Idris Kharroubi

Progressive enlargement of ltrations and Backward SDEs with jumps

This work deals with backward stochastic differential equations (BSDEs for short) with random marked jumps, and their applications to default risk. We show that these BSDEs are linked with Brownian BSDEs through the decomposition of processes with respect to the progressive enlargement of filtrations. We prove that the equations have solutions if the associated Brownian BSDEs have solutions. We also provide a uniqueness theorem for BSDEs with jumps by giving a comparison theorem based on the comparison for Brownian BSDEs. We give in particular some results for quadratic BSDEs. As applications, we study the pricing and the hedging of a European option in a market with a single jump, and the utility maximization problem in an incomplete market with a finite number of jumps.

BSDE representations for optimal switching problems with controlled volatility

This paper provides two different strong BSDE representations for optimal switching problems in the case where the dynamics of the underlying diffusion process depends on the current value of the switching mode. These new representations make use of either one-dimensional constrained BSDEs with jumps or multidimensional BSDEs with oblique reflections, thus extending the framework considered by Hu and Tang [11]. In particular, the numerical resolution of the corresponding switching problem can therefore be treated via the entirely probabilistic schemes presented in [3] or [7].

Discrete-time Approximation of Multidimensional BSDEs with oblique reflections

In this paper, we study the discrete-time approximation of multi-dimensional reflected BSDEs of the type of those presented by Hu and Tang and generalized by Hamadene and Zhang. In comparison to the penalizing approach followed by Hamadene and Jeanblanc or Elie and Kharroubi, we study a more natural scheme based on oblique projections. We provide a control on the error of the algorithm by introducing and studying the notion of multidimensional discretely reflected BSDE. In the particular case where the driver does not depend on the variable

When terminal facelift enforces delta constraints

Z

A decomposition approach for the discrete-time approximation of FBSDEs with a jump

, the error on the grid points is of order

Optimal Management of an Oil Exploitation

\frac{1}{2} - \epsilon

Adding constraints to BSDEs with jumps: an alternative to multidimensional reflections

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Mean-Variance Hedging on Uncertain Time Horizon in a Market with a Jump

\epsilon>0

Time discretization and quantization methods for optimal multiple switching problem

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Comparison theorem for Brownian multidimensional BSDEs via jump processes

This paper deals with the superreplication of non-path-dependent European claims under additional convex constraints on the number of shares held in the portfolio. The corresponding superreplication price of a given claim has been widely studied in the literature, and its terminal value, which dominates the claim of interest, is the so-called facelift transform of the claim. We investigate under which conditions the superreplication price and strategy of a large class of claims coincide with the exact replication price and strategy of the facelift transform of this claim. In one dimension, we observe that this property is satisfied for any local volatility model. In any dimension, we exhibit an analytical necessary and sufficient condition for this property, which combines the dynamics of the stock together with the characteristics of the closed convex set of constraints. To obtain this condition, we introduce the notion of first order viability property for linear parabolic PDEs. We investigate in detail several practical cases of interest: multidimensional Black–Scholes model, non-tradable assets, and short-selling restrictions.