Representing filtration consistent nonlinear expectations as $g$-expectations in general probability spaces

TL;DR

This paper proves that filtration consistent nonlinear expectations in general probability spaces can be represented as solutions to BSDEs with jumps and infinite-dimensional martingale representation, under a domination condition.

Contribution

It extends the representation of nonlinear expectations as BSDE solutions to general spaces with jumps and infinite-dimensional martingales, under a domination assumption.

Findings

Nonlinear expectations can be expressed as BSDE solutions with jumps.

Domination condition ensures the comparison theorem for BSDEs holds.

Generalizes nonlinear Doob-Meyer decomposition to broader contexts.

Abstract

We consider filtration consistent nonlinear expectations in probability spaces satisfying only the usual conditions and separability. Under a domination assumption, we demonstrate that these nonlinear expectations can be expressed as the solutions to Backward Stochastic Differential Equations with Lipschitz continuous drivers, where both the martingale and the driver terms are permitted to jump, and the martingale representation is infinite dimensional. To establish this result, we show that this domination condition is sufficient to guarantee that the comparison theorem for BSDEs will hold, and we generalise the nonlinear Doob-Meyer decomposition of Peng to a general context.