Nonlinear Feynman–Kac Formulas

• Nonlinear Feynman–Kac formulas are a probabilistic representation for solving nonlinear, path-dependent PDEs, using forward–backward SDEs with delay and jumps.
• The framework bridges stochastic analysis and PDE theory by ensuring existence, uniqueness, and regularity under global Lipschitz and small delay assumptions.
• It underpins mesh-free numerical schemes for high-dimensional problems and finds practical applications in finance, biology, and physics.

A nonlinear Feynman–Kac formula is a probabilistic representation of solutions to nonlinear partial differential equations (PDEs), typically parabolic or path-dependent Kolmogorov equations. Unlike the classical Feynman–Kac formula, which expresses solutions to linear PDEs as expectations of functionals over stochastic processes, the nonlinear counterpart accommodates nonlinearities—including dependence on the solution, its gradient, delay arguments, and jumps—by employing forward–backward stochastic differential equations (FBSDEs), often with extended features like time delay or jumps. This framework provides a crucial bridge between stochastic analysis and the theory of nonlinear PDEs, enabling existence, uniqueness, and regularity results, and underpinning probabilistic, mesh-free numerical schemes for high-dimensional PDEs.

1. Probabilistic Representation and Scope

Nonlinear Feynman–Kac formulas generalize the classical connection between linear PDEs and Markov processes to semilinear, quasilinear, or fully nonlinear nonlocal PDEs, even when the coefficients or nonlinearities depend on the path, include jumps, or have time delay. The canonical framework involves a decoupled or coupled system of SDEs and (backward) stochastic equations, often driven by Brownian motion and Poisson random measures to capture both continuous and discontinuous stochastic dynamics. The solution to the nonlinear PDE is then characterized as the solution (typically the Y component) of an associated FBSDE evaluated at the initial time.

Key features include:

• Accommodation of path-dependence (functionals depending on the entire path of the forward process).
• Integration of time delay (generator depending on past values via a delayed argument).
• Extension to jump-diffusions, via Poisson random measures in the stochastic calculus.
• Representation as unique solutions (in a mild, viscosity, or classical sense) to nonlocal, nonlinear PDEs driven by general nonlinearities, including gradient- and path-dependent terms.

2. Forward–Backward SDEs with Delay and Jumps

A prototypical nonlinear Feynman–Kac setting for path-dependent PDEs with jumps and delayed generator is given by the following FBSDE system:

• Forward SDE:

$$\begin{aligned} X^{t,\phi}(s) &= \phi(t) + \int_t^s b\bigl(r, X^{t,\phi}_{[0,r]}\bigr)\,\mathrm{d}r + \int_t^s \sigma\bigl(r, X^{t,\phi}_{[0,r]}\bigr)\,\mathrm{d}W(r) \ &\quad+ \int_t^s\!\!\int_{\mathbb{R}_0} \gamma\bigl(r, X^{t,\phi}_{[0,r]}, z\bigr)\, \tilde N(\mathrm{d}r,\mathrm{d}z), \qquad X^{t,\phi}(s) = \phi(s), ~ s < t. \end{aligned}$$

• Backward SDE with delay and jumps:

$$\begin{aligned} Y^{t,\phi}(s) &= h\bigl(X^{t,\phi}_{[0,T]}\bigr) + \int_s^T f\Bigl(r, X^{t,\phi}_{[0,r]}, Y^{t,\phi}(r), Z^{t,\phi}(r), \widetilde U^{t,\phi}(r), Y^{t,\phi}_r\Bigr) \, \mathrm{d}r \ & \quad - \int_s^T Z^{t,\phi}(r) \, \mathrm{d}W(r) - \int_s^T \int_{\mathbb{R}_0} U^{t,\phi}(r,z) \, \tilde N(\mathrm{d}r,\mathrm{d}z), \end{aligned}$$

with delay segment $Y_r^{t,\phi}$, compensated jump $\widetilde U^{t,\phi}(r)$, and initial condition $Y^{t,\phi}(s)=Y^{s,\phi}(s)$, $Z^{t,\phi}(s)=U^{t,\phi}(s,\cdot)=0$ for $s<t$.

