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Question: Is there a known path-dependent Black-Scholes PDE?

To be a little more precise, let $S$ be a stock price under a risk-neutral measure such that $S$ satisfies the SDE with path-dependent volatility: $$dS_t = r S_t dt + \sigma(t, S_.) S_t dB_t.$$ Here $\sigma: [0,\infty)\times C([0, \infty), (0,\infty)) \to \mathbb{R}$ is a previsible path functional, i.e. for any $t\geq 0$, $\sigma(t, S_.) \in \mathscr{F}_t$. In other words $\sigma$ depends on the path of $S$ up to time $t$ and is known at time $t$.

When $\sigma(t, S_.)=\sigma(t, S_t)$ only depends on the current state $S_t$, the ordinary Feynman-Kac formula says that $$u(t, s) = \mathbb{E}(e^{-r(T-t)}h(S_T) | S_t=s),$$ if and only if $u$ solves the PDE $$u_t+r s u_s + \frac12 \sigma(t,s)^2 s^2 u_{ss}-ru=0$$ with terminal condition $u(T, s) = h(s)$.

What can we say about this path-dependent case? $$U_t = \mathbb{E}(e^{-r(T-t)} h(S_T) | \mathscr{F}_t)?$$

My only hunch is that Bruno Dupire's functional Ito calculus might be applied to this particular case but it is taking me some time to read through this paper and check everything. Also his Feynman Kac has a slightly different form, 1) it is more general as it lets $h$ also be a functional, in my notation $h(t, S_.)$, and 2) his condition in the conditional expectation is $S_t$ is a little different. It involves conditioning on the path rather than the filtration.

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    $\begingroup$ I think you might find the following article useful (and references therein): royalsocietypublishing.org/doi/abs/10.1098/rspa.2004.1370 $\endgroup$
    – Frido
    Commented Sep 7, 2023 at 20:48
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    $\begingroup$ @Frido Interesting. My motivation was from using EMA-estimates for volatility (more so for portfolio optimization rather than option-pricing but the frameworks are related since the former can be solved via HJB-PDE which is related to the Feynman-Kac PDE) and the Hobson-Rogers model treats volatility as a function of "the deviation from the trend and is defined as the difference between the current value and a geometric, exponentially weighted average of past prices." So definitely useful, thank you for the reference. $\endgroup$ Commented Sep 8, 2023 at 16:39
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    $\begingroup$ You're welcome. Note that recently Julien Guyon has written some papers about path dependent vol as well. You may want to read his papers too. But afaik Hobson Rogers were one of the first to analyze these models. $\endgroup$
    – Frido
    Commented Sep 9, 2023 at 7:28

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