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Feynman-Kac formula for BSDEs with jumps and time delayed generators associated to path-dependent nonlinear Kolmogorov equations

Luca Di Persio, Matteo Garbelli, Adrian Zălinescu

TL;DR

This work develops a nonlinear Feynman–Kac representation for forward–backward stochastic differential equations with jumps and a time-delayed generator, linking their solution to a path-dependent Kolmogorov equation with delay and jump terms. By lifting the path space to a Delfour–Mitter Markov framework, the authors derive Y^{t,\phi}(s)=u(s,X^{t,\phi}) with u(t,\phi)=Y^{t,\phi}(t), and prove the existence of a mild solution to the associated PPIDE under suitable Lipschitz and smallness conditions. The results extend classical Feynman–Kac theory to non-Markovian, memory-including dynamics with discontinuities, and they are illustrated by a financial application in a generalized Large Investor problem with jump-diffusion stock prices. The framework provides a rigorous bridge between memoryful stochastic systems and path-dependent PDEs, with potential impact on pricing, hedging, and risk management in markets with jumps and delays.

Abstract

We consider a system of Forward Backward Stochastic Differential Equations (FBSDEs), with time delayed generator and driven by Lèvy-type noise. We establish a non linear Feynman Kac representation formula associating the solution given by the FBSDEs-system to the solution of a path dependent nonlinear Kolmogorov equation with both delay and jumps. Obtained results are then applied to study a generalization of the so-called Large Investor Problem where the stock price evolves according to a jump-diffusion dynamic.

Feynman-Kac formula for BSDEs with jumps and time delayed generators associated to path-dependent nonlinear Kolmogorov equations

TL;DR

This work develops a nonlinear Feynman–Kac representation for forward–backward stochastic differential equations with jumps and a time-delayed generator, linking their solution to a path-dependent Kolmogorov equation with delay and jump terms. By lifting the path space to a Delfour–Mitter Markov framework, the authors derive Y^{t,\phi}(s)=u(s,X^{t,\phi}) with u(t,\phi)=Y^{t,\phi}(t), and prove the existence of a mild solution to the associated PPIDE under suitable Lipschitz and smallness conditions. The results extend classical Feynman–Kac theory to non-Markovian, memory-including dynamics with discontinuities, and they are illustrated by a financial application in a generalized Large Investor problem with jump-diffusion stock prices. The framework provides a rigorous bridge between memoryful stochastic systems and path-dependent PDEs, with potential impact on pricing, hedging, and risk management in markets with jumps and delays.

Abstract

We consider a system of Forward Backward Stochastic Differential Equations (FBSDEs), with time delayed generator and driven by Lèvy-type noise. We establish a non linear Feynman Kac representation formula associating the solution given by the FBSDEs-system to the solution of a path dependent nonlinear Kolmogorov equation with both delay and jumps. Obtained results are then applied to study a generalization of the so-called Large Investor Problem where the stock price evolves according to a jump-diffusion dynamic.
Paper Structure (12 sections, 3 theorems, 116 equations)

This paper contains 12 sections, 3 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Notation and Problem formulation
  3. The forward-backward delayed system
  4. The forward path-dependent SDE with jumps
  5. The backward delayed path-dependent SDE with jumps
  6. The Well Posedness of the BSDE
  7. The Feynman-Kac formula
  8. The Delfour-Mitter space
  9. Main theorem
  10. Mild solution of the Kolmogorov Equation
  11. Financial Application
  12. Conclusions and Future Development

Key Result

Theorem 3.2

Let assumptions $A_4, A_5, A_6$ hold. If condition condition_KT is satisfied, then there exists a unique solution $(Y^{t,\phi}, Z^{t,\phi}, U^{t, \phi})$ of the BSDE described in backward such that $(Y^{t,\phi}, Z^{t,\phi}, U^{t, \phi}) \in \mathbb{S}_t^2 (\mathbb{R}) \times \mathbb{H}^2_t (\mathbb

Theorems & Definitions (11)

  • Remark 2.1
  • Remark 3.1
  • Theorem 3.2
  • Remark 4.1
  • Theorem 4.2: Feynman-Kac formula
  • proof
  • Definition 5.1
  • Theorem 5.1: Existence
  • proof
  • Remark 5.2: Uniqueness
  • ...and 1 more