= Solution
The <space-time Hermite polynomials> obtained by scaling the <Probabilists' Hermite polynomials> are genuine <polynomials> in both variables, including at $t=0$:
$$
H_n(x,t)=n!\sum_{j=0}^{\lfloor n/2\rfloor}\frac{(-1)^j x^{n-2j}t^j}{2^j j!(n-2j)!}.
$$
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 gives
$$
dH_n(B_t,t)=nH_{n-1}(B_t,t)\,dB_t\qquad(n\geq1).
$$
The integrand is continuous and <adapted>, hence <previsible>, and stopping the <Brownian motion> and time makes it bounded. Thus \b[$H_n(B_t,t)$ is a <local martingale> for every $n\geq0$.] 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
$$
\boxed{H_1(B_t,t)=B_t,\qquad H_2(B_t,t)=B_t^2-t,\qquad H_3(B_t,t)=B_t^3-3tB_t.}
$$
Back to article page
