Well-posedness of Backward Stochastic Differential Equations with General Filtration

TL;DR

This paper introduces a new notion of solution called transposition solution for backward stochastic differential equations with general filtration, enabling well-posedness analysis without relying on the Martingale Representation Theorem.

Contribution

It proposes the transposition solution concept, extending solution analysis to general filtrations and providing a comparison theorem, thus broadening the applicability of BSDE theory.

Findings

Transposition solutions coincide with strong solutions under natural filtration.

Established well-posedness of linear and semilinear BSDEs with general filtration.

Provided a comparison theorem for transposition solutions.

Abstract

This paper is addressed to the well-posedness of some linear and semilinear backward stochastic differential equations with general filtration, without using the Martingale Representation Theorem. The point of our approach is to introduce a new notion of solution, i.e., the transposition solution, which coincides with the usual strong solution when the filtration is natural but it is more flexible for the general filtration than the existing notion of solutions. A comparison theorem for transposition solutions is also presented.