arXiv subjects

Zakaria Bensaid

Publications and source records attributed to Zakaria Bensaid.

3 recordsLinked to original sources

arXiv2026

Forward recursive aggregator systems and the forward Epstein-Zin recursiveparadigm

We introduce forward recursive aggregator systems in It__o-diffusion markets, bridging the theories of recursive utilities and of forward performance criteria. We derive the associated HJB SPDE and sufficient conditions for consistency and admissibility. We also develop a convex duality theory based on state price densities and identify the density generated by the optimal wealth process. For homothetic aggregators, we characterize a class of forward Epstein--Zin recursive utility processes, derive the optimal investment and consumption policies, and obtain explicit solutions in the Black and Scholes and Heston models.

math.PR↗

arXiv2026

Law-invariant BSDEs and dynamic risk measures: new characterizations

We provide a new characterization of law-invariant backward stochastic differential equations (i.e. BSDEs) with quadratic growth. This answers the open question raised in Xu--Xu--Zhou (2022) on necessary conditions for law-invariance of g-expectations, and extends the analysis to general (possibly non-deterministic) generators. We also introduce and compare several dynamic notions of law-invariance in continuous time, establishing precise relationships among them. As an application, we study dynamic risk measures. For cash-additive, normalized risk measures, we recover and extend to continuous time the Kupper--Schachermayer (2009) characterization obtained in discrete time, showing that law-invariance and strong time-consistency force an entropic structure. We further obtain a new characterization of cash non-additive law-invariant risk measures generated by BSDEs via a time-dependent certainty equivalent representation.

math.OC↗

arXiv2024

Deep learning algorithms for FBSDEs with jumps: Applications to option pricing and a MFG model for smart grids

In this paper, we introduce various machine learning solvers for (coupled) forward-backward systems of stochastic differential equations (FBSDEs) driven by a Brownian motion and a Poisson random measure. We provide a rigorous comparison of the different algorithms and demonstrate their effectiveness in various applications, such as cases derived from pricing with jumps and mean-field games. In particular, we show the efficiency of the deep-learning algorithms to solve a coupled multi-dimensional FBSDE system driven by a time-inhomogeneous jump process with stochastic intensity, which describes the Nash equilibria for a specific mean-field game (MFG) problem for which we also provide the complete theoretical resolution. More precisely, we develop an extension of the MFG model for smart grids introduced in Alasseur, Campi, Dumitrescu and Zeng (Annals of Operations Research, 2023) to the case when the random jump times correspond to the jump times of a doubly Poisson process. We first provide an existence result of an equilibria and derive its semi-explicit characterization in terms of a system of FBSDEs in the linear-quadratic setting. We then compare the MFG solution to the optimal strategy of a central planner and provide several numerical illustrations using the deep-learning solvers presented in the first part of the paper.

math.NA↗