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Absolute continuity under time shift of trajectories and related stochastic calculus / Jorg-Uwe Lobus.

By: Material type: TextTextSeries: Memoirs of the American Mathematical Society ; v 249, no 1185.Publication details: Providence : American Mathematical Society, 2017.Description: v, 135 pages ; 26 cmISBN:
  • 9781470426033 (alk. paper)
Subject(s): DDC classification:
  • 510 23 Am512
Contents:
1. Introduction, Basic Objects, and Main Result -- 2. Flows and Logarithmic Derivative Relative to X under Orthogonal Projection -- 3. The Density Formula -- 4. Partial Integration -- 5. Relative Compactness of Particle Systems -- Appendix A: Basic Malliavin Calculus for Brownian Motion with Random Initial Data -- References -- Index.
Summary: The text is concerned with a class of two-sided stochastic processes of the form X=W+A. Here W is a two-sided Brownian motion with random initial data at time zero and A\equiv A(W) is a function of W. Elements of the related stochastic calculus are introduced. In particular, the calculus is adjusted to the case when A is a jump process. Absolute continuity of (X,P) under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, m, and on A with A_0=0 we verify \frac{P(dX_{\cdot -t})}{P(dX_\cdot)}=\frac{m(X_{-t}.
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Includes bibliographical references and index.

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

The text is concerned with a class of two-sided stochastic processes of the form X=W+A. Here W is a two-sided Brownian motion with random initial data at time zero and A\equiv A(W) is a function of W. Elements of the related stochastic calculus are introduced. In particular, the calculus is adjusted to the case when A is a jump process. Absolute continuity of (X,P) under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, m, and on A with A_0=0 we verify \frac{P(dX_{\cdot -t})}{P(dX_\cdot)}=\frac{m(X_{-t}.

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