# Quadratic BSDEs with random terminal time and elliptic PDEs in infinite dimension

## TL;DR

This paper investigates quadratic backward stochastic differential equations with random, possibly infinite, terminal times, establishing existence, uniqueness, and regularity results, and applies these findings to solve elliptic PDEs in infinite-dimensional spaces.

## Contribution

It extends the theory of quadratic BSDEs to infinite horizon cases with random terminal times and links these to elliptic PDEs in Hilbert spaces, providing new existence and regularity results.

## Key findings

- Existence and uniqueness of solutions for quadratic BSDEs with infinite horizon
- Regular dependence of solutions on parameters in infinite horizon case
- Application to existence and uniqueness of elliptic PDEs in Hilbert spaces

## Abstract

In this paper we study one dimensional backward stochastic differential equations (BSDEs) with random terminal time not necessarily bounded or finite when the generator F(t,Y,Z) has a quadratic growth in Z. We provide existence and uniqueness of a bounded solution of such BSDEs and, in the case of infinite horizon, regular dependence on parameters. The obtained results are then applied to prove existence and uniqueness of a mild solution to elliptic partial differential equations in Hilbert spaces.