# Ito's formula and Feynman Kac formula for path-dependent SDEs

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.

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?

• 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.
– SBF
Dec 24, 2022 at 20:07
• @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. Jan 1, 2023 at 17:56
• That I am not familiar with
– SBF
Jan 2, 2023 at 17:09