Searcharxiv⌕ Search

arXiv subjects

Said Hamadène

Publications and source records attributed to Said Hamadène.

At least 19 recordsLinked to original sources
arXiv2026

Viscosity solution of systems of integral-partial differential equations with interconnected obstacles without Monotonicity Conditions and infinite L{é}vy measure

In this paper, we study a system of second-order integral-partial differential equations with interconnected obstacles. A particular case, is the Hamilton-Jacobi-Bellman (HJB for short) system associated with the optimal switching problem in the jump-diffusion model. Getting rid of the monotonicity condition on the generators with respect to the jump component, we construct a continuous viscosity solution of the system, which is unique in the class of continuous bounded functions. The L{é}vy measure $λ$(.) is of infinite activity. The main tool we use is the associated system of reflected backward stochastic differential equations with jumps and interconnected obstacles for which we also study existence and uniqueness of the solution.

math.AP↗
arXiv2026

Optimal stochastic impulse control problem with delay with actions decided at the execution time

In this paper, we consider a class of stochastic impulse control problem when there is a fixed delay $Δ$ between the decision and execution times. The dynamics of the controlled system between two impulses is an arbitrary adapted stochastic process. Unlike the most existing literature, we consider the problem when the impulse sizes are decided at the execution time in both risk-neutral and risk-sensitive cases. This model fits more, in the real life, for some problems such as the pricing of swing options. The horizon T of the problem can be finite or infinite. In each case we show the existence of an optimal strategy. The main tools we use are the notions of reflected Backward Stochastic Differential Equations (BSDEs for short) and the Snell envelope of processes.

math.PR↗
arXiv2025

Discrete time stochastic impulse control with delay

We study a class of infinite-horizon impulse control problems with execution delay in discrete time. Using probabilistic methods, particularly the notion of the Snell envelope of processes, we construct an optimal strategy among all admissible strategies for both risk-neutral and risk-sensitive utility functions. Furthermore, we establish the existence of bounded $ε$-optimal strategies. This framework provides a robust approach to handling execution delays in discrete-time stochastic systems.

math.OC↗
arXiv2024

Viscosity solution of system of integro-partial differential equations with interconnected obstacles of non-local type without Monotonicity Conditions

In this paper, we study a system of second order integro-partial differential equations with interconnected obstacles with non-local terms, related to an optimal switching problem with the jump-diffusion model. Getting rid of the monotonicity condition on the generators with respect to the jump component, we construct a continuous viscosity solution which is unique in the class of functions with polynomial growth. In our study, the main tool is the notion of reflected backward stochastic differential equations with jumps with interconnected obstacles for which we show the existence of a solution.

math.AP↗
arXiv2022

$\varepsilon$-Nash Equilibria of a Multi-player Nonzero-sum Dynkin Game in Discrete Time

We study the infinite horizon discrete time N-player nonzero-sum Dynkin game ($N \geq 2$) with stopping times as strategies (or pure strategies). We prove existence of an $\varepsilon$-Nash equilibrium point for the game by presenting a constructive algorithm. One of the main features is that the payoffs of the players depend on the set of players that stop at the termination stage which is the minimal stage in which at least one player stops. The existence result is extended to the case of a nonzero-sum game with finite horizon. Finally, the algorithm is illustrated by two explicit examples in the specific case of finite horizon.

math.OC↗
arXiv2020

Zero-sum Switching Game, Systems of Reflected Backward SDEs and Parabolic PDEs with bilateral interconnected obstacles

In this paper we study a zero-sum switching game and its verification theorems expressed in terms of either a system of Reflected Backward Stochastic Differential Equations (RBSDEs in short) with bilateral interconnected obstacles or a system of parabolic partial differential equations (PDEs in short) with bilateral interconnected obstacles as well. We show that each one of the systems has a unique solution. Then we show that the game has a value.

math.PR↗
arXiv2020

On the Stochastic Control-Stopping Problem

We study the stochastic control-stopping problem when the data are of polynomial growth. The approach is based on backward stochastic dierential equations (BSDEs for short). The problem turns into the study of a specic reected BSDE with a stochastic Lipschitz coecient for which we show existence and uniqueness of the solution. We then establish its relationship with the value function of the control-stopping problem. The optimal strategy is exhibited. Finally in the Markovian framework we prove that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation.

math.OC↗
arXiv2019

Mean-field reflected backward stochastic differential equations

In this paper, we study a class of reflected backward stochastic differential equations (BSDEs) of mean-field type, where the mean-field interaction in terms of the distribution of the $Y$-component of the solution enters in both the driver and the lower obstacle. We consider in details the case where the lower obstacle is a deterministic function of $(Y,\E[Y])$ and discuss the more general dependence on the distribution of $Y$. Under mild Lipschitz and integrability conditions on the coefficients, we obtain the well-posedness of such a class of equations. Under further monotonicity conditions, we show convergence of the standard penalization scheme to the solution of the equation, which hence satisfies a minimality property. This class of equations is motivated by applications in pricing life insurance contracts with surrender options.

