Rough backward SDEs with discontinuous Young drivers

TL;DR

This paper develops a mathematical framework for solving backward stochastic differential equations driven by discontinuous rough paths and Brownian motion, establishing well-posedness, stability, and connections to backward doubly stochastic differential equations.

Contribution

It introduces a novel class of rough backward SDEs with discontinuous drivers, proves their global well-posedness, and links them to backward doubly stochastic differential equations with jump processes.

Findings

Established global existence and uniqueness of solutions.

Proved stability of solutions under rough noise perturbations.

Connected RBSDEs to backward doubly stochastic differential equations.

Abstract

We study solutions to backward differential equations that are driven hybridly by a deterministic discontinuous rough path $W$ of finite $q$-variation for $q \in [1, 2)$ and by Brownian motion $B$. To distinguish between integration of jumps in a forward- or Marcus-sense, we refer to these equations as forward- respectively Marcus-type rough backward stochastic differential equations (RBSDEs). We establish global well-posedness by proving global apriori bounds for solutions and employing fixed-point arguments locally. Furthermore, we lift the RBSDE solution and the driving rough noise to the space of decorated paths endowed with a Skorokhod-type metric and show stability of solutions with respect to perturbations of the rough noise. Finally, we prove well-posedness for a new class of backward doubly stochastic differential equations (BDSDEs), which are jointly driven by a Brownian martingale $B$ and an independent discontinuous stochastic process $L$ of finite $q$-variation. We explain, how our RBSDEs can be understood as conditional solutions to such BDSDEs, conditioned on the information generated by the path of $L$.