Under Lipschitz and small-delay conditions, this system admits a unique square-integrable adapted solution with continuous time parameter (Persio et al., 2022).

3. Associated Path-Dependent Nonlinear Kolmogorov Equation

The FBSDE system provides a mild or viscosity solution (depending on the regularity regime) to a path-dependent partial integro-differential equation (PIDE) of the form:

$$\begin{aligned} -\partial_t u(t,\phi) &- \mathcal{L} u(t,\phi) - f\Bigl(t, \phi, u(t,\phi), \partial_x u(t,\phi)\sigma(t,\phi), (u(\cdot,\phi))_t, \mathcal{J}u(t,\phi)\Bigr) = 0, \ u(T,\phi) &= h(\phi), \end{aligned}$$

where:

• $$\mathcal{L} u(t,\phi) = \tfrac{1}{2} \mathrm{Tr}\left[\sigma \sigma^\top(t,\phi)\,\partial^2_{xx} u(t,\phi)\right] + \langle b(t,\phi), \partial_x u(t,\phi)\rangle$$
• $$\mathcal{J}u(t,\phi) = \int_{\mathbb{R}_0} [u(t,\phi+\gamma(t,\phi,z))-u(t,\phi)] \lambda(z)\nu(dz)$$
• $$\begin{aligned} Y^{t,\phi}(s) &= h\bigl(X^{t,\phi}_{[0,T]}\bigr) + \int_s^T f\Bigl(r, X^{t,\phi}_{[0,r]}, Y^{t,\phi}(r), Z^{t,\phi}(r), \widetilde U^{t,\phi}(r), Y^{t,\phi}_r\Bigr) \, \mathrm{d}r \ & \quad - \int_s^T Z^{t,\phi}(r) \, \mathrm{d}W(r) - \int_s^T \int_{\mathbb{R}_0} U^{t,\phi}(r,z) \, \tilde N(\mathrm{d}r,\mathrm{d}z), \end{aligned}$$0
• Derivatives are in Dupire's path-dependent calculus sense

Under stated regularity and small delay, the solution to the FBSDE system yields $$\begin{aligned} Y^{t,\phi}(s) &= h\bigl(X^{t,\phi}_{[0,T]}\bigr) + \int_s^T f\Bigl(r, X^{t,\phi}_{[0,r]}, Y^{t,\phi}(r), Z^{t,\phi}(r), \widetilde U^{t,\phi}(r), Y^{t,\phi}_r\Bigr) \, \mathrm{d}r \ & \quad - \int_s^T Z^{t,\phi}(r) \, \mathrm{d}W(r) - \int_s^T \int_{\mathbb{R}_0} U^{t,\phi}(r,z) \, \tilde N(\mathrm{d}r,\mathrm{d}z), \end{aligned}$$1, which is the unique mild solution to this PIDE (Persio et al., 2022).

4. Identification and Uniqueness of the Feynman–Kac Representation

The nonlinear Feynman–Kac formula provides, for all $$\begin{aligned} Y^{t,\phi}(s) &= h\bigl(X^{t,\phi}_{[0,T]}\bigr) + \int_s^T f\Bigl(r, X^{t,\phi}_{[0,r]}, Y^{t,\phi}(r), Z^{t,\phi}(r), \widetilde U^{t,\phi}(r), Y^{t,\phi}_r\Bigr) \, \mathrm{d}r \ & \quad - \int_s^T Z^{t,\phi}(r) \, \mathrm{d}W(r) - \int_s^T \int_{\mathbb{R}_0} U^{t,\phi}(r,z) \, \tilde N(\mathrm{d}r,\mathrm{d}z), \end{aligned}$$2,

