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Brahim El Asri

Publications and source records attributed to Brahim El Asri.

At least 19 recordsLinked to original sources
arXiv2026

Zero sum two-player differential game under three regimes

This paper investigates a two player zero-sum stochastic differential game characterized by three distinct regimes, reflecting the regime switching dynamics within the system. We aim to derive an explicit solution for a number of configurations of the switching system by means of the viscosity solutions approach. In particular, we analyze the associated Hamilton-Jacobi-Bellman-Isaacs equations and examine how the interactions between the different regimes affect the structure of the value function. The proposed framework provides a systematic approach to characterize the optimal strategies of the two players and to obtain explicit representations of the value function un der suitable assumptions. We also illustrate how to derive the value functions in case we know the qualitative structure of switching regions.

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arXiv2025

Explicit solution to an optimal two-player switching game in infinite horizon

In this paper we use viscosity approach to provide an explicit solution to the problem of a two - player switching game. We characterize the switching regions which reduce the switching problem into one of finding a finite number of threshold values in state process that would trigger switchings and then derive an explicit solution to this problem. The state process is a one dimensional Itô diffusion process and switching costs are allowed to be non-positive. We also suggest a numerical procedure to compute the value function in case we know the qualitative structure of switching regions and we illustrate our results by numerical simulations.

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arXiv2024

Numerical approximations of the value of zero-sum stochastic differential impulse controls game in finite horizon

In this paper, we consider a differential stochastic zero-sum game in which two players intervene by adopting impulse controls in a finite time horizon. We provide a numerical solution as an approximation of the value function, which turns out to be the same for both players. While one seeks to maximize the value function, the other seeks to minimize it. Thus we find a single numerical solution for the Nash equilibrium as well as the optimal impulse controls strategy pair for both player based on the classical Policy Iteration (PI) algorithm. Then, we perform a rigorous convergence analysis on the approximation scheme where we prove that it converges to its corresponding viscosity solution as the discretization step approaches zero, and under certain conditions. We showcase our algorithm by implementing a two-player almost analytically solvable game in which the players act through impulse control and compete over the exchange rate.

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arXiv2023

Stochastic optimal switching and systems of variational inequalities with interconnected obstacles

This paper studies a system of $m$ variational inequalities with interconnected obstacles in infinite horizon associated to optimal multi-modes switching problems. Our main result is the existence and uniqueness of a continuous solution in viscosity sense, for that system. The proof of the main result strongly relies on the connection between the systems of variational inequalities and reflected backward stochastic differential equations (RBSDEs) with oblique reflection, which will be characterized through a Feynman-Kac's formula. The main feature of our system of infinite horizon RBSDEs is that its components are interconnected through both the generators and the obstacles.

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arXiv2023

Infinite Horizon Multi-Dimensional BSDE with Oblique Reflection and Switching Problem

This paper studies a system of multi-dimensional reflected backward stochastic differential equations with oblique reflections (RBSDEs for short) in infinite horizon associated to switching problems. The existence and uniqueness of the adapted solution is obtained by using a method based oa combination of penalization, verification method and contraction property.

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arXiv2022

Continuous and Impulse Controls Differential Game in Finite Horizon with Nash-Equilibrium and Application

This paper considers a new class of deterministic finite-time horizon, two-player, zero-sum differential games (DGs) in which the maximizing player is allowed to take continuous and impulse controls whereas the minimizing player is allowed to take impulse control only. We seek to approximate the value function, and to provide a verification theorem for this class of DGs. We first, by means of dynamic programming principle (DPP) in viscosity solution (VS) framework, characterize the value function as the unique VS to the related Hamilton-Jacobi-Bellman-Isaacs (HJBI) double-obstacle equation. Next, we prove that an approximate value function exists, that it is the unique solution to an approximate HJBI double-obstacle equation, and converges locally uniformly towards the value function of each player when the time discretization step goes to zero. Moreover, we provide a verification theorem which characterizes a Nash-equilibrium for the DG control problem considered. Finally, by applying our results, we derive a new continuous-time portfolio optimization model, and we provide related computational algorithms.

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arXiv2022

Deterministic Differential Games in Infinite Horizon Involving Continuous and Impulse Controls

We consider a two-player zero-sum deterministic differential game where each player uses both continuous and impulse controls in infinite-time horizon. We assume that the impulses supposed to be of general term and the costs depend on the state of the system. We use the dynamic programming principle and viscosity solutions approach to show existence and uniqueness of a solution for the Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equations (PDEs) of the game. We prove under Isaacs condition that the upper and lower value functions coincide.

