Suppose we have the following stochastic integral equation (we can make it an SDE) where $W$ is a standard Brownian motion $$ X_t = 1 + \int_0^t X_s d W_s. $$
I want to show that by using Picard Lindelof iteration that the $n$-th iterative is given by $$X_t^n = \sum_{k=0}^n \frac{1}{k!}H_k(W_t,t),$$ where the functions $H_k$ are given by $H_n(x,y) := y^{n/2} h_n(x/\sqrt{y})$ where $h_n(x) = (-1)^n \exp((1/2) x^2) \frac{d^n}{dx^n} \exp(-(1/2) x^2)$.
I have no idea how to show this and how to use the iteration w.r.t. $X_t$ as we are dealing with the BM. Could anyone help me?
We will prove this by induction, first the induction basis $n=0$: $$ X_t^0 \equiv 1 = H_0(W_t,t), $$ hence the induction basis holds.
Now suppose it holds for $n$, we will prove it for $n+1$:
Then \begin{align*} X_t^{n+1} &= 1 + \int_0^t X_s^n d W_s \\ &= 1+ \int_0^t \sum_{k=0}^n \frac{1}{k!} H_k(W_s,s) dW_s \\ &= 1 + \sum^n_{k=0} \int_0^t \frac{1}{k!} H_k(W_s,s) dW_s \\ &= 1 + \sum_{k=0}^n \frac{1}{k!} \frac{1}{k+1} (k+1) \int_0^t H_k(W_s,s)dW_s \\ &= 1 + \sum_{k=0}^n \frac{1}{(k+1)!} H_{k+1}(W_t,t) = \sum_{k=0}^{n+1} \frac{1}{k!} H_k(W_t,t), \end{align*} by using that we can interchange sum and integral and by using properties of the $h$ functions which are Hermite polynomials.