$$\begin{aligned} Y^{t,\phi}(s) &= h\bigl(X^{t,\phi}_{[0,T]}\bigr) + \int_s^T f\Bigl(r, X^{t,\phi}_{[0,r]}, Y^{t,\phi}(r), Z^{t,\phi}(r), \widetilde U^{t,\phi}(r), Y^{t,\phi}_r\Bigr) \, \mathrm{d}r \ & \quad - \int_s^T Z^{t,\phi}(r) \, \mathrm{d}W(r) - \int_s^T \int_{\mathbb{R}_0} U^{t,\phi}(r,z) \, \tilde N(\mathrm{d}r,\mathrm{d}z), \end{aligned}$$3

with $$\begin{aligned} Y^{t,\phi}(s) &= h\bigl(X^{t,\phi}_{[0,T]}\bigr) + \int_s^T f\Bigl(r, X^{t,\phi}_{[0,r]}, Y^{t,\phi}(r), Z^{t,\phi}(r), \widetilde U^{t,\phi}(r), Y^{t,\phi}_r\Bigr) \, \mathrm{d}r \ & \quad - \int_s^T Z^{t,\phi}(r) \, \mathrm{d}W(r) - \int_s^T \int_{\mathbb{R}_0} U^{t,\phi}(r,z) \, \tilde N(\mathrm{d}r,\mathrm{d}z), \end{aligned}$$4 given by the jump increment of $$\begin{aligned} Y^{t,\phi}(s) &= h\bigl(X^{t,\phi}_{[0,T]}\bigr) + \int_s^T f\Bigl(r, X^{t,\phi}_{[0,r]}, Y^{t,\phi}(r), Z^{t,\phi}(r), \widetilde U^{t,\phi}(r), Y^{t,\phi}_r\Bigr) \, \mathrm{d}r \ & \quad - \int_s^T Z^{t,\phi}(r) \, \mathrm{d}W(r) - \int_s^T \int_{\mathbb{R}_0} U^{t,\phi}(r,z) \, \tilde N(\mathrm{d}r,\mathrm{d}z), \end{aligned}$$5,

$$\begin{aligned} Y^{t,\phi}(s) &= h\bigl(X^{t,\phi}_{[0,T]}\bigr) + \int_s^T f\Bigl(r, X^{t,\phi}_{[0,r]}, Y^{t,\phi}(r), Z^{t,\phi}(r), \widetilde U^{t,\phi}(r), Y^{t,\phi}_r\Bigr) \, \mathrm{d}r \ & \quad - \int_s^T Z^{t,\phi}(r) \, \mathrm{d}W(r) - \int_s^T \int_{\mathbb{R}_0} U^{t,\phi}(r,z) \, \tilde N(\mathrm{d}r,\mathrm{d}z), \end{aligned}$$6

This identification follows from Itô calculus adapted to the path-dependent and jump structure, showing that $$\begin{aligned} Y^{t,\phi}(s) &= h\bigl(X^{t,\phi}_{[0,T]}\bigr) + \int_s^T f\Bigl(r, X^{t,\phi}_{[0,r]}, Y^{t,\phi}(r), Z^{t,\phi}(r), \widetilde U^{t,\phi}(r), Y^{t,\phi}_r\Bigr) \, \mathrm{d}r \ & \quad - \int_s^T Z^{t,\phi}(r) \, \mathrm{d}W(r) - \int_s^T \int_{\mathbb{R}_0} U^{t,\phi}(r,z) \, \tilde N(\mathrm{d}r,\mathrm{d}z), \end{aligned}$$7 solves the non-local, path-dependent nonlinear Kolmogorov equation (Persio et al., 2022).

Uniqueness is guaranteed under global Lipschitz assumptions on coefficients and small delay (ensuring contraction in an appropriate weighted space), and further, if the regularity is sufficient for the pathwise derivatives, the representation yields classical solutions as well.

