Nonlinear Feynman–Kac Formula Overview

• The nonlinear Feynman–Kac formula represents solutions of complex, path-dependent PDEs via BSDEs that integrate jumps and delays.
• It extends classical methods by mapping non-Markovian processes into a probabilistic framework, aiding analysis in stochastic control and quantitative finance.
• The framework’s use of infinite-dimensional lifting and well-posedness results enables robust numerical strategies and inspires further research in dynamic systems.

The nonlinear Feynman–Kac formula is a fundamental extension of the classical Feynman–Kac representation which bridges solutions of (possibly path-dependent, fully nonlinear, or infinite-dimensional) partial differential equations (PDEs) with corresponding stochastic processes, typically formulated as backward stochastic differential equations (BSDEs) or more general forward–backward systems (FBSDEs). Unlike the linear Feynman–Kac formula—where the solution of a linear parabolic PDE is given as the expectation of a functional along the path of a Markov process—the nonlinear version accommodates nonlinearities in the PDE, dependence on the full path (i.e., non-Markovianity), jumps, delays, or infinite-dimensional features, and connects these analytically to stochastic control, finance, and rough path analysis.

1. Probabilistic Representation and Path-Dependent Formulation

The essential innovation of the nonlinear Feynman–Kac formula is its capacity to represent the solution $u(t, \phi)$ of a path-dependent, nonlinear Kolmogorov-type PDE via a probabilistic system—most typically a forward–backward SDE with jumps and delays. Specifically, for $\phi$ an initial path in the space of càdlàg functions $\Lambda = D([0,T]; \mathbb{R}^d)$, and with a nonlinear generator depending on the current value, the path, and delayed segments, one has the representation: $u(t, \phi) = Y^{(t, \phi)}(t)$ where $Y^{(t, \phi)}(\cdot)$ is the BSDE component of a coupled FBSDE system (see equation (feynman)), with the forward process $X^{(t,\phi)}$ satisfying a path-dependent SDE incorporating both diffusion and jump (Lévy) terms, and the generator $f$ of the BSDE depending as

$$f\left(r, X^{(t, \phi)}, Y^{(t, \phi)}(r), Z^{(t, \phi)}(r), \tilde{I}^{(t, \phi)}(r), Y_r^{(t, \phi)}\right),$$

with $Y_r^{(t, \phi)}$ denoting the delayed segment $$\{Y^{(t, \phi)}(r+\theta): \theta \in [-\delta,0]\}$$.

The forward and backward SDEs encode both continuous dynamics (via standard Brownian motion $W$) and jump dynamics (via a compensated Poisson random measure $\tilde{N}$), thus creating a highly general framework for semimartingale-driven systems with memory.

2. Analytical Structure of the Path-Dependent Nonlinear Kolmogorov Equation

The associated PDE becomes a path-dependent (integro-)differential equation with both delay and jump terms: $$-\partial_t u(t, \phi) - \mathcal{L} u(t, \phi) - f\left(t, \phi, u(t, \phi), \partial_x u(t, \phi)\sigma(t, \phi), (u(\cdot, \phi))_t, \mathcal{J}u(t,\phi)\right) = 0,$$ with terminal condition $u(T, \phi) = h(\phi)$.

• $\mathcal{L}$ is the second-order differential operator:

$$\mathcal{L}u(t, \phi) = \frac{1}{2}\mathrm{Tr}[\sigma(t, \phi)\sigma^*(t, \phi)\partial_{xx}^2 u(t, \phi)] + \langle b(t, \phi), \partial_x u(t, \phi) \rangle,$$

• The jump operator $\mathcal{J}$ is defined by

$$\mathcal{J}u(t, \phi) = \int_{\mathbb{R}\setminus\{0\}} [u(t, \phi + \gamma(t, \phi, z)) - u(t, \phi)]\ \lambda(z)\,\nu(dz).$$

• The delay term $(u(\cdot, \phi))_t$ refers to the segment restriction to $[t-\delta, t]$.

This setting captures a general class of path-dependent, non-Markovian problems where both historical trajectories and jump effects crucially affect present evolution.

3. FBSDE Systems with Jumps and Delay: Structure and Well-Posedness

The FBSDE system underpinning the probabilistic representation is as follows:

Forward SDE:

$$X^{(t,\phi)}(s) = \phi(t) + \int_t^s b(r, X^{(t, \phi)}) dr + \int_t^s \sigma(r, X^{(t, \phi)}) dW(r) + \int_{t}^{s}\int_{\mathbb{R}\setminus\{0\}} \gamma(r, X^{(t, \phi)}, z)\, \tilde{N}(dr, dz),$$

for $s \in [t,T]$, capturing both diffusion and jumps.

Backward SDE with Delayed Generator:

$\begin{split} Y^{(t, \phi)}(s) = &~ h(X^{(t,\phi)}) + \int_s^T f\left(r, X^{(t, \phi)}, Y^{(t, \phi)}(r), Z^{(t, \phi)}(r), \tilde{I}^{(t, \phi)}(r), Y_r^{(t, \phi)}\right) dr \ & - \int_s^T Z^{(t, \phi)}(r) dW(r) - \int_s^T \int_{\mathbb{R}\setminus\{0\}} U^{(t, \phi)}(r, z)\, \tilde{N}(dr, dz), \end{split}$

where $\tilde{I}^{(t, \phi)}(r)$ is an appropriate integral over the jump sizes.

