Laurent Denis

A theoretical framework for the pricing of contingent claims in the presence of model uncertainty

The aim of this work is to evaluate the cheapest superreplication price of a general (possibly path-dependent) European contingent claim in a context where the model is uncertain. This setting is a generalization of the uncertain volatility model (UVM) introduced in by Avellaneda, Levy and Paras. The uncertainty is specified by a family of martingale probability measures which may not be dominated. We obtain a partial characterization result and a full characterization which extends Avellaneda, Levy and Paras results in the UVM case.

A criterion of density for solutions of Poisson-driven SDEs

Abstract. We construct a Dirichlet structure related to a Poisson measure on ℝ+×M, where M is a general measured space, with compensator dt⊗dv. We obtain a criterion of density for variables in the domain of the Dirichlet form and we apply it to S.D.E. driven by this Poisson measure.

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.

Dirichlet forms methods for Poisson point measures and Lévy processes : with emphasis on the creation-annihilation techniques

Introduction.- Notations and Basic Analytical Properties.- 1.Reminders on Poisson Random Measures, Levy Processes and Dirichlet Forms.- 2.Dirichlet Forms and (EID).- 3.Construction of the Dirichlet Structure on the Upper Space.- 4.The Lent Particle Formula and Related Formulae.- 5.Sobolev Spaces and Distributions on Poisson Space.- 6.- Space-Time Setting and Processes.- 7.Applications to Stochastic Differential Equations driven by a Random Measure.- 8.Affine Processes, Rates Models.- 9.Non Poissonian Cases.- A.Error Structures.- B.The Co-Area Formula.- References.

Dirichlet Forms for Poisson Measures and Lévy Processes: The Lent Particle Method

We present a new approach to absolute continuity of laws of Poisson functionals. The theoretical framework is that of local Dirichlet forms as a tool for studying probability spaces. The argument gives rise to a new explicit calculus that we present first on some simple examples: it consists in adding a particle and taking it back after computing the gradient. Then we apply the method to SDE’s driven by Poisson measure.

Malliavin calculus for Markov chains using perturbations of time

In this article, we develop a Malliavin calculus associated to a time-continuous Markov chain with finite state space. We apply it to get a criterion of density for solutions of stochastic differential equation involving the Markov chain and also to compute greeks.

Iteration of the Lent Particle Method for Existence of Smooth Densities of Poisson Functionals

In previous works (Bouleau and Denis, J Funct Anal 257:1144–1174, 2009, Probab Theory Relat Fields, 2011) we have introduced a new method called the lent particle method which is an efficient tool to establish existence of densities for Poisson functionals. We now go further and iterate this method in order to prove smoothness of densities. More precisely, we construct Sobolev spaces of any order and prove a Malliavin-type criterion of existence of smooth density. We apply this approach to SDE’s driven by Poisson random measures and also present some non-trivial examples to which our method applies.

The existence and uniqueness result for quasilinear stochastic PDEs with obstacle under weaker integrability conditions

We prove an existence and uniqueness result for quasilinear Stochastic PDEs with Obstacle (in short OSPDE) under a weaker integrability condition on the coefficient and the barrier.

Space-Time Setting and Examples

We are now going to focus on examples involving a Levy process, where the time does play a role. This means that the bottom space is of the form \(X={\mathbb R} _+ \times \Xi \). We will be able to define a filtration and the notion of predictability, and to apply our tools to the setting of stochastic processes with jumps. Let us give more precisely the hypotheses.

Reminders on Poisson Random Measures

This chapter is devoted to the notions which constitute the foundation of our inquiry. They are presented briefly together with references allowing a deeper investigation. Poisson random measures are random distributions of points in abstract spaces widely used in applications in order to represent spatial independence (see for instance references in Neveu Processus Ponctuels, 1997, [270]). Levy processes i.e. processes with independent increments, may be defined through Poisson random measures so that equipping Poisson random measures with Dirichlet forms make this tool available for studying Levy functionals . Next we present the framework adopted in the book and the famous chaos decomposition of the \(L^2\) space of Poisson random measures.