Mind the jumps: when 2BSDEs meet semi-martingales

TL;DR

This paper develops a unified framework for stochastic control problems involving semi-martingale BSDEs, enabling analysis of controlled diffusions, jump processes, and discrete-time models with non-dominated uncertainty.

Contribution

It introduces an aggregated value process for semi-martingale BSDE control problems, providing a semi-martingale decomposition and a characterization as a second-order BSDE.

Findings

Constructed an aggregated value process for semi-martingale BSDEs.

Derived the semi-martingale decomposition of the value function.

Characterized the value function as a solution to a semi-martingale second-order BSDE.

Abstract

We construct an aggregated version of the value processes associated with stochastic control problems, where the criterion to optimise is given by solutions to semi-martingale backward stochastic differential equations (BSDEs). The results can be applied to control problems where the triplet of semi-martingale characteristics is controlled in a possibly non-dominated case or where uncertainty about the characteristics is present in the optimisation. The construction also provides a time-consistent system of fully nonlinear conditional expectations on the Skorokhod space. We find the semi-martingale decomposition of the value function and characterise it as the solution to a semi-martingale second-order BSDE. The generality we seek allows for the treatment of controlled diffusions, pure-jump processes, and discrete-time processes in a unified setting.