A general non-existence result for linear BSDEs driven by Gaussian processes

TL;DR

This paper demonstrates that linear backward stochastic differential equations driven by certain Gaussian processes, including fractional Brownian motion with Hurst parameter not equal to 1/2, may lack solutions for specific terminal conditions.

Contribution

It establishes a general non-existence result for solutions of linear BSDEs driven by Gaussian non-martingales, extending understanding of their solvability.

Findings

Existence of terminal conditions with no solutions for certain Gaussian-driven BSDEs

Non-martingale Gaussian processes can lead to non-solvability of linear BSDEs

Results apply to fractional Brownian motion with Hurst parameter in (0,1) except 1/2

Abstract

In this paper, we study linear backward stochastic differential equations driven by a class of centered Gaussian non-martingales, including fractional Brownian motion with Hurst parameter $H\in (0,1)\setminus \{\frac12\}$. We show that, for every choice of deterministic coefficient functions, there is a square integrable terminal condition such that the equation has no solution.