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

TL;DR

This paper establishes the existence of viscosity solutions for path-dependent nonlinear Kolmogorov equations using a stochastic approach, introducing a novel nonlinear Feynman-Kac formula linked to BSDEs with time delays, with applications in finance.

Contribution

It introduces a new stochastic method to solve path-dependent nonlinear PDEs by connecting them to BSDEs with time-delayed generators, expanding the theoretical framework.

Findings

Proves existence of viscosity solutions for complex path-dependent PDEs.

Develops a nonlinear Feynman-Kac representation for non-Markovian BSDEs.

Applies results to finance, including large investor problems and risk measures.

Abstract

We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation: \[ \begin{cases} \partial_{t}u(t,\phi)+\mathcal{L}u(t,\phi)+f(t,\phi,u(t,\phi),\partial_{x}u(t,\phi) \sigma(t,\phi),(u(\cdot,\phi))_{t})=0,\;t\in[0,T),\;\phi\in\mathbb{\Lambda}\, ,u(T,\phi)=h(\phi),\;\phi\in\mathbb{\Lambda}, \end{cases} \] where $\mathbb{\Lambda}=\mathcal{C}([0,T];\mathbb{R}^{d})$, $(u(\cdot ,\phi))_{t}:=(u(t+\theta,\phi))_{\theta\in[-\delta,0]}$ and \[ \mathcal{L}u(t,\phi):=\langle b(t,\phi),\partial_{x}u(t,\phi)\rangle+\dfrac {1}{2}\mathrm{Tr}\big[\sigma(t,\phi)\sigma^{\ast}(t,\phi)\partial_{xx} ^{2}u(t,\phi)\big]. \] 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.