= Iterated Itô integral
{c}
{title2=$I_n(t)=\int_0^tI_{n-1}(s)\,dB_s,\quad I_0=1$}
= Ordered iterated Itô integral
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{synonym}
For <Brownian motion>, an ordered iterated Itô integral is defined by the displayed recursion. It represents integration on $0\leq s_1\leq\cdots\leq s_n\leq t$, with time increasing toward the outermost <Itô integral>. The <Itô isometry> proves inductively that
$$
\mathbb E I_n(t)^2=\int_0^t\mathbb E I_{n-1}(s)^2ds=\frac{t^n}{n!}.
$$
Thus every recursive integrand is a <square-integrable stochastic integrand> on each finite interval, and $I_n$ is a <martingale> for $n\geq1$. For $n=1,2,3$, increment algebra gives $I_1=B_t$, $I_2=(B_t^2-t)/2$ and $I_3=(B_t^3-3tB_t)/6$. These are the corresponding <space-time Hermite polynomials> divided by $n!$.
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