On changing time for two-parameter strong martingales: A counterexample (Q1078910)

Brownian motion owes part of its significance to the fact that any continuous martingale may be considered as a Brownian motion running with a different clock. This means that by a suitable time change any continuous martingale can be transformed into Brownian motion. Since its multiparameter analogue, the Wiener sheet, is a strong martingale and the strong martingale property is preserved under transformations by stopping domains, it could only be expected to play a similarly central role among strong martingales. In this paper a strong martingale is constructed which is measurable w.r.t. the Wiener sheet's filtration, but allows no transformation into the Wiener sheet by changing time with stopping domains.