# How can we describe explicitly the “infinitely complex differentiable” complex-valued local martingales?

Let $\mathcal{F}_t$ be a continuous filtration on a probability space, and let $B$ be a standard $\mathbb{C}$-valued $\mathcal{F}_t$-Brownian motion. Let's call a complex-valued process $X$, possibly defined only up to a stopping time, analytic, if it is a continuous $\mathcal{F}_t$-local martingale with stochastic differential $dX_t = X^{(1)}_t dB_t$, such that $X^{(1)}$ is again a continuous local martingale, $dX^{(1)}_t = X^{(2)}_t dB_t$, and so on. Equivalently, for every $n$ $X_t$ is a polynomial in $B_t$ of degree $n$ up to an $(n+1)$-fold stochastic integral.

The question is: can we describe such processes explicitly?

More precisely, the obvious examples are $f(B_t)$ for complex analytic $f$, and the analytic continuations along $B_t$ of complex analytic functions, defined up to the time when their radius of convergence shrinks to $0$. Note that these may be called strongly analytic, in the sense that they are analytic with respect to the filtration induced by $B_t$. Are these the only ones? Do there exist analytic processes which are not strongly analytic?

An obvious remark: if all $X^{(n)}$ are, say, $L^p,p>1$ integrable martingales then $X_t$ is an entire function of $B_t$ (since we can reconstruct the chaos decomposition, which happens to be a power series in $B_t$), so I'm interested in cases when they're not.

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