Past exam of the continuous martingales and stochastic calculus course of the University of Oxford / 2023 / 2 / c

Put $M_t=B_t^3-3t{B}_t$. It is an adapted process and is integrable at every finite time, because normal random variables have finite absolute moments of every order. Fix $0\le s<t$ and let $D=B_t-B_s$. Then $D$ is independent of ${\mathcal{F}}_s$, has mean zero, variance $t-s$, and third moment zero. Expanding the cube inside the conditional expectation givesAlso $\mathbb{E}[B_t\mid {\mathcal{F}}_s]=B_s$. ThereforeThese are exactly the defining conditions for a martingale.

Interpret the ordered Iterated Itô integral recursively as $I_0(t)=1$ andThe diagonal boundary of the ordered region does not affect this recursive definition. For $n=1$, $I_1(t)=B_t$. For $n=2$, part (a)(iii) gives $I_2(t)=\frac{1}{2}(B_t^2-t)$. For $n=3$, use parts (b)(ii) and (b)(iii):Thus the three Probabilists' Hermite polynomials giveThe right sides are interpreted as their polynomial extensions at $t=0$. These space-time Hermite polynomials arise here directly from ordered Iterated Itô integrals.

Part (ii) represents the process as an Iterated Itô integral:For a centered normal random variable $B_s$ of variance $s$,Consequently, for every finite $T$,The Itô integral of this square-integrable stochastic integrand is a true square-integrable martingale on $[0,T]$. Since $T$ is arbitrary,The last identity is the Itô isometry.