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A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance

Francesco Cordoni, Luca Di Persio, Lucian Maticiuc, Adrian Zălinescu

TL;DR

The paper addresses proving the existence of viscosity solutions to path-dependent nonlinear Kolmogorov equations with time-delayed generators by developing a stochastic representation via a forward SDE with delay and a backward SDE with time-delayed generator, yielding a nonlinear, non-Markovian Feynman–Kac formula. It combines functional Itô calculus with a viscosity-solution framework to establish the PDKE result under small delay or Lipschitz constants, and provides continuity of the value functional $u(t,\phi)=Y^{t,\phi}(t)$ as well as the delay-aware Feynman–Kac link. The work further specializes to finance, offering two models: a large-investor market-impact scenario and memory-based dynamic risk measures through $g$-expectations, illustrating path-dependent pricing and risk management with memory effects. Overall, the results deliver existence, representation, and applications of path-dependent PDEs in a delayed, non-Markovian setting, expanding the toolkit for pricing and risk assessment in finance with memory.

Abstract

We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation: \[ \begin{cases} \partial_{t}u(t,φ)+\mathcal{L}u(t,φ)+f(t,φ,u(t,φ),\partial_{x}u(t,φ) σ(t,φ),(u(\cdot,φ))_{t})=0,\;t\in[0,T),\;φ\in\mathbbΛ\, ,u(T,φ)=h(φ),\;φ\in\mathbbΛ, \end{cases} \] where $\mathbbΛ=\mathcal{C}([0,T];\mathbb{R}^{d})$, $(u(\cdot ,φ))_{t}:=(u(t+θ,φ))_{θ\in[-δ,0]}$ and \[ \mathcal{L}u(t,φ):=\langle b(t,φ),\partial_{x}u(t,φ)\rangle+\dfrac {1}{2}\mathrm{Tr}\big[σ(t,φ)σ^{\ast}(t,φ)\partial_{xx} ^{2}u(t,φ)\big]. \] The result is obtained by a stochastic approach. In particular we prove a new type of nonlinear Feynman-Kac representation formula associated to a backward stochastic differential equation with time-delayed generator which is of non-Markovian type. Applications to the large investor problem and risk measures via $g$-expectations are also provided.

A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance

TL;DR

The paper addresses proving the existence of viscosity solutions to path-dependent nonlinear Kolmogorov equations with time-delayed generators by developing a stochastic representation via a forward SDE with delay and a backward SDE with time-delayed generator, yielding a nonlinear, non-Markovian Feynman–Kac formula. It combines functional Itô calculus with a viscosity-solution framework to establish the PDKE result under small delay or Lipschitz constants, and provides continuity of the value functional as well as the delay-aware Feynman–Kac link. The work further specializes to finance, offering two models: a large-investor market-impact scenario and memory-based dynamic risk measures through -expectations, illustrating path-dependent pricing and risk management with memory effects. Overall, the results deliver existence, representation, and applications of path-dependent PDEs in a delayed, non-Markovian setting, expanding the toolkit for pricing and risk assessment in finance with memory.

Abstract

We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation: where , and \[ \mathcal{L}u(t,φ):=\langle b(t,φ),\partial_{x}u(t,φ)\rangle+\dfrac {1}{2}\mathrm{Tr}\big[σ(t,φ)σ^{\ast}(t,φ)\partial_{xx} ^{2}u(t,φ)\big]. \] The result is obtained by a stochastic approach. In particular we prove a new type of nonlinear Feynman-Kac representation formula associated to a backward stochastic differential equation with time-delayed generator which is of non-Markovian type. Applications to the large investor problem and risk measures via -expectations are also provided.

Paper Structure

This paper contains 13 sections, 7 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preliminaries
  4. Pathwise derivatives and functional Itô's formula
  5. Path--dependent PDEs
  6. The forward--backward delayed system
  7. The forward path-dependent SDE
  8. The backward delayed SDE
  9. Path--dependent PDE -- proof of the existence theorem
  10. Financial applications
  11. The large investor problem
  12. Risk measures via g-expectations
  13. Appendix: Proofs of Theorem \ref{['theorem 3']} and Theorem \ref{['theorem 2']}

Key Result

Theorem 1

Let $A$ be a $d$--dimensional Itô process, i.e. $A:\left[ 0,T\right] \times \mathbb{\Lambda }\rightarrow \mathbb{R}^{d}$ is a continuous $\mathbb{R}^{d}$--valued semimartingale defined on the probability space $\left( \mathbb{\Lambda },\mathbb{F},\mathbb{P}\right)$ which admits the representation where $b,\sigma$ are progressively measurable stochastic processes such that $\int_{0}^{T}(\left\vert

Theorems & Definitions (23)

  • Theorem 1: Functional Itô's formula
  • Definition 2
  • Remark 3
  • Theorem 4
  • Definition 5
  • Remark 6
  • Theorem 7
  • Remark 8
  • Remark 9
  • Remark 10
  • ...and 13 more