Representation theorems for generators of BSDEs and the extended g-expectations in probability spaces with general filtration

TL;DR

This paper develops representation theorems for BSDE generators in general filtrations, providing new characterizations and extending g-expectations, thereby advancing the theoretical understanding of BSDEs in complex probability spaces.

Contribution

It introduces novel representation theorems for BSDE generators and extends g-expectations to general filtrations, enriching the theoretical framework of stochastic analysis.

Findings

Converse comparison theorem for BSDE generators.

Characterizations of generator properties like homogeneity and convexity.

Extension of g-expectations to general filtrations.

Abstract

In this paper, we establish representation theorems for generators of backward stochastic differential equations (BSDEs in short) in probability spaces with general filtration from the perspective of transposition solutions of BSDEs. As applications, we give a converse comparison theorem for generators of BSDEs and also some characterizations to positive homogeneity, independence of y, subadditivity and convexity of generators of BSDEs. Then, we extend concepts of g-expectations and conditional g-expectations to the probability spaces with general filtration and investigate their properties.