Said Hamadène

A Finite Horizon Optimal Multiple Switching Problem

-1We consider the problem of optimal multiple switching in a finite horizon when the state of the system, including the switching costs, is a general adapted stochastic process. The problem is formulated as an extended impulse control problem and solved using probabilistic tools such as the Snell envelope of processes and reflected backward stochastic differential equations. Finally, when the state of the system is a Markov process, we show that the associated vector of value functions provides a viscosity solution to a system of variational inequalities with interconnected obstacles.

Mixed Zero-Sum Stochastic Differential Game and American Game Options

In this paper we solve the mixed zero-sum stochastic differential game problem in the general case. The main tool is the notion of a local solution of backward stochastic differential equations (BSDEs) with two reflecting barriers. As an application we deal with the American game options.

The Continuous Time Nonzero-Sum Dynkin Game Problem and Application in Game Options

In this paper we study the nonzero-sum Dynkin game in continuous time, which is a two-player noncooperative game on stopping times. We show that it has a Nash equilibrium point for general stochastic processes. As an application, we consider the problem of pricing American game contingent claims by the utility maximization approach.

Backward-forward SDE's and stochastic differential games

In this paper, the first part is concerned with the study of backward-forward stochastic differential equations without the non-degeneracy condition for the forward equation. We show existence and unicity of the solution to such equations under weaker monotonicity assumptions than those of Hu and Peng (1990). In a second part, we apply the results of the first part for studying the problem of existence of open-loop Nash equilibrium points for nonzero sum linear-quadratic stochastic differential games with random coefficients. We show existence, and give their expression, of such points without any limitation of the duration of the game.

Backward stochastic differential equation with local time

In this paper we deal with the following backward stochastic differential equation: where W is a d-dimensional Brownian motion is the symmetric local time of Fat the level a, v is a signed measure on is a -measurable random variable in and is an adapted map from to . If h is continuous with linear growth, we show the existence of a solution (Y,Z) for this backward equation. Some applications of this result, in connection with partial differential equations, and with linear quadratic stochastic control problem, are also given

On a finite horizon Starting and Stopping Problem with Risk of Abandonment

We address the issue of finding a strategy to sustain structural profitability of an investment project, whose production activity depends on the market price of a number of underlying commodities. Depending on the fluctuating prices of these commodities, the activity will either continue until the projects profitability reaches a critical low level at which it is stopped and starts again when it becomes profitable. But, if the structural nonprofitability remains for a while, the investment project will face the risk to be abandoned or be definitely closed. We suggest a general probabilistic set up to model profitability as a function of the market price of a set of commodities, and find the related optimal strategy to sustain it, under the constraint that the project faces the abandonment risk when being nonprofitable under a fixed finite time interval. When the market price dynamics is described by a diffusion process, we show that the optimal strategy is related to viscosity solutions of a system of two variational inequalities with inter-connected obstacles.

Lp-SOLUTIONS FOR REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

This paper deals with the problem of existence and uniqueness of a solution for a backward stochastic differential equation (BSDE for short) with one reflecting barrier in the case when the terminal value, the generator and the obstacle process are Lp-integrable with p ∈ ]1, 2[. To construct the solution we use two methods: penalization and Snell envelope. As an application we broaden the class of functions for which the related obstacle partial differential equation problem has a unique viscosity solution.

Existence of Nash equilibrium points for Markovian non-zero-sum stochastic differential games with unbounded coefficients

This paper is related to non-zero-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.

The multi-player nonzero-sum Dynkin game in discrete time

We study the infinite horizon discrete time N-player nonzero-sum Dynkin game (