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For Brownian motion, an ordered iterated Itô integral is defined by the displayed recursion. It represents integration on $0\le s_1\le \cdots \le s_n\le t$, with time increasing toward the outermost Itô integral. The Itô isometry proves inductively thatThus every recursive integrand is a square-integrable stochastic integrand on each finite interval, and $I_n$ is a martingale for $n\ge 1$. For $n=1,2,3$, increment algebra gives $I_1=B_t$, $I_2=(B_t^2-t)/2$ and $I_3=(B_t^3-3t{B}_t)/6$. These are the corresponding space-time Hermite polynomials divided by $n!$.

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.

The space-time Hermite polynomials obtained by scaling the Probabilists' Hermite polynomials are genuine polynomials in both variables, including at $t=0$:This expansion follows by differentiating the defining Gaussian exponential, and shows that $H_n(x,0)=x^n$ extends smoothly through $t=0$. Consequently the Itô formula is applicable from time zero. The given backward heat equation cancels the drift, and the derivative identity givesThe integrand is continuous and adapted, hence previsible, and stopping the Brownian motion and time makes it bounded. Thus $H_n(B_t,t)$ is a local martingale for every $n\ge 0$. The $n=0$ case is the constant $H_0=1$; in fact Gaussian moments make the displayed Itô integral square-integrable on every finite horizon, so these are true martingales as well.

The first three Probabilists' Hermite polynomials are $h_1(x)=x$, $h_2(x)=x^2-1$ and $h_3(x)=x^3-3x$. Hence