Stable processes with opposing drifts

Abstract

A strong Markov process, $W\sp0$, is constructed by a natural linking together of two independent stable processes of type ($\alpha,\ \beta\sb1$) and ($\alpha,\ \beta\sb2$). The drift for a stable process X of type ($\alpha,\ \beta$) can be measured by $\beta$ since$$\rm P\sp0(X\sb{t}>0)={1\over2}+{1\over\pi\alpha}\ tan\sp{-1}({-} \beta\ tan({\pi\alpha\over2})).$$Conditions for when $W\sp0$ will hit 0 are determined and asymptotics for $\sigma$, the time it reaches 0, are obtained.We then consider the extension problem for $W\sp0$ which is to describe all, or at least important classes, of processes W, defined for all time, that agree with $W\sp0$ until time $\sigma$. It is customary to require that the extension have no sojourn at 0. Our interest is in scale-invariant extensions since $W\sp0$ is scale-invariant. Extensions of the stable processes ($\alpha,\ \beta$) killed at $\sigma$ are also considered.