5. Implications for Delay, Jumps, and Path-Dependence

The Feynman–Kac formula in this regime is not restricted to Markovian models or standard diffusions:

• Jumps are incorporated through integration against compensated Poisson random measures and directly in the integro-differential operator $$\begin{aligned} Y^{t,\phi}(s) &= h\bigl(X^{t,\phi}_{[0,T]}\bigr) + \int_s^T f\Bigl(r, X^{t,\phi}_{[0,r]}, Y^{t,\phi}(r), Z^{t,\phi}(r), \widetilde U^{t,\phi}(r), Y^{t,\phi}_r\Bigr) \, \mathrm{d}r \ & \quad - \int_s^T Z^{t,\phi}(r) \, \mathrm{d}W(r) - \int_s^T \int_{\mathbb{R}_0} U^{t,\phi}(r,z) \, \tilde N(\mathrm{d}r,\mathrm{d}z), \end{aligned}$$8. Uniqueness and existence rely on square-integrability assumptions and compensator bounds on the Lévy measure (Persio et al., 2022).
• The delayed generator structure—where $$\begin{aligned} Y^{t,\phi}(s) &= h\bigl(X^{t,\phi}_{[0,T]}\bigr) + \int_s^T f\Bigl(r, X^{t,\phi}_{[0,r]}, Y^{t,\phi}(r), Z^{t,\phi}(r), \widetilde U^{t,\phi}(r), Y^{t,\phi}_r\Bigr) \, \mathrm{d}r \ & \quad - \int_s^T Z^{t,\phi}(r) \, \mathrm{d}W(r) - \int_s^T \int_{\mathbb{R}_0} U^{t,\phi}(r,z) \, \tilde N(\mathrm{d}r,\mathrm{d}z), \end{aligned}$$9 depends on past trajectories $Y_r^{t,\phi}$0—enters both the BSDE and the PDE, requiring specialized contraction mappings (small delay restriction), see also general frameworks in the time delay SDE literature (Persio et al., 2022).
• Path-dependence is essential in both the forward SDE and the PIDE via non-anticipative functionals, and the PDE operator acts on the entire history, compatible with modern functional Itô calculus.

A plausible implication is that these models subsume a large class of systems in finance, biology, and physics where memory, hereditary, or feedback effects are present and the underlying risk is discontinuous and/or non-Markovian.

6. Financial Applications: The Large Investor Problem

A key application is to financial models of large traders whose actions impact price:

• The risky asset price follows a jump–diffusion with coefficients depending on both current and past investor wealth.
• The wealth process $Y_r^{t,\phi}$1 under a self-financing trading strategy $Y_r^{t,\phi}$2 satisfies an FBSDE with both jump and delay terms, and the nonlinear value function $Y_r^{t,\phi}$3 solves a path-dependent PIDE incorporating all relevant features:

$Y_r^{t,\phi}$4

where $Y_r^{t,\phi}$5 encodes the dependences on wealth, trading, and market coefficients. The nonlinear Feynman–Kac formula identifies the FBSDE solution components: - $Y_r^{t,\phi}$6, - $Y_r^{t,\phi}$7, - $Y_r^{t,\phi}$8.

This approach allows the explicit computation and numerical approximation of the value function for optimal trading strategies in models with jumps, delay, and feedback (Persio et al., 2022).

7. Further Implications and Pathwise Fractional Calculus Perspective

The probabilistic foundation of the nonlinear Feynman–Kac representation admits further generalizations:

• Pathwise (deterministic) versions based on nonlinear Young integrals via fractional calculus extend the probabilistic representation to heat-type equations with random, Hölder-continuous media, where the stochastic integral becomes a deterministic, pathwise nonlinear Young integral $Y_r^{t,\phi}$9 (Hu et al., 2015).
• This suggests that nonlinear Feynman–Kac representations are robust under rough path perturbations and likely extendable to SPDEs and PDEs with complex noise or stochastic parameter dependence, provided suitable integrability and regularity hold.

These developments unify and substantially generalize classical connections between SDEs/BSDEs and PDEs, enabling analysis of high-dimensional, non-Markovian, and nonlocal systems via probabilistic and pathwise stochastic calculus (Persio et al., 2022, Hu et al., 2015).