H. Mete Soner

The Dynamic Programming Equation for the Problem of Optimal Investment Under Capital Gains Taxes

This paper considers an extension of the Merton optimal investment problem to the case where the risky asset is subject to transaction costs and capital gains taxes. We derive the dynamic programming equation in the sense of constrained viscosity solutions. We next introduce a family of functions

The Problem of Super-replication under Constraints

(V_varepsilon)_{varepsilon>0}

A STOCHASTIC REPRESENTATION FOR THE LEVEL SET EQUATIONS

, which converges to our value function uniformly on compact subsets, and which is characterized as the unique constrained viscosity solution of an approximation of our dynamic programming equation. In particular, this result justifies the numerical results reported in the accompanying paper [I. Ben Tahar, H. M. Soner, and N. Touzi (2005), Modeling Continuous-Time Financial Markets with Capital Gains Taxes, preprint, http://www.cmap.polytechnique.fr/

Small time path behavior of double stochastic integrals and applications to stochastic control

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Stochastic Representations for Nonlinear Parabolic PDEs

touzi/bst06.pdf].

HEDGING UNDER GAMMA CONSTRAINTS BY OPTIMAL STOPPING AND FACE-LIFTING

These notes present an overview of the problem of super-replication under portfolio constraints. We start by examining the duality approach and its limitations. We then concentrate on the direct approach in the Markov case which allows to handle general large investor problems and gamma constraints. In the context of the Black and Scholes model, the main result from the practical view-point is the so-called face-lifting phenomenon of the payoff function.

Dynamic Programming for Stochastic Target Problems and Geometric Flows

ABSTRACT A Feynman-Kac representation is proved for geometric partial differential equations. This representation is in terms of a stochastic target problem. In this problem the controller tries to steer a controlled process into a given target by judicial choices of controls. The sublevel sets of the unique level set solution of the geometric equation is shown to coincide with the reachability sets of the target problem whose target is the sublevel set of the final data.

Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs

We study the small time path behavior of double stochastic integrals of the form R t 0 ( R r 0 b(u)dW(u)) T dW(r), where W is a d-dimensional Brownian motion and b an integrable progressively measurable stochastic process taking values in the set of d£dmatrices. We prove a law of the iterated logarithm that holds for all bounded progressively measurable b and give additional results under continuity assumptions on b. As an application, we discuss a stochastic control problem that arises in the study of the super-replication of a contingent claim under gamma constraints.

A Viscosity Solution Approach to the Asymptotic Analysis of Queueing Systems

Abstract We discuss several different representations of nonlinear parabolic partial differential equations in terms of Markov processes. After a brief introduction of the linear case, different representations for nonlinear equations are discussed. One class of representations is in terms of stochastic control and differential games. An extension to geometric equations is also discussed. All of these representations are through the appropriate expected values of the data. Different type of representations are also available through backward stochastic differential equations. A recent extension to second-order backward stochastic differential equations allow us to represent all fully nonlinear scalar parabolic equations.

A stochastic representation for mean curvature type geometric flows

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.