Vlad Bally

Université Paris-Est, Marne-la-Vallée

Malliavin calculus and applications to the 2D space homogenuous Bolzmann equation view abstract Vlad Bally
Malliavin calculus and applications to the 2D space homogenuous Bolzmann equation
We establish some integration by parts formulas of Malliavin type and we use them in order to study the regularity of the law of the solution of non linear jump type stochastic equations driven by Poisson point measures. In particular we study the regularization efect of the semigroup associated to the two dimensional space homogenuous Bolzmann equation. download slides

Mireille Bossy

INRIA Sophia Antipolis

On conditional McKean Langevin processes view abstract Mireille Bossy
On conditional McKean Langevin processes
This talk is based on joint work with Jean Francois Jabir. We construct a McKean nonlinear confined Langevin process aimed to satisfy a mean no-permeability condition at the boundary. This confined process is a first example of solutions to the class of stochastic Lagrangian equations with boundary conditions, issued from the so-called PDF methods for Computational Fluid Dynamics. download slides

Paul Feehan

Rutgers, The State University of New Jersey

American-style options, stochastic volatility, and degenerate parabolic variational inequalities view abstract Paul Feehan
American-style options, stochastic volatility, and degenerate parabolic variational inequalities
Elliptic and parabolic partial differential equations arising in option pricing problems involving the Cox-Ingersoll-Ross or Heston stochastic processes are well-known to be degenerate parabolic. We provide a report on our work on the existence, uniqueness, and regularity questions for variational inequalities involving degenerate parabolic differential operators and applications to American-style option pricing problems for the Heston model. This is joint work with Panagiota Daskalopoulos (Department of Mathematics, Columbia University). download slides

Valentin Konakov

Academy of Science, Moscow

Discrete “parametrix” method and its applications view abstract Valentin Konakov
Discrete “parametrix” method and its applications
We discuss a discrete counterpart to the well known “parametrix” method introduced by E. Levy. Parametrix is a powerful technique appeared in second order partial differential equations about one hundred years ago. It relates to some special representation for fundamental solutions of PDE’s or, in probability terms, for transition densities of stochastic differential equations. This representation helps reduce usual Malliavin calculus smoothness assumptions. We recently launched with co-authors (S.Molchanov, E.Mammen and S.Menozzi) an advanced discrete version of this technique suitable for approximations of PDEs or SDEs. We develop this result for proving new local limit theorems for families of Markov chains weakly converging to diffusions, for proving Edgeworth type expansions, for Euler scheme corresponding to SDEs driven by symmetric stable process, for degenerate diffusions corresponding to Kolmogorov equation, for transport processes in R^d (random walk over ellipsoids in R^d ). download slides

Damien Lamberton

Université Paris-Est, Marne-la-Vallée

American options and integro-differential equations view abstract Damien Lamberton
American options and integro-differential equations
This talk is based on joint work with Mohammed Mikou. We present some results on American option pricing in exponential Lévy models. In particular, we study regularity properties of the price function, which solves a parabolic integro-differential variational inequality, and discuss the behaviour of the exercise boundary close to maturity. download slides

Peter Laurence

Sapienza Università di Roma - Courant Institute, New York

Asymptotics for local-stochastic volatility models view abstract Peter Laurence
Asymptotics for local-stochastic volatility models
We discuss recent work using the heat kernel expansion to obtain asymptotic approximations for call prices and implied volatility in local-stochastic volatility models. Parts of this work are in collaboration with Gatheral, Hsu, Ouyang and Wang and others with Ben Arous. download slides

Luca Lorenzi

Università di Parma

Compactness and invariance properties of evolution operators associated with Kolmogorov operators with unbounded coefficients download abstract download slides

Alessandra Lunardi

Università di Parma

Dirichlet problems for Ornstein-Uhlenbeck operators in Hilbert spaces download abstract download slides

Carlo Marinelli

Università di Bolzano

Semilinear perturbations of Kolmogorov operators, obstacle problems, and optimal stopping view abstract Carlo Marinelli
Semilinear perturbations of Kolmogorov operators, obstacle problems, and optimal stopping
We prove well-posedness for a class of semilinear parabolic equations with discontinuous nonlinearity that are related to the pricing of American options, or more generally to the characterization of the value function in optimal stopping problems. Our setting covers both finite and infinite dimensional situations. (Joint work with Viorel Barbu and Zeev Sobol) download slides

