Jean-Michel Lasry

Applications of Malliavin calculus to Monte Carlo methods in finance

Abstract. This paper presents an original probabilistic method for the numerical computations of Greeks (i.e. price sensitivities) in finance. Our approach is based on the {\it integration-by-parts} formula, which lies at the core of the theory of variational stochastic calculus, as developed in the Malliavin calculus. The Greeks formulae, both with respect to initial conditions and for smooth perturbations of the local volatility, are provided for general discontinuous path-dependent payoff functionals of multidimensional diffusion processes. We illustrate the results by applying the formula to exotic European options in the framework of the Black and Scholes model. Our method is compared to the Monte Carlo finite difference approach and turns out to be very efficient in the case of discontinuous payoff functionals.

A remark on regularization in Hilbert spaces

We present here a simple method to approximate uniformly in Hilbert spaces uniformly continuous functions byC1,1 functions. This method relies on explicit inf-sup-convolution formulas or equivalently on the solutions of Hamilton-Jacobi equations.

CONVEX VISCOSITY SOLUTIONS AND STATE CONSTRAINTS

Abstract We establish the convexity of a viscosity solution of some general second order fully nonlinear elliptic equation with state constraints boundary conditions. Our method combines a comparison principle with the observation that, under suitable assumptions, the convex envelope of the solution is a supersolution. This property relies on the characterization of the viscosity subject of the convex envelope of a lower semicontinuous coercive function. The equation solved by the conjugate of a convex solution as well as partial convexity are topics we also discuss.

Long time average of mean field games

We consider a model of mean field games system defined on a time interval

Long time average of mean field games with a nonlocal coupling

[0,T]

Partial differential equation models in macroeconomics

and investigate its asymptotic behavior as the horizon

Large investor trading impacts on volatility

T

A system of non-linear functional differential equations arising in an equilibrium model of an economy with borrowing constraints

tends to infinity. We show that the system, rescaled in a suitable way, converges to a stationary ergodic mean field game. The convergence holds with exponential rate and relies on energy estimates and the Hamiltonian structure of the system.

Mean field games

We study the long time average, as the time horizon tends to infinity, of the solution of a mean field game system with a nonlocal coupling. We show an exponential convergence to the solution of the associated stationary ergodic mean field game. Proofs rely on semiconcavity estimates and smoothing properties of the linearized system.

Jeux à champ moyen. I - Le cas stationnaire

The purpose of this article is to get mathematicians interested in studying a number of partial differential equations (PDEs) that naturally arise in macroeconomics. These PDEs come from models designed to study some of the most important questions in economics. At the same time, they are highly interesting for mathematicians because their structure is often quite difficult. We present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research.