Consider an extension to the (one-dimensional) geometric Brownian motion model, $$dS_t = \mu(t,S_.)S_t dt + \sigma(t, S_.)S_t dB_t,$$ where $\mu$ and $\sigma$ are previsible path functionals, i.e. they depend on the path, denoted by $S_.$ of the stock price, at least up to the current time $t$. Then consider the function $$u(t,s) = \mathbb{E}\left[\int_t^T c(w,S_.) dw \big\lvert S_t=s\right]$$ where $$c(w,S_.)=\frac12 \left(\frac{\mu(w, S_.)}{\sigma(w, S_.)}\right)^2.$$

What is known:

When the coefficients depend only on the current price, i.e. (slight abuse of notation ahead) $$\mu(t, S.)=\mu(t, S_t) \text{ and } \sigma(t, S_.)=\sigma(t, S_t),$$ then it is known that $u(t,s)$ solves the PDE $$u_t + \mathscr{L} u +c(t,s) = 0,$$ with vanishing terminal condition $u(T,s)=0$ and where $\mathscr{L}$ is the infinitesimal generator of $S$, i.e. $$\mathscr{L} f(t,s) = \mu(t, s)s f_s + \frac12 \sigma(t, s)^2s^2 f_{ss}.$$

This actually gives us the expected log return in a portfolio that optimizes the terminal log wealth, and when $\mu(t,s)=\mu$, $\sigma(t,s)=\sigma$ are constant, we get the expected log-growth equal to $\frac12 \frac{\mu^2}{\sigma^2}(T-t).$

My question:

Is there a generalization to the above PDE that $u$ satisfies when the coefficients only depend on the current price, to the case when the coefficients are previsible path functionals?

Some thoughts: The standard approaches are no longer valid, I believe, as we lose Markov property. Bruno Dupire's paper on functional Ito calculus seems promising but I have yet to work out a meaningful application of it to this specific problem.

Please comment for clarifications, corrections, or questions.

Edit/Update 1/2/2023 Just realized a subtlety I overlooked. In the case I am concerned with, where the coefficients depend on the path up to time $t$, I am not actually trying to compute the conditional expectation conditional on $S_t=s$, $$\mathbb{E}\left(\int_t^T c(w, S_.)\big\vert S_t=s\right),$$ but rather the conditional expectation conditional on the entire history up to time $t$: $\mathscr{F}_t^S = \sigma(S_u: u\leq t)$ $$\mathbb{E}\left(\int_t^T c(w, S_.)\big\vert \mathscr{F}_t^S\right),$$ which complicates things greatly. First of all, in the former case the conditional expectation is a function of $(t,s)$. In the latter, the conditional expectation is a random process itself. An amateur guess is that then this latter expression should satisfy a stochastic PDE, analogous to the PDE that $u(t,s)$ solves in the Markov case.



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