Nizar Touzi

Law Invariant Risk Measures Have the Fatou Property

S. Kusuoka [K 01, Theorem 4] gave an interesting dual characterizationof law invariant coherent risk measures, satisfying the Fatou property.The latter property was introduced by F. Delbaen [D 02]. In thepresent note we extend Kusuokas characterization in two directions, thefirst one being rather standard, while the second one is somewhat surprising. Firstly we generalize — similarly as M. Fritelli and E. Rossaza Gianin [FG05] — from the notion of coherent risk measures to the more general notion of convex risk measures as introduced by H. F¨ollmer and A. Schied [FS 04]. Secondly — and more importantly — we show that the hypothesis of Fatou property may actually be dropped as it is automatically implied by the hypothesis of law invariance.We also introduce the notion of the Lebesgue property of a convex risk measure, where the inequality in the definition of the Fatou property is replaced by an equality, and give some dual characterizations of this property.

Martingale Representation Theorem for the G-Expectation

This paper considers the nonlinear theory of G-martingales as introduced by Peng in [16, 17]. A martingale representation theorem for this theory is proved by using the techniques and the results established in [20] for the second order stochastic target problems and the second order backward stochastic differential equations. In particular, this representation provides a hedging strategy in a market with an uncertain volatility.

Dual formulation of second order target problems.

This paper provides a new formulation of second order stochastic target problems introduced in [19] by modifying the reference probability so as to allow for difierent scales. This new ingredient enables us to prove a dual formulation of the target problem as the supremum of the solutions of standard backward stochastic difierential equations. In particular, in the Markov case, the dual problem is known to be connected to a fully nonlinear, parabolic partial difierential equation and this connection can be viewed as a stochastic representation for all nonlinear, scalar, second order, parabolic equations with a convex Hessian dependence.

Optimal stochastic control, stochastic target problems, and backward SDE

Preface.- 1. Conditional Expectation and Linear Parabolic PDEs.- 2. Stochastic Control and Dynamic Programming.- 3. Optimal Stopping and Dynamic Programming.- 4. Solving Control Problems by Verification.- 5. Introduction to Viscosity Solutions.- 6. Dynamic Programming Equation in the Viscosity Sense.- 7. Stochastic Target Problems.- 8. Second Order Stochastic Target Problems.- 9. Backward SDEs and Stochastic Control.- 10. Quadratic Backward SDEs.- 11. Probabilistic Numerical Methods for Nonlinear PDEs.- 12. Introduction to Finite Differences Methods.- References.

On viscosity solutions of path dependent PDEs

In this paper we propose a notion of viscosity solutions for path dependent semilinear parabolic PDEs. This can also be viewed as viscosity solutions of non-Markovian Backward SDEs, and thus extends the well known nonlinear Feynman-Kac formula to non-Markovian case. We shall prove the existence, uniqueness, stability, and comparison principle for the viscosity solutions. The key ingredient of our approach is a functional It^ o’s calculus recently introduced by Dupire [6].

Optimal Investment Under Relative Performance Concerns

We consider the problem of optimal investment when agents take into account their relative performance by comparison to their peers. Given N interacting agents, we consider the following optimization problem for agent i,: where is the utility function of agent i, his portfolio, his wealth, the average wealth of his peers, and is the parameter of relative interest for agent i. Together with some mild technical conditions, we assume that the portfolio of each agent i is restricted in some subset. We show existence and uniqueness of a Nash equilibrium in the following situations: We also investigate the limit when the number of agents N goes to infinity. Finally, when the constraints sets are vector spaces, we study the impact of the s on the risk of the market.

Stochastic Target Problems

In this section, we study a special class of stochastic target problems which avoids facing some technical difficulties, but reflects in a transparent way the main ideas and arguments to handle this new class of stochastic control problems.

HEDGING UNDER GAMMA CONSTRAINTS BY OPTIMAL STOPPING AND FACE-LIFTING

A super-replication problem with a gamma constraint, introduced in Soner and Touzi, is studied in the context of the one-dimensional Black and Scholes model. Several representations of the minimal super-hedging cost are obtained using the characterization derived in Cheridito, Soner, and Touzi. It is shown that the upper bound constraint on the gamma implies that the optimal strategy consists in hedging a conveniently face-lifted payoff function. Further an unusual connection between an optimal stopping problem and the lower bound is proved. A formal description of the optimal hedging strategy as a succession of periods of classical Black-Scholes hedging strategy and simple buy-and-hold strategy is also provided.

Dynamic Programming Equation in the Viscosity Sense

We now turn to the stochastic control problem introduced in Sect. 3.1. The chief goal of this section is to use the notion of viscosity solutions in order to relax the smoothness condition on the value function V in the statement of Propositions 3.4 and 3.5. Notice that the following proofs are obtained by slight modification of the corresponding proofs in the smooth case.

Introduction to Viscosity Solutions

Throughout this chapter, we provide the main tools from the theory of viscosity solutions for the purpose of our applications to stochastic control problems. For a deeper presentation, we refer to the excellent overview paper by Crandall et al. [14].