[Paper Review] Mean-Field Delayed BSDEs with Jumps

Nacira Agram|arXiv (Cornell University)|Jan 10, 2018

Stochastic processes and financial applications14 references3 citations

TL;DR

This paper establishes existence and uniqueness of solutions to mean-field backward stochastic differential equations with jumps and time-delayed generators, where the generator depends on the past values of the solution process up to a delay constant δ. Using a fixed-point argument in a weighted normed space, the authors prove unique solvability for sufficiently small δ and any finite time horizon T, under Lipschitz conditions on the driver and terminal condition.

ABSTRACT

We establish sufficient conditions for the existence and uniqueness of mean-field backward stochastic differential equations with time delayed generator in the sense that at t, the generator may depend on previous values up to a delay constant δ not on the hole past as in Delong and Imkeller [10], [13]. For sufficiently small delay constant δ and for any finite time horizon, we get a unique solution.

Motivation & Objective

• To extend mean-field backward stochastic differential equations (BSDEs) to include time delays in the generator and jumps.
• To address the challenge of solving BSDEs where the generator depends on past values of the solution process, not just current values.
• To establish sufficient conditions for existence and uniqueness of solutions under delayed and mean-field dependence with jumps.
• To generalize prior work on delayed BSDEs by restricting dependence to a finite delay δ rather than the full past path.
• To provide a framework applicable to finance and stochastic control where agents face delayed information.

Proposed method

• Formulates a mean-field delayed BSDE (MF-DBSDE) driven by Brownian motion and a Lévy process with a compensated Poisson random measure.
• Introduces a weighted normed space using an exponential weight e^{βt} to control growth in the solution process.
• Applies a fixed-point argument in a complete metric space of adapted processes to prove existence and uniqueness.
• Uses Itô's formula on the squared norm of the difference of two solutions, leading to an a priori estimate.
• Applies a generalized Young’s inequality to bound cross-terms in the Itô formula expansion.
• Demonstrates that the contraction mapping condition holds when the delay δ is sufficiently small, ensuring a unique fixed point.

Experimental results

Research questions

• RQ1 Under what conditions does a mean-field delayed BSDE with jumps admit a unique solution when the generator depends on past values of the solution process up to a finite delay δ?
• RQ2 Can the existence and uniqueness of solutions be guaranteed for any finite time horizon T when the delay δ is small enough, even with jump and mean-field components?
• RQ3 How does the inclusion of a Lévy process with a Poisson random measure affect the solvability of delayed mean-field BSDEs compared to the diffusion-only case?
• RQ4 What role does the delay structure (finite vs. full path dependence) play in ensuring solvability of MF-DBSDEs with jumps?
• RQ5 Can the contraction mapping principle be applied to MF-DBSDEs with jumps and finite delay, and under what conditions on δ does it yield a unique solution?

Key findings

• A unique solution exists for the MF-DBSDE with jumps and finite time delay δ when δ is sufficiently small, regardless of the finite time horizon T.
• The solution is guaranteed under Lipschitz conditions on the driver and terminal condition, with the driver depending on the past values of (Y, Z, K) and the law of the solution process.
• The contraction mapping principle is applied via a weighted norm with exponential factor e^{βt}, ensuring convergence of the iterative scheme.
• The critical condition for uniqueness is that the integral of e^{-βr} over [-δ, 0] multiplied by 1/ρ is less than 1, which holds for small δ.
• The method relies on Itô's formula and a generalized Young’s inequality to bound the difference of two solutions, leading to a contraction in the weighted L² space.
• The result holds for the Hilbert space of measures introduced by Agram and Øksendal, allowing for mean-field dependence through the law of the solution process.