{{varvec{L}}}^{{varvec{p}}}-Solutions and Comparison Results for L\xe9vy-Driven Backward Stochastic Differential Equations in a Monotonic, General Growth Setting

Stefan Kremsner,Alexander Steinicke

doi:10.1007/s10959-020-01056-3

Abstract

We present a unified approach to L^p-solutions (p > 1) of multidimensional backward stochastic differential equations (BSDEs) driven by Lévy processes and more general filtrations. New existence, uniqueness and comparison results are obtained. The generator functions obey a time-dependent extended monotonicity (Osgood) condition in the y-variable and have general growth in y. Within this setting, the results generalize those of Royer, Yin and Mao, Yao, Kruse and Popier, and Geiss and Steinicke.

Highlights

The existence and uniqueness of solutions to a backward stochastic differential equation (BSDE) have been extensively investigated in many, and various chosen settings, partly due to certain applications in practice and partly for theoretically interesting reasons. We both unify and simplify the approach for a general backward stochastic differential equations (BSDEs) framework driven by a Lévy process with a straightforward extension to more general filtrations
We show new comparison results and relax the assumptions known so far for guaranteeing unique L p-solutions, p > 1, to a BSDE with terminal
Recent and most relevant for the present paper are the results by Kruse and Popier [13] considering L p-solutions for BSDEs driven by Brownian motion, a Poisson random measure and an additional martingale term under a monotonicity condition

Summary

Introduction

The existence and uniqueness of solutions to a backward stochastic differential equation (BSDE) have been extensively investigated in many, and various chosen settings, partly due to certain applications in practice and partly for theoretically interesting reasons. We both unify and simplify the approach for a general BSDE framework driven by a Lévy process with a straightforward extension to more general filtrations. We show new comparison results and relax the assumptions known so far for guaranteeing unique L p-solutions, p > 1, to a BSDE with terminal. Journal of Theoretical Probability condition ξ and generator f that satisfies a monotonicity condition. An L p-solution is a triplet of processes (Y , Z , U ) from suitable L p-spaces The BSDE (1) itself will be denoted by (ξ, f )

Related Works

Main Contribution

Structure of the Paper

Setting

Notation

Lévy Process with Finite Measure

Main Theorem

A Priori Estimates and Stability

Proof of the Main Theorem 1

Comparison Results

Inequalities