Mohamed Mnif

A model of optimal portfolio selection under liquidity risk and price impact

We study a financial model with one risk-free and one risky asset subject to liquidity risk and price impact. In this market, an investor may transfer funds between the two assets at any discrete time. Each purchase or sale policy decision affects the rice of the risky asset and incurs some fixed transaction cost. The objective is to maximize the expected utility from terminal liquidation value over a finite horizon and subject to a solvency constraint. This is formulated as an impulse control problem under state constraints and we characterize the value function as the unique constrained viscosity solution to the associated quasi-variational Hamilton–Jacobi–Bellman inequality.

Optimal risk control and dividend policies under excess of loss reinsurance

We study the optimal reinsurance policy and dividend distribution of an insurance company under excess of loss reinsurance. The objective of the insurer is to maximize the expected discounted dividends. We suppose that in the absence of dividend distribution, the reserve process of the insurance company follows a compound Poisson process. We first prove existence and uniqueness results for this optimization problem by using singular stochastic control methods and the theory of viscosity solutions. We then compute the optimal strategy of reinsurance, the optimal dividend strategy and the value function by solving the associated integro-differential Hamilton–Jacobi–Bellman Variational Inequality numerically.

OPTIMAL MULTIPLE STOPPING AND VALUATION OF SWING OPTIONS IN LÉVY MODELS

In this paper, we extend the results of Carmona and Touzi [6] for an optimal multiple stopping problem to a market where the price process is allowed to jump. We also generalize the problem of valuation swing options to the context of a Levy market. We prove the existence of multiple exercise policies under an additional condition on Snell envelops. This condition emerges naturally in the case of Levy processes. Then, we give a constructive solution for perpetual put swing options when the price process has no negative jumps. We use the Monte Carlo approximation method based on Malliavin calculus in order to solve the finite horizon case. Numerical results are given in the last two sections. We illustrate the theoretical results of the perpetual case and give the numerical solution for the finite horizon case.

Maximization of Recursive Utilities: A Dynamic Maximum Principle Approach

We study a maximization problem from terminal wealth and consumption for a class of robust utility functions introduced in Bordigoni, Matoussi, and Schweizer [A stochastic control approach to a robust utility maximization problem, in Stochastic Analysis and Applications, Abel Symp. 2, F. E. Benth, G. Di Nunno, T. Lindstrom, B. Oksendal, and T. Zhang, eds., Springer, Berlin, 2007, pp. 125-151]. Our method is based on backward stochastic differential equation theory techniques. We prove a dynamic maximum principle for the optimal control. We study the existence and the uniqueness of the consumption-investment strategy which is characterized as the unique solution of a forward-backward system.

A General Optimal Multiple Stopping Problem with an Application to Swing Options

In their paper, Carmona and Touzi [8] studied an optimal multiple stopping time problem in a market where the price process is continuous. In this article, we generalize their results when the price process is allowed to jump. Also, we generalize the problem associated to the valuation of swing options to the context of jump diffusion processes. We relate our problem to a sequence of ordinary stopping time problems. We characterize the value function of each ordinary stopping time problem as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman variational inequality.

Portfolio Optimization with Stochastic Volatilities: A Backward Approach

In this article, we are interested in an investment problem with stochastic volatilities and Portfolio constraints on proportions. We model the risky assets by jump diffusion processes and we consider a power utility function. The objective is to maximize the expected utility from terminal wealth. The value function is known to be a viscosity solution of an integro-differential Hamilton-Jacobi-Bellman (HJB) equation. Thanks to an exponential transformation, we reduce the nonlinearity of the HJB equation to a semi-linear equation. We prove existence of a smooth solution to the latter equation and we state a verification theorem which relates this solution to the value function. We derive the backward stochastic differential equation (BSDE) associated with the semi-linear equation. We propose a numerical scheme for the resolution of the BSDE based on iterative regressions on function bases and Monte-Carlo method. We illustrate an example which shows the importance of this reduction for the optimal portfolio numerical study. Therefore, we compute the optimal investment strategy by solving the associated BSDE.

Optimal contract with moral hazard for Public Private Partnerships

Abstract Public–Private Partnership (PPP) is a contract between a public entity and a consortium, in which the public outsources the construction and the maintenance of an equipment (hospital, university, prison ...). One drawback of this contract is that the public may not be able to observe the effort of the consortium but only its impact on the social welfare of the project. We aim to characterize the optimal contract for a PPP in this setting of asymmetric information between the two parties. This leads to a stochastic control under partial information and it is also related to principal-agent problems with moral hazard. Considering a wider set of information for the public and using martingale arguments in the spirit of Sannikov, the optimization problem can be reduced to a standard stochastic control problem, that is solved numerically. We then prove that for the optimal contract, the effort of the consortium is explicitly characterized. In particular, it is shown that the optimal rent is not a linear function of the effort, contrary to some models of the economic literature on PPP contracts.

Optimal Market Dealing Under Constraints

We consider a market dealer acting as a liquidity provider by continuously setting bid and ask prices for an illiquid asset in a quote-driven market. The market dealer may benefit from the bid–ask spread, but has the obligation to permanently quote both prices while satisfying some liquidity and inventory constraints. The objective is to maximize the expected utility from terminal liquidation value over a finite horizon and subject to the above constraints. We characterize the value function as the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation, and further enrich our study with numerical results. The contributions of our study concern both the modelling aspects and the dynamic structure of the control strategies. Important features and constraints characterizing market making problems are no longer ignored.

OPTIMAL STOCHASTIC CONTROL PROBLEM UNDER MODEL UNCERTAINTY WITH NONENTROPY PENALTY

In this paper, a stochastic control problem under model uncertainty with general penalty term is studied. Two types of penalties are considered. The first one is of type f-divergence penalty treated in the general framework of a continuous filtration. The second one called consistent time penalty studied in the context of a Brownian filtration. In the case of consistent time penalty, we characterize the value process of our stochastic control problem as the unique solution of a class of quadratic backward stochastic differential equation with unbounded terminal condition.

Euler time discretization of backward doubly SDEs and application to semilinear SPDEs

This paper investigates a numerical probabilistic method for the solution of some semilinear stochastic partial differential equations (SPDEs in short). The numerical scheme is based on discrete time approximation for solutions of systems of decoupled forward–backward doubly stochastic differential equations. Under standard assumptions on the parameters, the convergence and the rate of convergence of the numerical scheme is proven. The proof is based on a generalization of the result on the path regularity of the backward equation.