Density estimates for the solutions of backward stochastic differential equations driven by Gaussian processes

TL;DR

This paper establishes bounds for the probability densities of solutions to backward stochastic differential equations driven by Gaussian processes, including fractional Brownian motion, using advanced probabilistic and PDE techniques.

Contribution

It introduces novel bounds for densities of BSDE solutions driven by fractional Brownian motion and extends Gaussian estimates to solutions driven by general Gaussian processes.

Findings

Derived non-Gaussian bounds for fractional BSDEs densities.

Obtained Gaussian bounds for solutions driven by Gaussian processes.

Connected solutions to auxiliary Brownian-driven BSDEs for estimation.

Abstract

The aim of this paper is twofold. Firstly, we derive upper and lower non-Gaussian bounds for the densities of the marginal laws of the solutions to backward stochastic differential equations (BSDEs) driven by fractional Brownian motions. Our arguments consist of utilising a relationship between fractional BSDEs and quasilinear partial differential equations of mixed type, together with the profound Nourdin-Viens formula. In the linear case, upper and lower Gaussian bounds for the densities and the tail probabilities of solutions are obtained with simple arguments by their explicit expressions in terms of the quasi-conditional expectation. Secondly, we are concerned with Gaussian estimates for the densities of a BSDE driven by a Gaussian process in the manner that the solution can be established via an auxiliary BSDE driven by a Brownian motion. Using the transfer theorem we succeed in deriving Gaussian estimates for the solutions.