Well-posedness of this system (existence, uniqueness, regularity) is ensured under Lipschitz and smallness conditions on the delay (cf. the smallness of the delay parameter $K_T$; see Remark "small" and condition (condition_KT)). The presence of both path dependence and delayed generator necessitates a functional-analytic approach, often via lifting to an infinite-dimensional space—specifically, a Delfour–Mitter Hilbert space $$M^2 = L^2([−T,0]; \mathbb{R}^d) \times \mathbb{R}^d$$—to manage pathwise differentiation and mild solution formulations.

4. Jump and Delay Effects: Analytical and Applied Context

The incorporation of both jumps (through Lévy-type noise via $\tilde{N}$ and the operator $\mathcal{J}$) and delays (in the form of dependencies on past segments) extends the classical Feynman–Kac paradigm to accommodate memory and abrupt transitions. These features are crucial for modeling:

• Non-Markovianity: The solution depends on the full path (or segments), not just the present state, modeling memory effects.
• Jump-Diffusion: Realistic market, physical, or biological phenomena often exhibit abrupt, discontinuous changes.
• Delay: Many systems exhibit after-effects; the future evolution depends explicitly on past trajectories, which plays a significant role in optimal control and mathematical finance.

The analytical structure supports existence and uniqueness of (mild) solutions in the infinite-dimensional space, enabling rigorous treatment of delayed and jump-affected systems.

5. Applications: The Large Investor Problem and Market Dynamics

An important application highlighted is the generalization of the Large Investor Problem. In such models, an investor's actions (portfolio changes) affect asset trajectories—modeled by SDEs with both delay and jumps:

• The risky asset $S$ and the riskless asset $S_0$ are modeled as jump-diffusions:

$$\frac{dS_0(t)}{S_0(t)} = r(t, X^\pi(t), X_t^\pi) dt,\qquad \frac{dS(t)}{S(t)} = \mu(t, X^\pi(t), X_t^\pi) dt + \sigma(t) dW(t) + \int_{\mathbb{R}\setminus\{0\}} \gamma(t, z)\,\tilde{N}(dt, dz),$$

where $X^\pi$ is the dynamic wealth process, $X^\pi_t$ its path segment, and the impact of investment strategy $\pi(t)$ is explicit.

• The BSDE formulation is employed in replicating contingent claims, with the FBSDE/BSDE system capturing price dynamics influenced by both current and historical trading strategies, as well as sudden market jumps.

This framework directly models practical scenarios in mathematical finance where markets exhibit both feedback effects and non-Gaussian noise.

6. Implications and Directions for Future Research

The extension of the nonlinear Feynman–Kac formula to path-dependent, delay, and jump-diffusion settings augments the scope of probabilistic representations for nonlinear PDEs and brings new analytic and computational challenges. Key implications include:

• Theoretical Enrichment: BSDEs with jump and delayed generators form a flexible toolkit for the analysis of nonlinear, path-dependent PDEs, supporting both existence and regularity theory in high dimensions and infinite dimensions.
• Mild Solutions and Functional Lifting: Use of Hilbert space lifting enables analysis of non-Markovian processes as Markovian in extended spaces, supporting the derivation and paper of mild solutions.
• Numerical Analysis: The framework suggests directions for developing robust numerical schemes (potentially leveraging discretizations or neural network-based techniques) for approximating path-dependent PDE solutions in the presence of jumps and delays.
• Control and Game-Theoretic Extensions: The comprehensive state-space dynamics admit generalizations to optimal control and game-theoretic contexts, such as stochastic differential games in delayed, jump-diffusive environments.

Potential areas for further development include allowing delays in all components of the BSDE (including $Z$ and $U$), investigation of more general jump processes, and the design of efficient simulation-based schemes for high-dimensional, path-dependent systems.

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Summary Table: Core Elements in the Nonlinear Feynman–Kac Formula with Path, Delay, and Jump Dependence

Component	Mathematical Structure	Main Role
Path-dependent PDE	$$-\partial_t u - \mathcal{L} u - f(t, \phi, u, \partial_x u\,\sigma, (u(\cdot, \phi))_t, \mathcal{J}u) = 0$$	Target equation
Forward SDE	$X^{(t, \phi)}$ with Brownian and jump (Poisson) noise, and path-dependent coefficients	Models the forward trajectory
Backward SDE	$Y^{(t, \phi)}$ with generator depending on both delays and jumps	Provides probabilistic representation
Jump Operator $\mathcal{J}$	$\mathcal{J}u(t, \phi)$ as an integral over jump terms	Models nonlocal, jump-induced effects
Delay in BSDE/PDE	Generator and PDE depend on past segments $Y_r^{(t, \phi)}$, $u(\cdot, \phi)_t$	Incorporates memory effects
Infinite-dimensional lifting	Functional space (e.g., Delfour–Mitter) for reformulating the problem	Ensures differentiability/Markovianity
Application: Large Investor Problem	SDE/BSDE for risky and riskless assets with feedback from path and jumps	Market impact model with path/jump

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The nonlinear Feynman–Kac formula for path-dependent, jump-diffusion, and delayed systems thus represents a unifying stochastic-analytic principle for a large class of nonlinear PDEs and SPDEs. The framework naturally supports rigorous analysis and is a foundation for advanced modeling in applied probability, stochastic control, and quantitative finance (Persio et al., 2022).