Some Classes of Continuous-Time Stochastic Processes

Abstract

So far we have studied random processes in discrete time. We now turn to studying random processes in continuous time, for which the unformal definitions of filtration, conditional expectations, martingales, submartingales, supermartingales, stopping times, Markov processes… are essentially the same as in discrete time—but continuous-time entails some technicalities! By the way did you first believe, judging by the names, that a submartingale should be nondecreasing on average and a supermartingale nonincreasing? If not, that’s because you forgot to turn your head and look backward, i.e. we want that a submartingale and a supermartingale attached to the same terminal condition (random variable ξ) are “in the right order” (so sub- under super-, as should be). Did you forget that we have financial derivatives in mind, which are defined in terms of a terminal payoff ξ at a future time point (maturity) T and will be studied later in the book through backward SDEs?