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# Finance Mathématique et Probabilités Numériques

Abstract : This document presents my work in mathematical finance and numerical probability since 1998.

The first Chapter deals with theoretical finance. The first part concerns discrete time financial markets with proportional transaction costs. We study two extensions of the usual models: markets with incomplete information, models with non-linear returns. We also discuss the American option pricing problem. The second part presents some advances in the theory of utility maximization in continuous time which includes: transaction costs, random time horizon and non-smooth utility functions. We also provide some multivariate versions of the so-called "bipolar theorem on $L^0_+$" which allows to better understand the real nature of the dual problems in this litterature.

The second chapter contains some works on stochastic targets problems and PDEs. The two first sections make the links with parabolic PDEs. The third one deals with the hedging of European options in models with costs.

In the third chapter, we present some advances on the numerical approximation of BSDEs. We start with the study of a family of representations for conditional expectations which allow a quick computation of a large number of conditional expectations at the same time. This is then used to provide numerical schemes for BSDEs in a Markovian setting. We also present some extensions of these works to BSDEs with jumps or with reflection.

The last chapter deals with the "canadisation" of the time horizon in optimal control.
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https://tel.archives-ouvertes.fr/tel-00116933
Contributor : Bruno Bouchard <>
Submitted on : Wednesday, November 29, 2006 - 9:46:05 AM
Last modification on : Monday, December 14, 2020 - 9:48:36 AM
Long-term archiving on: : Tuesday, April 6, 2010 - 11:33:40 PM

### Identifiers

• HAL Id : tel-00116933, version 1

### Citation

Bruno Bouchard. Finance Mathématique et Probabilités Numériques. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2007. ⟨tel-00116933⟩

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