Consider the following multiple Ito integral,
$$I(n)=\int\limits_0^t \int\limits_0^{t_n} \int\limits_0^{t_{n-1}} \dots \int\limits_0^{t_2}dW_{t_1} \, dW_{t_2} \, \dots \, dW_{t_n},$$
where $W_t$ is a standard Brownian motion.
$I(1)=W_t$
$I(2)=\frac{1}{2}W_t^2-\frac{1}{2}t$
$I(3)=\frac{1}{6}W_t^3-\frac{1}{2}tW_t$
The first 3 terms seem to be elegant.
I wonder whether there is a general term formula.
Best Answer
Such iterated integrals are closely related to Hermite polynomials.
If we define
$$H_n(t,x) = \frac{(-t)^n}{n!} \exp \left( \frac{x^2}{2t} \right) \frac{d^n}{dx^n} \exp \left(- \frac{x^2}{2t} \right),$$
then a straight-forward (but messy) computation shows that
$$dH_{n+1}(t,W_t) = H_n(t,W_t) \, dW_t \tag{1}$$
see this answer for details. You can easily compute the first Hermite polynomials $H_n$: $$H_1(t,x)=x \qquad H_2(t,x) = \frac{x^2}{2}- \frac{t}{2} \qquad \ldots.$$
We prove the assertion by induction. For $n=1$ it is obvious that $I(1)=W_t = H_1(t,W_t)$. Now suppose that $(2)$ holds for $n=k$, then we obtain from $(1)$ that $$H_{k+1}(t,W_t) = \int_0^t H_k(s,W_s) \, dW_s \stackrel{\text{ind. hypothesis}}{=} \int_0^t \left( \int_0^s \dots \int_0^{s_2} dW_{s_1} \ldots \, dW_{s_k} \right) \, W_s,$$ i.e. $(2)$ holds for $n=k+1$.