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The Girsanov Theorem Without (So Much) Stochastic Analysis

  • Antoine Lejay
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)

Abstract

In this pedagogical note, we construct the semi-group associated to a stochastic differential equation with a constant diffusion and a Lipschitz drift by composing over small times the semi-groups generated respectively by the Brownian motion and the drift part. Similarly to the interpretation of the Feynman-Kac formula through the Trotter-Kato-Lie formula in which the exponential term appears naturally, we construct by doing so an approximation of the exponential weight of the Girsanov theorem. As this approach only relies on the basic properties of the Gaussian distribution, it provides an alternative explanation of the form of the Girsanov weights without referring to a change of measure nor on stochastic calculus.

Keywords

Girsanov theorem Lie-Trotter-Kato formula Feynman-Kac formula Stochastic differential equation Euler scheme Splitting scheme Flow Heat equation Cameron-Martin theorem 

Notes

Acknowledgements

I wish to thank K. Coulibaly-Pasquier, V. Bally and A. Kohatsu-Higa for some motivating and interesting discussions on this approach. This article is a follow-up of a talk given at the “groupe de travail” of the Probability and Statistics teams of Institut Élie Cartan de Lorraine (Nancy) on the link between the Trotter-Kato-Lie and the Feynman-Kac formula, and I am grateful to the audience for his/her patience.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université de Lorraine, CNRS, Inria, IECLNancyFrance

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