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Time-Delayed Generalized BSDEs

Luca Di Persio, Matteo Garbelli, Lucian Maticiuc, Adrian Zălinescu

TL;DR

The paper tackles time-delayed generalized BSDEs with a Stieltjes integral driven by an increasing process $A$, proving existence and uniqueness in the small-delay regime via a Banach fixed-point argument on weighted spaces, and extending to arbitrary delay in a monotone/linear generator special case. It also establishes stability of solutions under perturbations of terminal data and coefficients using a stochastic Helly-Bray-type result. An insurance application to a variable annuity hedging problem demonstrates the framework's practical relevance for delayed, path-dependent liabilities. Overall, the work provides a rigorous probabilistic treatment of delayed-path dependent BSDEs with Neumann-boundary PDE connections and lays groundwork for future FBSDE and PDE extensions.

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.

Time-Delayed Generalized BSDEs

TL;DR

The paper tackles time-delayed generalized BSDEs with a Stieltjes integral driven by an increasing process , proving existence and uniqueness in the small-delay regime via a Banach fixed-point argument on weighted spaces, and extending to arbitrary delay in a monotone/linear generator special case. It also establishes stability of solutions under perturbations of terminal data and coefficients using a stochastic Helly-Bray-type result. An insurance application to a variable annuity hedging problem demonstrates the framework's practical relevance for delayed, path-dependent liabilities. Overall, the work provides a rigorous probabilistic treatment of delayed-path dependent BSDEs with Neumann-boundary PDE connections and lays groundwork for future FBSDE and PDE extensions.

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.

Paper Structure

This paper contains 7 sections, 5 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Problem setting and notations
  3. Existence and uniqueness
  4. Dependence on parameters
  5. Hedging a stream of payments with time-delayed GBSDE
  6. Conclusions and further developments
  7. Appendix

Key Result

Theorem 4

Let us assume that $\mathrm{(A}_{0}\mathrm{)}$--$\mathrm{(A}_{3}\mathrm{)}$ hold true and $\beta>2\sqrt{2}\tilde{L}$. If conditions $\mathrm{(H}_{1}\mathrm{)}$ and $\mathrm{(H}_{2}\mathrm{)}$ are satisfied then there exists a unique solution $\left( Y,Z\right) \in\mathcal{S}_{\beta }^{2,m}\times\m

Theorems & Definitions (9)

  • Definition 1
  • Remark 2
  • Remark 3
  • Theorem 4
  • Remark 5
  • Proposition 6
  • Theorem 7
  • Proposition 8
  • Lemma 9