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In Rogers and Williams Volume 2, Chapter V Section 2 Subsection 8, we have introduced there the notion of a general form of SDEs where the coefficients are taken to be previsible path functionals: $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\E}{\mathbb{E}}$

$$X_t=X_0+\int_0^t \mu(s, X_.)ds+\int_0^t \sigma(s, X_.)dB_s$$ where $\mu$ and $\sigma$ are previsible path functionals. If $W$ is a suitable path space then $\mu, \sigma$ are functions on $[0,\infty)\times W \to \mathbb{R}$ that are adapted to relevant filtrations. I am purposely omitting describing the various sigma-algebras and canonical spaces, since I do not think they are relevant to my question, at least to the intuition and formal manipulations. All that matters is that $\mu(t, X_.)$ and $\sigma(t, X_.)$ are previsible with respect to $\mathscr{F}_t$ (we assume the existence of a space $(\Omega, \mathscr{F}_t, \mathbb{P})$ such that the above SDE for $X$ has a weak solution).

I apologize if this is being reckless. I have a handful of questions but they all really stem from one underlying desire: generalizing known facts about diffusions to SDEs with path-dependent coefficients.

Questions:

Is Ito's formula, and the form of the infinitesimal generator the same as it is in the simpler case when $\mu(t, X_.)=\mu(t, X_t)$ depends on only the current state $X_t$? i.e.

  1. Do we have $$df(X_t) = f'(X_t) dX_t+\frac12 f''(X_t)d[X]_t,$$ for smooth $f$? Note: $f:\mathbb{R}\to \mathbb{R}$ is an ordinary function.

Answer: Yes, right? Here is a sketch of the justification I thought of: Ito's formula applies to any semimartingale. We know $X_t = X_0 + A_t +M_t$ where $A_t = \int_0^t \mu(s, X_.) ds$ is an adapted process of finite variation (since $X$ is adapted and $\mu$ is a previsible path functional), and $M_t = \int_0^t \sigma(s, X_.) dB_s$ is a $\mathscr{F}_t$-martingale provided $\mathbb{E}(\int_0^t |\sigma(s, X_.)|^2 ds)<\infty$ a.s. for all $t$. We may assume this last condition so that the SDE has a weak solution to begin with.

  1. If we carry through with the analogous computations in a formal manner, we would find the generator to be $$\mathscr{L}f(x) = \mu(t, X_.) f'(x)+\frac12 \sigma(t, X_.)^2 f''(x).$$ Now we have some suspcions. The above operator is now, apparently, a stochastic process itself, since the coefficients depend on the path and vary at each time $t$, starting at $X_t=x$.

  2. If the above generator makes sense, do we still have a Feynman-Kac formula, i.e. $$u(t,x)=\mathbb{E}(g(X_T) |X_t=x),$$ solves $$\frac{\partial u}{\partial t} + \mu(t, X_.) \frac{\partial u}{\partial x}+\frac12 \sigma(t, X_.)^2 \frac{\partial^2 u}{\partial x^2}=0,$$ with $u(T,x)=g(x)$? Again, there must be some mistake here, because the above seems to be a Stochastic Partial Differential Equation, while the definition of $u(t,x)$ should average out all randomness.

Some thoughts:

Ito's lemma is proven for semimartingales in the book, but it does not exactly clarify these issues. There is an interesting section on Martingale Problems. The law $\mathbb{P}^y$ (a measure on a relevant path space) of the process $X$ starting form $y$ and solving the SDE $$dX_t = \mu(t, X_.)dt+\sigma(t, X_.)dB_t,$$ solves the martingale problem, i.e. it has the following decisive properties:

  1. $\mathbb{P}^y(X_0=y)=1$

  2. Under the law of $X$, for each $f\in C_{K}^\infty(\mathbb{R}^n)$, $$C_t^f := f(X_t)-f(X_0)-\int_0^t \mathscr{L}(s,X_.) ds$$ is a martingale (with respect to the filtration of the path space). Here R&W define $$\mathscr{L}f(s,X_.) =\mu(s, X_.)\partial_x f(X_s)+ \frac12 \sigma(s, X_.)^2 \partial_{xx}f(X_s).$$ And it is stated that this was proven in Volume 1 under the assumption of diffusion type SDEs: $\mu(t,X.)=\mu(X_t)$, $\sigma(t,X_.)=\sigma(X_t)$, edit: (forgot to add this) but that the calculations are the same in the present case. (I have stated the results here in one-dimension). This is from Section 4, subsection 19, page 158.

