Generalization of the Feynman-Kac formula for Markov processes

Area of Science:

• Stochastic Processes
• Mathematical Physics
• Computational Finance

Background:

• Feynman-Kac equations are crucial for solving stochastic differential equations.
• Existing models often struggle with complex Markov processes like jumps and regime switching.
• A unified framework is needed to handle diverse stochastic dynamics.

Purpose of the Study:

• Extend forward and backward Feynman-Kac equations for general Markov processes.
• Specialize these equations for various models including diffusion, jump-diffusion, and systems with resetting.
• Illustrate the formalism's power with diverse stochastic process models.

Main Methods:

• Generalization of forward and backward Feynman-Kac equations.
• Application to differential Chapman-Kolmogorov equations for Markov processes.
• Derivation of closed-form solutions for various stochastic models.

Main Results:

• Closed-form solutions are provided for arbitrary initial/final conditions.
• The framework successfully models diffusion, jump-diffusion with drift, regime switching, and hybrid systems with resetting.
• A novel model with mean reversion and uncorrelated states is analyzed, departing from conventional assumptions.

Conclusions:

• The extended Feynman-Kac formalism offers a unified approach to diverse stochastic processes.
• The method naturally incorporates resetting phenomena in stochastic hybrid systems.
• This work provides a powerful tool for analyzing complex systems in physics, finance, and beyond.