Iterated Itô integral ($I_n(t)={\int }_0^t{I}_{n-1}(s)d{B}_s,I_0=1$)

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 $n!$.