Additional questions:

a) The notation seems a little inconsistent with treating the LHS $f$ as a function of $(t, X_.)$ and the RHS as only of $X_t$. What is the meaning of this?

b) Is this enough to work out the analogs of Feynman-Kac for path dependent SDEs?

Update August 24th 2023: Here is a formal attempt at getting the ordinary Feynman-Kac formula (at least in one direction). It proceeds in exactly the usual fashion, I believe, as in the case of coefficients depending only on the latest state. Assume $u_t + \mathscr{L}_xu(t, x_.)=0$ i.e., for any continuous path $x_. \in C(\R)$ with $x_t=x$, $$u_t+\mu(t, x_.) u_x(t,x)+\frac12 \sigma(t, x_.)^2 u_{xx}(t, x)=0$$ and $u(T, x) = h(x)$. Here $$\mathscr{L}_x u(t, x_.) := \mu(t, x_.) u_x(t, x)+\frac12 \sigma(t, x_.)^2 u_{xx}(t,x),$$ for any path $x_.\in C(\R)$ with $x_t=x$.

Applying Ito's formula to $Y_t = u(t, X_t)$ gives $$dY_t = [u_t(t, X_t)+\mathscr{L}_x u(t, X_.)] dt + u_x(t, X_t) \sigma(t, X_.) dB_t$$ but the drift is zero by assumption since $u$ solves $u_t+\mathscr{L}_xu(t, x_.)=0$ for any path $x_.$ Thus we are left with $$h(X_T) = u(T, X_T) = u(t, X_t) + \int_t^T u_x(s, X_s) \sigma(s, X_.) dB_s$$ which upon taking the conditional expectation conditional on $X_t=x$ we obtain $$u(t,x) = \E(h(X_T) |X_t=x),$$ since $M_t = \int_t^T u_x(s, X_s) \sigma(s, X_.)dB_s$ is a (local) martingale.

Does this hold up to scrutiny? and, if so, what changes if we take $\E(\cdot|\mathscr{F}_t)$ instead?

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    $\begingroup$ My guess would be that FK and similar things are related to the (strong) Markov property of the Ito diffusions. If you have path-dependent funcitonals as $\mu$ and $\sigma$ you lose that property and it would not make sense to talk about generators etc. I do now know whether Ito formula works as your wrote, there you might be right. $\endgroup$
    – SBF
    Dec 24, 2022 at 20:07
  • $\begingroup$ @Ilya Sounds like a sensible guess to me. But can we write $$\lim_{\delta \to 0^+} \frac{\mathbb{E}(f(X_{t+\delta})|\mathscr{F}_t)-f(X_t)}{\delta} = \mathscr{L}f(t, X_.)$$ where $$\mathscr{L}f(t, X_.)=\mu(t, X_.) f'(X_t)+\frac12 \sigma(t, X_.)^2 f''(X_t)$$ if we assume $\mu, \sigma$ are known conditional on the history up to time $t$? i.e. we have a sort of conditional generator? I did find Bruno Dupire's very nice paper (papers.ssrn.com/sol3/papers.cfm?abstract_id=1435551) which has an Ito's lemma and FK for functionals $f$ and $g$; I think this is more general than what I desired. $\endgroup$ Jan 1, 2023 at 17:56
  • $\begingroup$ That I am not familiar with $\endgroup$
    – SBF
    Jan 2, 2023 at 17:09

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