Absolute Continuity under Time Shift of Trajectories and Related Stochastic Calculus

TL;DR

This paper investigates the absolute continuity of two-sided stochastic processes of the form X=W+A under time shifts, developing stochastic calculus for jump processes and establishing conditions for measure change and process compactness.

Contribution

It introduces a stochastic calculus framework for processes with jumps, analyzes measure change under time shifts, and proves partial integration and compactness results.

Findings

Derived explicit Radon-Nikodym derivatives for time-shifted processes

Established partial integration formulas relative to the process generator

Proved relative compactness of sequences of such stochastic processes

Abstract

The paper 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 $(X,P_{\sbnu})$ under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, $m=d\bnu/dx$, and on $A$ with $A_0=0$ we verify % {eqnarray*} \frac{P_{\sbnu}(dX_{\cdot -t})}{P_{\sbnu}(dX_\cdot)}=\frac{m(X_{-t})} {m(X_0)}\cdot\prod_i|\nabla_{W_0}X_{-t}|_i {eqnarray*} % a.e. where the product is taken over all coordinates. Here $\sum_i(\nabla_{W_0}X_{-t})_i$ is the divergence of $X_{-t}$ with respect to the initial position. Crucial for this is the {\it temporal homogeneity} in the sense that $X(W_{\cdot +v}+A_v\1)=X_{\cdot+v}(W)$, $v\in {\Bbb R}$, where $A_v\1$ is the trajectory taking the constant value $A_v (W)$. By means of such a density, partial integration relative to the generator of the process $X$ is established. Relative compactness of sequences of such processes is established.