On two numerical problems in applied probability : discretization of Stochastic Differential Equations and optimization of an expectation depending on a parameter

¹ Université Paris-Est, CERMICS, project team MathRisk ENPC-INRIA-UMLV, Ecole des Ponts, 6-8 avenue Blaise Pascal, 77455 Marne-la-vallée, France.
e-mail: alfonsi@cermics.enpc.fr
² Université Paris-Est, CERMICS, project team MathRisk ENPC-INRIA-UMLV, Ecole des Ponts, 6-8 avenue Blaise Pascal, 77455 Marne-la-vallée, France.
e-mail: jourdain@cermics.enpc.fr
³ Laboratoire de Mathématiques Appliquées aux Systèmes, école Centrale Paris, Grande Voie des Vignes, F-92 295 Châtenay-Malabry Cedex, France.
e-mail: sophie.laruelle@ecp.fr
⁴ TOSCA project-team, INRIA Sohpia Antipolis - Mediterrane, France, .
Email: sebastian.niklitschek-soto@inria.fr
⁵ Credit Quantitative Analysis, Citigroup Global Market Limited, Citigroup Center, 33 Canada Square, Canary Wharf, London, United Kingdom.
e-mail: victor.reutenauer@citi.com

Abstract

In the present paper, we first deal with the discretization of stochastic differential equations. We elaborate on the analysis of the weak error of the Euler scheme by Talay and Tubaro 31 to contruct schemes with quicker weak rate of convergence for SDEs corresponding to an infinitesimal generator with smooth coefficients. We also extend this analysis to the case of a discontinuous drift coefficient. In a second part, we present two applications of stochastic gradient algorithms in finance.