Time-Delayed Generalized BSDEs

TL;DR

This paper establishes existence, uniqueness, and continuity results for time-delayed generalized backward stochastic differential equations (BSDEs) with applications in insurance, extending the theory to small delays and arbitrary delays under certain conditions.

Contribution

It introduces new existence and uniqueness results for time-delayed BSDEs with stochastic integrals, including continuity with respect to the increasing process and solutions for arbitrary delays.

Findings

Proved existence and uniqueness in small delay setting.

Established continuity of solutions with respect to the increasing process.

Applied the theoretical results to an insurance example.

Abstract

We prove the existence and uniqueness of the solution of a BSDE with time-delayed generators in the small delay setting (or equivalently small Lipschitz constant), which employs the Stieltjes integral with respect to an increasing continuous stochastic process. Moreover, we obtain a result of continuity of the solution with regard to the increasing process, assuming only uniform convergence, but not in variation. We also prove the existence in the case of an arbitrary delay by imposing monotonicity and linearity on generators. Lastly, we provide an application of the theoretical framework within an insurance based example.