arXiv subjects

Wissal Sabbagh

Publications and source records attributed to Wissal Sabbagh.

10 recordsLinked to original sources

arXiv2026

Deep numerical schemes for systems of Ergodic BSDEs with applications to regime-switching forward utilities

In this paper, we introduce two neural-network-based numerical schemes for solving systems of coupled ergodic Backward Stochastic Differential Equations (eBSDEs), motivated by the approximation of optimal strategies within the framework of forward utilities in a regime-switching stochastic factor model. Our approach builds on the representation of such models through systems of eBSDEs introduced in [HLT20]. We first establish a link between the solution of the system of ergodic BSDEs and that of an associated multidimensional BSDE with random terminal time, given by the hitting time of the positive recurrent stochastic factor. Building on this representation, we introduce a locally additive deep learning scheme obtained by minimizing aggregated local error terms. We then present a new Deep Galerkin Method (DGM) inspired algorithm that minimizes the residual of the associated ergodic PDE system, relying on a representation of the ergodic cost. Finally, we apply this framework to regime-switching forward utilities in a stochastic factor model. We first derive a general consistency SPDE that characterizes regime-switching forward utilities and retrieve their representation with systems of ergodic BSDEs in the homothetic case. Numerical experiments demonstrate the performance of the proposed methods, with a particular focus on the impact on forward preferences of taking into account regime switches.

math.NA↗

arXiv2026

Optimal Impulse Control for Cyber Risk Management

We explore an optimal impulse control problem wherein an electronic device owner strategically calibrates protection levels against cyber attacks. Utilizing epidemiological compartment models, we qualitatively characterize the dynamics of cyber attacks within the network. We determine the optimal protective measures against effective hacking by formulating and solving a stochastic control problem with optimal switching. We demonstrate that the value function for the cluster owner constitutes a viscosity solution to a system of coupled variational inequalities associated with a fully coupled reflected backward stochastic differential equation (BSDE). Furthermore, we devise a comprehensive algorithm alongside a verification procedure to ascertain the optimal timing for network protection across various cyber attack scenarios. Our findings are illustrated through numerical approximations employing deep Galerkin methods for partial differential equations (PDEs). We visualize the optimal protection strategies in the context of two distinct attack scenarios: (1) a constant cyber attack, (2) an exogenous cyber attack strategy modeled with a Poisson process.

math.OC↗

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 scheme for forward utilities using ergodic BSDEs

In this paper, we present a probabilistic numerical method for a class of forward utilities in a stochastic factor model. For this purpose, we use the representation of dynamic consistent utilities with mean of ergodic Backward Stochastic Differential Equations (eBSDEs) introduced by Liang and Zariphopoulou in [27]. We establish a connection between the solution of the ergodic BSDE and the solution of an associated BSDE with random terminal time $τ$ , defined as the hitting time of the positive recurrent stochastic factor V . The viewpoint based on BSDEs with random horizon yields a new characterization of the ergodic cost $λ$ which is a part of the solution of the eBSDEs. In particular, for a certain class of eBSDEs with quadratic generator, the Cole-Hopf transform leads to a semi-explicit representation of the solution as well as a new expression of the ergodic cost $λ$. The latter can be estimated with Monte Carlo methods. We also propose two new deep learning numerical schemes for eBSDEs, where the ergodic cost $λ$ is optimized according to a loss function at the random horizon $τ$ or taking into account the whole trajectory. Finally, we present numerical results for different examples of eBSDEs and forward utilities along with the associated investment strategies.

math.PR↗

arXiv2023

A new Mertens decomposition of $\mathscr{Y}^{g,ξ}$-submartingale systems. Application to BSDEs with weak constraints at stopping times

We first introduce the concept of $\mathscr{Y}^{g,ξ}$-submartingale systems, where the nonlinear operator $\mathscr{Y}^{g,ξ}$ corresponds to the first component of the solution of a reflected BSDE with generator $g$ and lower obstacle $ξ$. We first show that, in the case of a left-limited right-continuous obstacle, any $\mathscr{Y}^{g,ξ}$-submartingale system can be aggregated by a process which is right-lower semicontinuous. We then prove a \textit{Mertens decomposition}, by using an original approach which does not make use of the standard penalization technique. These results are in particular useful for the treatment of control/stopping game problems and, to the best of our knowledge, they are completely new in the literature. As an application, we introduce a new class of \textit{Backward Stochastic Differential Equations (in short BSDEs) with weak constraints at stopping times}, which are related to the partial hedging of American options. We study the wellposedness of such equations and, using the $\mathscr{Y}^{g,ξ}$-Mertens decomposition, we show that the family of minimal time-$t$-values $Y_t$, with $(Y,Z)$ a supersolution of the BSDE with weak constraints, admits a representation in terms of a reflected backward stochastic differential equation.

math.OC↗

arXiv2018

Systems of reflected quasilinear stochastic PDEs in a convex domain

This paper presents existence and uniqueness results for reflected system of quasilinear stochastic partial differential equations in a convex domain D from Rk. The method is based on the probabilistic interpretation of the solution by using the backward doubly stochastic differential equation. The solution is expressed as a pair (u,ν) where u is a predictable continuous process which takes values in a proper Sobolev space and ν is a random signed regular measure satisfying the minimal Skohorod condition.

math.PR↗

arXiv2017

Large Deviation Principles of Obstacle Problems for Quasilinear Stochastic PDEs

In this paper, we present a sufficient condition for the large deviation criteria of Budhiraja, Dupuis and Maroulas for functionals of Brownian motions. We then establish a large deviation principle for obstacle problems of quasi-linear stochastic partial differential equations. It turns out that the backward stochastic differential equations will play an important role.

math.PR↗

arXiv2016

Probabilistic interpretation for solutions of fully Nonlinear Stochastic PDEs

In this article, we propose a wellposedness theory for a class of second order backward doubly stochastic differential equation (2BDSDE). We prove existence and uniqueness of the solution under a Lipschitz type assumption on the generator, and we investigate the links between our 2BDSDEs and a class of parabolic fully nonLinear Stochastic PDes. Precisely, we show that the Markovian solution of 2BDSDEs provide a probabilistic interpretation of the classical and stochastic viscosity solution of fully nonlinear SPDEs.

math.PR↗

arXiv2016

Numerical Computation for Backward Doubly SDEs with random terminal time

In this article, we are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time tau. The main motivations are giving a probabilistic representation of the Sobolev's solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when tau is the first exit time of a forward SDE from a cylindrical domain. Euler schemes and bounds for the discrete-time approximation error are provided.

math.PR↗

arXiv2014

The obstacle problem for semilinear parabolic partial integro-differential equations

This paper presents a probabilistic interpretation for the weak Sobolev solution of the obstacle problem for semilinear parabolic partial integro-differential equations (PIDEs). The results of Leandre (1985) concerning the homeomorphic property for the solution of SDEs with jumps are used to construct random test functions for the variational equation for such PIDEs. This results in the natural connection with the associated Reflected Backward Stochastic Differential Equations with jumps (RBSDEs), namely Feynman Kac's formula for the solution of the PIDEs. Moreover it gives an application to the pricing and hedging of contingent claims with constraints in the wealth or portfolio processes in financial markets including jumps.

math.PR↗