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arXiv2022

One dimensional reflected BSDEs with two barriers under logarithmic growth and applications

In this paper we deal with the problem of the existence and the uniqueness of a solution for one dimensional reflected backward stochastic differential equations with two strictly separated barriers when the generator is allowing a logarithmic growth $(|y||\ln|y||+|z|\sqrt{|\ln|z||})$ in the state variables $y$ and $z$. The terminal value $ξ$ and the obstacle processes $(L_t)_{0\leq t\leq T}$ and $(U_t)_{0\leq t\leq T}$ are $L^p$-integrable for a suitable $p > 2$. The main idea is to use the concept of local solution to construct the global one. As applications, we broaden the class of functions for which mixed zero-sum stochastic differential games admit an optimal strategy and the related double obstacle partial differential equation problem has a unique viscosity solution.

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arXiv2022

Reflected BSDEs with Logarithmic Growth and Applications in Mixed Stochastic Control Problems

In this article we study the existence and the uniqueness of a solution for reflected backward stochastic differential equations in the case when the generator is logarithmic growth in the $z$-variable $(|z|\sqrt{|\ln(|z|)|})$, the terminal value and obstacle are an $L^p$-integrable, for a suitable $p > 2$. To construct the solution we use localization method. We also apply these results to get the existence of an optimal control strategy for the mixed stochastic control problem in finite horizon.

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arXiv2021

Mixed Zero-Sum Stochastic Differential Game and Doubly Reflected BSDEs with a Specific Generator

This paper studies the mixed zero-sum stochastic differential game problem. We allow the functionals and dynamics to be of polynomial growth. The problem is formulated as an extended doubly reflected BSDEs with a specific generator. We show the existence of solution for this doubly reflected BSDEs and we prove the existence of a saddle-point of the game. Moreover, in the Markovian framework we prove that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation.

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arXiv2021

A Zero-Sum Deterministic Impulse Controls Game in Infinite Horizon with a New HJBI QVI

In the present paper, we study a two-player zero-sum deterministic differential game with both players adopting impulse controls, in infinite time horizon, under rather weak assumptions on the cost functions. We prove by means of the dynamic programming principle (DPP) that the lower and upper value functions are continuous and viscosity solutions to the corresponding Hamilton-Jacobi-Bellman-Isaacs (HJBI) quasi-variational inequality (QVI). We define a new HJBI QVI for which, under a proportional property assumption on the maximizer cost, the value functions are the unique viscosity solution. We then prove that the lower and upper value functions coincide.

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arXiv2018

Stochastic differential switching game in infinite horizon

We study a zero-sum stochastic differential switching game in infinite horizon. We prove the existence of the value of the game and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities with bilateral obstacles. We also obtain a verification theorem which provides an optimal strategy of the game. Finally, some numerical examples with two regimes are given.

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arXiv2017

Zero-sum stochastic differential game in finite horizon involving impulse controls

This paper considers the problem of two-player zero-sum stochastic differential game with both players adopting impulse controls in finite horizon under rather weak assumptions on the cost functions ($c$ and $χ$ not decreasing in time). We use the dynamic programming principle and viscosity solutions approach to show existence and uniqueness of a solution for the Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equation (PDE) of the game. We prove that the upper and lower value functions coincide.

math.OC↗
arXiv2016

Viscosity Solutions for a System of PDEs and Optimal Switching

In this paper, we study the $m$-states optimal switching problem in finite horizon, when the switching cost functions are arbitrary and can be positive or negative. This has an economic incentive in terms of central evaluation in cases where such organizations or state grants or financial assistance to power plants that promotes green energy in their production activity or what uses less polluting modes in their production. We show existence for optimal strategy via a verification theorem then we show existence and uniqueness of the value processes by using an approximation scheme. In the markovian framework we show that the value processes can be characterized in terms of deterministic continuous functions of the state of the process. Those latter functions are the unique viscosity solutions for a system of $m$ variational partial differential inequalities with inter-connected obstacles.

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arXiv2013

Minimax Impulse Control Problems in Finite Horizon

We consider the problem of impulse control minimax in finite horizon, when cost functions $(C(t,x,ξ)>0)$. We show existence of value function of the problem. Moreover, the value function is characterized as the unique viscosity solution of an Isaacs quasi-variational inequality. This problem is in relation with an application in mathematical finance.

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arXiv2012

Uniqueness of Viscosity Solutions for Optimal Multi-Modes Switching Problem with Risk of default

In this paper we study the optimal m-states switching problem in finite horizon as well as infinite horizon with risk of default. We allow the switching cost functionals and cost of default to be of polynomial growth and arbitrary. We show uniqueness of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. This system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem with risk of default. This problem is connected with the valuation of a power plant in the energy market.

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arXiv2011

Stochastic Optimal Control and BSDEs with Logarithmic Growth

In this paper, we study the existence of an optimal strategy for the stochastic control of diffusion in general case and a saddle-point for zero-sum stochastic differential games. The problem is formulated as an extended BSDE with logarithmic growth in the $z$-variable and terminal value in some $L^p$ space. We also show the existence and uniqueness of solution of this BSDE.

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