Anis Matoussi

Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison

It is now established that under quite general circumstances, including in models with jumps, the existence of a solution to a reflected BSDE is guaranteed under mild conditions, whereas the existence of a solution to a doubly reflected BSDE is essentially equivalent to the so-called Mokobodski condition. As for uniqueness of solutions, this holds under mild integrability conditions. However, for practical purposes, existence and uniqueness are not enough. In order to further develop these results in Markovian set-ups, one also needs a (simply or doubly) reflected BSDE to be well posed, in the sense that the solution satisfies suitable bound and error estimates, and one further needs a suitable comparison theorem. In this paper, we derive such estimates and comparison results. In the last section, applicability of the results is illustrated with a pricing problem in finance.

The obstacle problem for quasilinear stochastic PDE’s

We prove an existence and uniqueness result for the obstacle problem of quasilinear parabolic stochastic PDEs. The method is based on the probabilistic interpretation of the solution by using the backward doubly stochastic differential equation.

The obstacle Problem for Quasilinear Stochastic PDEs: Analytical approach

We prove existence and uniqueness of the solution of quasilinear stochas- tic PDEs with obstacle. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as pair (u,�) where u is a pre- dictable continuous process which takes values in a proper Sobolev space andis a random regular measure satisfying minimal Skohorod condition.

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.

Empirical Regression Method for Backward Doubly Stochastic Differential Equations

In this paper we design a numerical scheme for approximating backward doubly stochastic differential equations which represent a solution to stochastic partial differential equations. We first use a time discretization and then we decompose the value function on a functions basis. The functions are deterministic and depend only on time-space variables, while decomposition coefficients depend on the external Brownian motion

Probabilistic interpretation for solutions of fully nonlinear stochastic PDEs

B

The obstacle problem for semilinear parabolic partial integro-differential equations

. The coefficients are evaluated through an empirical regression scheme, which is performed conditionally to

OPTIMAL STOCHASTIC CONTROL PROBLEM UNDER MODEL UNCERTAINTY WITH NONENTROPY PENALTY

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

. We establish nonasymptotic error estimates, conditionally to

Numerical computation for backward doubly SDEs with random terminal time

B