Stopping times with given laws

Given a stochastic process Xt, t b T CR, and s ~ R, then a) iff b): a) For every probability measure p on there is a stopping time T for Xt with law L(T) = p: b) If At is the smallest a-algebra for which Xu are measurable for all u ~ t, then P restricted to At is nonatomic for all t > s. This note began with a question of G. Shiryaev, connected with the following example. Let Wt be a standard Wiener process, t ~ T = Any exponential distribution on ]0,oo] will be shown to be the law of a stopping time. Using this, one can obtain a standard Poisson process Pt from Wt by a nonanticipating transformation, Pt = s Definitions. A probability space (Q,A,P), or A (for P), is nonatomic iff for every A E A and 0 p P(A) there is a A, B E A, with P(B) = p. A stochastic process (here) is a map X: (t,~) >Xt(~)’ t ~ where (Q,A,P) is a complete probability