_{ t \in [0,T]}$ adapted real valued process such that \begin{align*} E \left[ \exp \left( \frac{1}{2} \int_{0}^{T} \left|X_{t} \right|^{2}dt \right)\right]< \infty \tag{1} \end{align*} Then $\displaystyle M_{t}:= \exp \left( \int_{0}^{t}X_{s}dW_{s}-\frac{1}{2} \int_{0}^{t} X_{s}^{2}ds \right)$ is a martingale with respect to $(\Omega, \mathcal{F}, \left(\mathcal{F}_{t}\right)_{t \in [0,T]}, P)$.
My question
I am looking for $b: \mathbb{R} \to \mathbb{R}$ be a unbounded function such that square integrable i.e. $\displaystyle \int_{\mathbb{R}}\left| b(x) \right|^{2}dx < \infty $ and \begin{align*} \exp \left( \int_{0}^{t}b \left(W_{s} \right)dW_{s}-\frac{1}{2} \int_{0}^{t} b \left(W_{s} \right)^{2}ds \right) \end{align*} is not a martingale. At the beginning, I have looked for $b$ which $E \left[ \exp \left( \frac{1}{2} \int_{0}^{T} \left|b \left(B_{t} \right) \right|^{2}dt \right)\right]=\infty$. But I don't know how to calculate $E \left[ \exp \left( \frac{1}{2} \int_{0}^{T} \left|b \left(B_{t} \right) \right|^{2}dt \right)\right]$.
Please tell me if you know an example of $b$.
Thank you in advance.