Stéphane Menozzi

Universite de Paris VII, Paris

Density Estimates and Martingale Problem for a Random Noise Propagating through a Chain of Differential Equations view abstract Stéphane Menozzi
Density Estimates and Martingale Problem for a Random Noise Propagating through a Chain of Differential Equations
We will give two sided bounds for the density of the solution of a system of n differential equations of dimension d, the first one being forced by a non-degenerate random noise and the (n-1) other ones being degenerate. The system formed by the n equations satisfies a suitable Hörmander condition. When the coefficients of the system are Lipschitz continuous, we show that the density of the solution satisfies Gaussian bounds with non-diffusive time scales. The proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem. The associated martingale problem will be discussed as well. download slides

Kaj Nyström

Umeå University

Obstacle problems and boundary behaviour of non-negative solutions for operators of Kolmogorov type view abstract Kaj Nyström
Obstacle problems and boundary behaviour of non-negative solutions for operators of Kolmogorov type
In this talk I will describe recent joint results with Chiara Cinti, Andrea Pascucci and Sergio Polidoro concerning obstacle problems for operators of Kolmogorov type and concerning the boundary behaviour of non-negative solutions to operators of Kolmogorov type. These results are part of an on going project which has two purposes. One purpose is to develop a complete theory for the obstacle problem, including in particular a regularity theory for the free boundary, for operators of Kolmogorov type with subsequent applications to classes of American options. The other purpose is to develop a theory concerning the boundary behaviour of non-negative solutions to operators of Kolmogorov type in appropriate domains. The long-term target for this part of the project is to develop a line of research parallel to the line of research concerning the boundary behaviour of non-negative solutions to second order uniformly elliptic parabolic operators completed by the work of Fabes, Safonov and Yuan. download slides

Marco Papi

Engineering School, Università CBM, Roma

Regularity for singular risk-neutral valuation equations download abstract download slides

Enrico Priola

Università di Torino

Pathwise uniqueness for singular SDEs driven by stable processes view abstract Enrico Priola
Pathwise uniqueness for singular SDEs driven by stable processes
We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric α-stable Lévy processes with values in R^d having a bounded and β-Hölder continuous drift term. We assume β > 1 - α/2 and α ∈ [1; 2). The proof requires analytic regularity results for the associated non-local operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property of solutions. download slides

Sandro Salsa

Politecnico di Milano

Regularity in free boundary problems: recent results and open questions view abstract Sandro Salsa
Regularity in free boundary problems: recent results and open questions
We present some recent results on the regularity for elliptic and evolution two-phase free boundary problems. We discuss some questions that remain open. download slides

Henrik Shahgholian

KTH - Royal Institute of Technology, Stockholm

Some problems in Mathematical Finance, with free boundaries view abstract Henrik Shahgholian
Some problems in Mathematical Finance, with free boundaries
In this talk I shall present two problems from finance and economics, that boil down to free boundaries. The first problem, is the well-known model of pricing american option, which is an obstacle problem with Lipschitz obstacle. I shall discuss the case of max-option, with two underlying assets, and the behavior of the value-function and the exercise region close to maturity. The second topic concerns decision under uncertainty. Such problems, widely known as optimal switching problems are well studied from a stochastic point of view. They lead to a system of variational inequalities, with inter-obstacles. Here we shall present the problem from a pde point of view and show regularity of the solution functions. I shall also discuss general existence theory. If time permits I shall also discuss forthcoming work on convertible bonds, and the behavior of the free boundary in such problems. download slides

Denis Talay

INRIA Sophia Antipolis - École Polytechnique Paris Tech

Probabilistic approaches to divergence form operators with a discontinuous coefficient view abstract Denis Talay
Probabilistic approaches to divergence form operators with a discontinuous coefficient
In this lecture we present two recent works on elliptic and parabolic equations driven by divergence form operators with discontinuous coefficients. We establish probabilistic interpretations which are suitable to design stochastic numerical methods and obtain accurate convergence rate estimates. The first work concerns the Poisson-Bolztmann equation in Molecular Dynamics and is co-authored with M. Bossy, N. Champagnat, S. Maire. The second work concerns parabolic problems and is co-authored with M. Martinez. download slides

Speakers

Giuseppina Guatteri

Politecnico di Milano

Weak solutions of forward-backward systems view abstract Giuseppina Guatteri
Weak solutions of forward-backward systems
In this talk we present an existence and uniqueness result for a class of forward- backward stochastic systems with non smooth coefficients. Our approach exploits the connection between such stochastic systems and a class of quasilinear deterministic Pde. download slides

Federica Masiero

Università di Milano Bicocca

Some results on BSDEs and related HJB equations download abstract download slides

Norayr Matevosyan

University of Cambrdige

Monotonicity formulas for operators with variable coefficients download abstract download slides

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