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Absolute Continuity under Time Shift of Trajectories and Related Stochastic Calculus

About this Title

Jörg-Uwe Löbus, Matematiska institutionen, Linköpings universitet, Linköping SE-581 83, Sverige

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
MSC: Primary 60H07; Secondary 60J65, 60J75

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Table of Contents

Chapters

• 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
• 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 $X=W+A$. 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 \[ \frac{P(dX_{⋅-t})}P(dX_{⋅})=\frac{m(X_{-t})}m(X_{0})⋅∏_{i}|∇_{d,W_{0}}X_{-t}|_{i} \] a.e. where the product is taken over all coordinates. Here $\sum _i \left (\nabla _{d,W_0}X_{-t}\right )_i$ is the divergence of $X_{-t}$ with respect to the initial position. Crucial for this is the temporal homogeneity of $X$ in the sense that $X\left (W_{\cdot +v}+A_v \mathbf {1}\right )=X_{\cdot +v}(W)$, $v\in {\mathbb R}$, where $A_v \mathbf {1}$ is the trajectory taking the constant value $A_v(W)$.

By means of such a density, partial integration relative to a generator type operator of the process $X$ is established. Relative compactness of sequences of such processes is established.