math.PR↗
arXiv2019

On a switching control problem with càdlàg costs

This work addresses a switching control problem under which the cost associated with the changes of regimes is allowed to have discontinuities in time. Our main contribution is to show several characterizations of the optimal cost function as well as the existence of "-optimal control policies. As a by-product, we also study the existence and uniqueness of solutions of a system of backward stochastic differential equations whose barriers (or obstacles) are discontinuous (in fact of càdlàg type) and depend itself on the unknown solution. At the last part of the paper, we study the case when an underlying diffusion is part of the dynamic of the system. In this special case, the optimal payoff becomes a weak solution of the HJB system of PDEs with obstacles which is of quasi-variational type. This paper is somehow a continuation of the papers [8, 17] that consider continuous costs.

math.OC↗
arXiv2018

Viscosity Solutions of Systems of PDEs with Interconnected Obstacles and Switching Problem without Monotonicity Condition

We show the existence and uniqueness of a continuous viscosity solution of a system of partial differential equations (PDEs for short) without assuming the usual monotonicity conditions on the driver function as in Hamadène and Morlais's article \cite{hamadene2013viscosity}. Our method strongly relies on the link between PDEs and reflected backward stochastic differential equations with interconnected obstacles for which we already know that the solution exists and is unique for general drivers.

math.OC↗
arXiv2016

On the Equality of Solutions of Max-Min and Min-Max Systems of Variational Inequalities with Interconnected Bilateral Obstacles

In this paper, we deal with the solutions of systems of PDEs with bilateral inter-connected obstacles of min-max and max-min types. These systems arise naturally in stochastic switching zero-sum game problems. We show that when the switching costs of one side are regular, the solutions of the min-max and max-min systems coincide. Then, this common viscosity solution is related to a multi-dimensional doubly reflected BSDE with bilateral interconnected obstacles. Finally, its relationship with the the values of a zero-sum switching game is studied.

math.PR↗
arXiv2015

Viscosity solutions for second order integro-differential equations without monotonicity conditions: The Probabilistic Approach

In this paper, we establish a new uniqueness result of a (continuous) viscosity solution for some integro-partial differential equation (IPDE in short). The novelty is that we relax the so-called monotonicity assumption on the driver, assumption which is classically assumed in the literature of viscosity solution of equation with a non local term. Our method strongly relies on the link between IPDEs and backward stochastic differential equations (BSDEs in short) with jumps for which we already know that the solution exists and is unique. In the second part of the paper, we deal with the IPDE with obstacle and we obtain similar results.

math.AP↗
arXiv2015

On the bang-bang type Nash equilibrium point for Markovian nonzero-sum stochastic differential game

In this paper, we study a nonzero-sum stochastic differential game in Markovian framework. We show the existence of the Nash equilibrium point which is discontinuous and of bang-bang type under natural conditions. The main tool is the notion of backward stochastic differential equations which, in our case, are multidimensional with discontinuous generators with respect to z component.

math.OC↗
arXiv2014

Risk-sensitive Nonzero-sum Stochastic Differential Game with Unbounded Coefficients

This article is related to risk-sensitive nonzero-sum stochastic differential games in the Markovian framework. This game takes into account the attitudes of the players toward risk and the utility is of exponential form. We show the existence of a Nash equilibrium point for the game when the drift is no longer bounded and only satisfies a linear growth condition. The main tool is the notion of backward stochastic differential equation, which in our case, is multidimensional with continuous generator involving both a quadratic term of Z and a stochastic linear growth component with respect to Z.

math.OC↗
arXiv2014

Existence of Nash Equilibrium Points for Markovian Nonzero-sum Stochastic Differential Games with Unbounded Coefficients

This paper is related to nonzero-sum stochastic differential games in the Markovian framework. We show existence of a Nash equilibrium point for the game when the drift is no longer bounded and only satisfies a linear growth condition. The main tool is the notion of backward stochastic differential equations which, in our case, are multidimensional with continuous coefficient and stochastic linear growth.

math.OC↗
arXiv2011

Viscosity solutions of systems of PDEs with interconnected obstacles and Multi modes switching problems

This paper deals with existence and uniqueness, in viscosity sense, of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. A particular case of this system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem. The switching cost functions are arbitrary. This problem is connected with the valuation of a power plant in the energy market. The main tool is the notion of systems of reflected BSDEs with oblique reflection.

math.OC↗