# Hermite polynomials as martingales [closed]

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.

• What have you tried yourself? Please don't dump all your homework on us. Apr 26 at 8:36

(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.