L. Rogers, und D. Williams. Cambridge Mathematical Library Cambridge University Press, Cambridge, (2000)Foundations,
Reprint of the second (1994) edition.
L. Rogers, und D. Williams. Cambridge Mathematical Library Cambridge University Press, Cambridge, (2000)Itô calculus,
Reprint of the second (1994) edition.
L. Rogers, und J. Pitman. Ann. Probab., 9 (4):
573--582(1981)Let $(G,G)$ and $(H,H)$ be two measurable spaces, and $f$ be a measurable function of $G$ into $H$. If $\X(t)\$ is a homogeneous Markov process with initial distribution $łambda$ and state space $G$, under what conditions is the process $f(X(t))$ Markov? Conditions are well known for $f(X)$ to be Markov for all the initial distributions $łambda$, or invariant $łambda$. This article provides conditions for $f(X)$ to be Markov when there may be no invariant $łambda$, or for some though not all initial $łambda$. The main result is then applied to present a simple proof of Williams' result: For a linear Brownian motion $B(t)$ starting from 0 and with drift $\mu$, the process $Y(t)=2M(t)-B(t)$ (where $M(t)=B(s)\ 0st$) is a diffusion following the same law as the radial part of a three-dimensional Brownian motion starting from the origin and with drift $|\mu|$..