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This study extends Feynman-Kac equations for general Markov processes, offering closed-form solutions for diffusion, jump-diffusion, and resetting systems. The generalized framework provides new insights into stochastic modeling with complex dynamics.

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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.