Representation Results for Stopping Times in Jump-with-Drift Processes

Abstract

While preparing the paper Dynkin and Yushkevich [6] 39 years ago, we discussed with Eugene B. Dynkin two possible approaches to the notion of a “random variable τ independent from the future,” now known under the name “stopping time.” The final adopted definition in terms of σ-algebras N _t generated by evaluations X _s , s ≤ t of the process competed with the desire to work in intuitively more visual terms of sample paths coinciding up to τ. The latter idea obtained an existence in terms of “saturated” concepts in Courrège and Priouret [3] and Meyer [9]; see Proposition 2.1 below.