I am cross-posting [this question from MSE][1] since I did not received any answer, furthermore I tried asking some professors in my university but still we could not find an answer.
The most surprising thing is that this exercise was taken from an introductory text on Stochastic Analysis. (Introduction to Stochastic Integration by Hui-Hsiung Kuo)
 

Assume we have the following stochastic process:

$$X_t=\int_0^t e^{B(s)^2}dB(s)\, ,0\leq t \leq 1$$

where $(B)_{t\geq 0}$ is a Brownian Motion.

I have to show that $X_t$ is not a martingale.

I know that if $t< \frac 1 4$ then $\int_0^t \mathbb E(e^{2B(s)^2})ds < \infty $ and then the process is a martingale, this makes me think that $X_t$ is actually a local martingale, but I don't see how to prove that it's not a *proper* martingale.

Thanks in advance.


  [1]: https://math.stackexchange.com/questions/3528142/how-to-show-this-is-not-a-martingale