Lemme itô and Martingale

Question

I want to to find values of $a$, $b$ such that the process:

$$e^{W_{t}^2+at+b\int_\limits{0}^{t}W_{s}^2\,ds}$$ be a martingale

Could you please help me do that

Thank you

Answer 1

Set $X_t = W_t^2 + at + b\int_0^t W_s^2\, ds$, so that $$dX_t = (1 + a + bW_t^2)\, dt + 2W_t\, dW_t$$ and $$d[X,X]_t = 4W_t^2\, dt.$$

Using Ito's lemma, we get $$d[e^{X_t}] = e^{X_t}\, dX_t + \frac{1}{2} e^{X_t} d[X, X]_t = e^{X_t}\bigl((1 + a + (b + 2)W_t^2)\,dt + 2W_t\, dW_t\bigr).$$ For $e^{X_t}$ to be a martingale, the drift term must vanish, i.e., $a = -1$ and $b = -2$.