Feynman–Kac Formula & Regime-Switching Diffusions

• Feynman–Kac formula is a probabilistic representation linking PIDEs with expectations over stochastic processes, particularly in regime-switching jump-diffusions.
• It provides a framework for modeling complex systems in finance, engineering, and applied mathematics using continuous dynamics coupled with discrete regime shifts.
• The methodology ensures analytical tractability and supports Monte Carlo simulations while establishing existence and uniqueness under Lipschitz and growth conditions.

The Feynman–Kac formula establishes a fundamental correspondence between linear parabolic (and certain partial integro-differential) equations and expectations of functionals of stochastic processes. In the context of regime-switching jump-diffusion processes—where both continuous Itô dynamics, Lévy jump events, and a finite-state Markov chain-driven regime-switching act jointly—the formula provides a probabilistic representation of solutions to a broad class of partial integro-differential equations (PIDEs) with terminal, initial, and boundary conditions. The regime-switching mechanism couples discrete Markovian environments with stochastic dynamics and jump events, admitting both analytical tractability and flexible modeling of complex evolution phenomena across finance, engineering, and applied mathematics (Zhu et al., 2017).

1. Regime-Switching Jump-Diffusions: Model Structure

A regime-switching jump-diffusion consists of a process $(X_t, \alpha_t)$, where $X_t$ is $\mathbb{R}^n$-valued and $\alpha_t$ is a finite-state Markov chain on $M = \{1,\ldots, m\}$. The continuous dynamics of $X_t$ are governed by coefficients $b$, $\sigma$, and $y$, which depend jointly on the current state $X_t$ and regime $X_t$0. Jumps occur according to a Poisson random measure $X_t$1 with Lévy compensator $X_t$2, respecting integrability conditions on $X_t$3 with respect to $X_t$4.

The SDE reads: $X_t$5 with the Markov chain $X_t$6 evolving according to state-dependent transition rates $X_t$7: $X_t$8 This hybrid construction enables the full generator (see Section 3) to capture both local differential, jump, and discrete switching effects (Zhu et al., 2017).

2. Feynman–Kac Representation for Integro-Differential Problems

The Feynman–Kac formula connects solutions of a backward Kolmogorov-type PIDE with expectations over the regime-switching jump-diffusion paths. Consider the backward Cauchy problem: $X_t$9 where $\mathbb{R}^n$0 is a nonnegative, continuous discount or killing rate, $\mathbb{R}^n$1 is a terminal datum, and $\mathbb{R}^n$2 is the full infinitesimal generator.

The Feynman–Kac representation asserts: $\mathbb{R}^n$3 where the expectation is taken over the solution path $\mathbb{R}^n$4 starting from $\mathbb{R}^n$5 (Zhu et al., 2017). This correspondence is justified by applying Itô’s formula to the process $\mathbb{R}^n$6.

3. Generator Structure and Integro-Differential Decomposition

The full generator $\mathbb{R}^n$7 acting on suitable test functions $\mathbb{R}^n$8 ($\mathbb{R}^n$9 in $\alpha_t$0 for each $\alpha_t$1) is: $\alpha_t$2 It is customary to split $\alpha_t$3, separating the continuous (diffusion and jump) and discrete (switching) contributions (Zhu et al., 2017).

4. Existence and Well-Posedness Conditions

Well-posedness of the SDE, moment bounds, and validity of the Feynman–Kac representation depend on global Lipschitz continuity in $\alpha_t$4 (uniform in $\alpha_t$5) for $\alpha_t$6 and $\alpha_t$7, quadratic and linear growth bounds on $\alpha_t$8, and boundedness of switching rates $\alpha_t$9. For boundary value problems, the domain $M = \{1,\ldots, m\}$0 must be $M = \{1,\ldots, m\}$1, $M = \{1,\ldots, m\}$2 continuous on $M = \{1,\ldots, m\}$3, and boundary data continuous and bounded.

Under these standing assumptions, the regime-switching SDE admits a unique strong solution without explosion, and the associated PIDE has a unique (classical or viscosity) solution matching the Feynman–Kac formula (Zhu et al., 2017).

5. Boundary, Initial, and Nonhomogeneous Value Problems

The Feynman–Kac approach extends to several PIDE scenarios:

• Backward Cauchy problem (pure terminal data):

$M = \{1,\ldots, m\}$4

admits the representation

$M = \{1,\ldots, m\}$5

• Nonhomogeneous terminal-value problem:

$M = \{1,\ldots, m\}$6

with solution

$M = \{1,\ldots, m\}$7

• Dirichlet (boundary) problem:

$M = \{1,\ldots, m\}$8

leads to

$M = \{1,\ldots, m\}$9

For each case, the Feynman–Kac formula provides not only a representation but, under analyticity assumptions, a means of verifying that the expectation satisfies the corresponding PIDE (Zhu et al., 2017).

6. Representative Examples and Application Domains

(a) Jump-diffusion Black–Scholes with switching:

In a risk-neutral regime-switching jump-diffusion Black–Scholes model, the log-stock $X_t$0 satisfies: $X_t$1 The no-arbitrage price $X_t$2 of a European contingent claim, with payoff $X_t$3, is

$X_t$4

which solves a generalized Black–Scholes integro-PDE under the operator $X_t$5 (Zhu et al., 2017).

(b) Two-time-scale switching/arcsine law:

In the fast-switching limit ($X_t$6 with generator $X_t$7, $X_t$8) and vanishing drift, the occupation-time functional

$X_t$9

appropriately normalized converges in law to the classical arcsine law, connecting the weak convergence of the process to universal limit theorems (Zhu et al., 2017).

7. Analytical and Computational Consequences

For regime-switching jump-diffusions, the Feynman–Kac formula serves as both (i) an existence/uniqueness tool (via stochastic representation and classical solution) and (ii) a pathway to practical Monte Carlo computation for local, global, and boundary-value problems. The underlying technical tools include extended Itô formulas for jump-diffusion-switching processes and careful moment and exit-time analysis under the stated Lipschitz and growth constraints (Zhu et al., 2017). This framework is indispensable in stochastic modeling and financial engineering for processes with discontinuities, regime shifts, and random switching mechanisms.