Deriving Hermite Polynomial Martingales from the Brownian Exponential

Summary

The document presents a proof strategy for showing that the time-indexed Hermite polynomials evaluated at standard Brownian motion are martingales. It defines the polynomials through the power-series expansion of the Brownian exponential martingale, then applies Itô’s lemma to that exponential to obtain its stochastic differential.

Matching coefficients of the power series yields a differential for each polynomial: its drift vanishes and its stochastic term is proportional to the preceding polynomial. This supports the martingale claim, with the first two examples identified as Brownian motion and Brownian motion squared minus time. The explanation is a hint rather than a fully rigorous proof: it does not discuss conditions for exchanging the infinite series and stochastic differential, or integrability details needed for a complete martingale argument. Its focus is a probability result, rather than a direct trading method.

Key ideas

• The Hermite polynomials are defined as coefficients in the expansion of an exponential involving Brownian motion.
• Itô’s lemma shows that this exponential has a stochastic differential with no drift term.
• Matching series coefficients gives each polynomial a stochastic differential driven by the previous polynomial.
• The first two evaluated polynomials recover familiar Brownian martingales.
• A rigorous proof would need to justify the series manipulations and martingale conditions.

Tags

# Hermite polynomials as martingales


# Hermite polynomials as martingales












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!

## Answer by ir7 (score 3)

https://quant.stackexchange.com/a/63605

(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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.