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Given a (one-dimensional) stochastic process $X_t$ and Wiener process $W_t$, the Itô isometry says that $$ \mathbb{E}\left(\left[\int_0^TX_tdW_t\right]^2\right)=\mathbb{E}\left(\int_{0}^T X_t^2 dt\right). $$ Is there a similar (or different!) trick to computing $$ \mathbb{E}\left(\left[\int_0^TX_tdW_t\right]^n\right), $$ where $n\in\mathbb{Z}_{\geq 3}$?

Mathmo
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1 Answers1

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As mentioned here 4th moment of a Wiener stochastic integral with Ito isometry property?, one can use that Hermite polynomials are martingales to at least estimate them.

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In case the link dies, the proof simply uses that Hermite polynomials $H_{2m}(x,t)$ of Itô integrals are martingales and that the leading coefficient is $x^{2m}$.

Thomas Kojar
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