Path-dependent backward stochastic Volterra integral equations with jumps, differentiability and duality principle

Ludger Overbeck,Jasmin A L Röder

doi:10.1186/s41546-018-0030-2

Abstract

We study the existence and uniqueness of a solution to path-dependent backward stochastic Volterra integral equations (BSVIEs) with jumps, where path-dependence means the dependence of the free term and generator of a path of a càdlàg process. Furthermore, we prove path-differentiability of such a solution and establish the duality principle between a linear path-dependent forward stochastic Volterra integral equation (FSVIE) with jumps and a linear path-dependent BSVIE with jumps. As a result of the duality principle we get a comparison theorem and derive a class of dynamic coherent risk measures based on path-dependent BSVIEs with jumps.

Highlights

Since Pardoux and Peng provided in Pardoux and Peng (1990) the existence and uniqueness to a non-linear backward stochastic differential equation (BSDE) with Lipschitz continuous generator and measurable terminal condition, this type of equation has attracted enormous attention in probability theory and its applications
We prove path-differentiability of this solution, where we use the functional Itoformula introduced by Dupire (2009) and extended by Cont and Fournie (2010) and Levental et al (2013)
For any γ ∈ D [0, T ], Rd and a fixed t ∈ [0, T ], we are looking for a unique solution (Yγt (·), Zγt (·, ·), Uγt (·, ·, ·)) to the following path-dependent backward stochastic Volterra integral equation (BSVIE) with jumps

Summary

Introduction

Since Pardoux and Peng provided in Pardoux and Peng (1990) the existence and uniqueness to a non-linear backward stochastic differential equation (BSDE) with Lipschitz continuous generator and measurable terminal condition, this type of equation has attracted enormous attention in probability theory and its applications. Ren (2010) studied the existence and uniqueness of a solution to BSVIEs driven by a cylindrical Brownian motion on a separable Hilbert space and a Poisson random measure with a non-Lipschitz coefficient He proved a duality principle between linear FSVIEs with jumps and linear BSVIEs with jumps.

Preliminaries

Well-posedness of path-dependent BSVIEs with jumps

Path-differentiability of path-dependent BSVIEs with jumps

Duality Principle and Comparison Theorem

Path dependent BSVIEs and dynamic risk measures