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Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite polynomial $H_{n}(t, x)$ by $$\exp \left(\theta x-\frac{1}{2} \theta^{2} t\right)=\sum_{n=0}^{\infty} \frac{\theta^{n}}{n !} H_{n}(t, x)$$ Prove that for each $n \in \mathbb{N}, H_{n}\left(t, W_{t}\right)$ is an $\left\{\mathcal{F}_{t}\right\}_{t \geq 0}$ martingale.

Thanks in advance!

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    $\begingroup$ What have you tried yourself? Please don't dump all your homework on us. $\endgroup$
    – Bob Jansen
    Apr 26 at 8:36
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(As said in the comments, you need to put down some of your thoughts regarding the question too, like specifying the tools/theorems you would use or actual attempts to apply them, even if you can only cover early steps, not just the question itself.)

Hints: We are given

$$X_t^\theta:=\exp \left(\theta W_t-\frac{1}{2} \theta^{2} t\right)=\sum_{n=0}^{\infty} \frac{\theta^{n}}{n !} H_{n}(t, W_t)$$

Show (using Ito Lemma again, of course, and the definition of $X_t^\theta$):

$$ dX_t^\theta = \theta X_t^\theta dW_t $$

Plug in the infinite summations on both sides:

$$\sum_{n=0}^{\infty} \frac{\theta^{n}}{n !} d H_{n}(t, W_t) =\theta \sum_{n=0}^{\infty} \frac{\theta^{n}}{n !} H_{n}(t, W_t) dW_t $$

Conclude

$$ d H_{n}(t, W_t) = n H_{n-1}(t, W_t) d W_t $$

and $H_{n}(t, W_t)$'s martingality.

Also, note that

$$ H_1(t,W_t) = W_t, $$ $$ H_2(t,W_t) = W_t^2 -t, $$ easily recognizable martingales.

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