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Absolute Continuity under Time Shift of Trajectories and Related Stochastic Calculus


About this Title

Jörg-Uwe Löbus

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 249, Number 1185
ISBNs: 978-1-4704-2603-3 (print); 978-1-4704-4137-1 (online)
DOI: https://doi.org/10.1090/memo/1185
Published electronically: August 9, 2017
Keywords:Non-linear transformation of measures, anticipative stochastic calculus, Brownian motion, jump processes

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Table of Contents


Chapters

  • Chapter 1. Introduction, Basic Objects, and Main Result
  • Chapter 2. Flows and Logarithmic Derivative Relative to $X$ under Orthogonal Projection
  • Chapter 3. The Density Formula
  • Chapter 4. Partial Integration
  • Chapter 5. Relative Compactness of Particle Systems
  • Appendix A. Basic Malliavin Calculus for Brownian Motion with Random Initial Data

Abstract


The text is concerned with a class of two-sided stochastic processes of the form . Here is a two-sided Brownian motion with random initial data at time zero and is a function of . Elements of the related stochastic calculus are introduced. In particular, the calculus is adjusted to the case when is a jump process. Absolute continuity of under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, , and on with we verifya.e. where the product is taken over all coordinates. Here is the divergence of with respect to the initial position. Crucial for this is the temporal homogeneity of in the sense that , , where is the trajectory taking the constant value .By means of such a density, partial integration relative to a generator type operator of the process is established. Relative compactness of sequences of such processes is established.

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