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Brownian Motion Martingales from Probabilists’ Hermite Polynomials

Article Quant Q&A · Author: Lawrence Chun
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Summary

The document explains how to construct higher-order martingales from Brownian motion. It connects the familiar low-order examples to the power-series expansion of an exponential martingale, identifying the resulting polynomial family as the probabilists’ Hermite polynomials. This gives a general route to expressions beyond the orders listed in the question.

For the fourth order, it provides the polynomial x⁴ − 6x² + 3 and applies the corresponding time scaling to obtain a Brownian martingale involving B_t⁴, tB_t², and t². The response is concise and offers a formula rather than a derivation or proof. It does not discuss convergence of the expansion, filtration assumptions, or other martingale constructions, so its scope is the Hermite-polynomial family generated by the stated exponential expansion.

Key ideas

  • The listed Brownian martingales arise from expanding an exponential martingale in a parameter.
  • The resulting polynomial sequence is the probabilists’ Hermite family.
  • The fourth-order martingale combines the fourth and second powers of Brownian motion with time terms.
  • The response gives a compact formula but does not show a proof or discuss broader assumptions.

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# 4th Order Brownian Motion Martingale


# 4th Order Brownian Motion Martingale












I understand the first order MG of brownian motion is Bt.. the second order is Bt^2 - t and the third order is bt^3 - 3tBt. How can I find the fourth and beyond order of a Brownian Motion Martingale?

## Answer by jaehyukchoi49 (score 5, accepted)

https://quant.stackexchange.com/a/74028

Those are the expansion of $$ \exp(\sigma B_t - \sigma^2t/2) $$ in the power of $\sigma$. The general $n$-th order martingale is expressed by the probabilist's Hermite polynomials.

The 4th order is polynomial is $x^4 - 6x^2 + 3$, so the margingale is $$ B_t^4 - 6t B_t^2 + 3 t^2.$$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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