A solution to Skorokhod's embedding for linear Brownian motion and its local time

ISSN： 0081-6906

Source： Studia Scientiarum Mathematicarum Hungarica, Vol.39, Iss.1-2, 2002-12, pp. : 97-127

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

Let $(B _t;tgeq 0)$ be a one dimensional Brownian motion started at $0$, $(L _t;tgeq 0)$ its local time at $0$, and $mu(dx,dl)= u(dl)M(l,dx)$ be a probability measure on $Bbb R times Bbb R _+$ disintegrated with respect to its second marginal $ u (dl)$. Then there exists a stopping time $T$ such that the law of $(B _T,L _T)$ is $mu$ and $(B _{twedge T};tgeq 0)$ is a uniformly integrable martingale if and only if (1) $int _{Bbb R times Bbb R _+} |x|mu(dx,dl) <+infty$, (2) $m(l):=int_{Bbb R}x ^+M(l,dx)=int_{Bbb R}x ^-M(l,dx)<+infty$, $ u(dl) $ a.e. and $ u$ solves (3) $m(l) u (dl)={1 over 2} u[l,+infty[ dl$. Two consequences are given: the first one concerns the distributions of $L _T,T$ running over the stopping times such that $(B _{twedge T};tgeq 0)$ is u.i.. The second one deals with the independence of $B_ T$ and $L _T$: $B_ T$ and $L _T$ are independent iff (1) the distribution of $B _T$ is centered, $b:= bold E[|B_ T|]<+infty$, (2) $L _T$ is exponentially distributed with parameter $1/b$ (i.e., $bold E[L _T]=b$), and $bold E[|B _T&#L _T=l]$ does not depend on $l$.