Absolute Continuity Under Time Shift of Trajectories and Related Stochastic Calculus (eBook)

Joerg-Uwe Loebus (Autor)

eBook Download: PDF

135 Seiten
American Mathematical Society (Verlag)
978-1-4704-4137-1 (ISBN)

Infos

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})}{m(X_0)}/cdot /prod_i/left/nabla_{d,W_0}X_{-t}/right_i $ i.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.

Produkt-Details

• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• ISBN-10: 1-4704-4137-3 / 1470441373
• ISBN-13: 978-1-4704-4137-1 / 9781470441371