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?
1 Answer
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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 $$ B_t^4 - 6t B_t^2 + 3 t^2.$$