Stochastic processes depending on a continuous parameter

Stochastic processes depending on a continuous parameter have been defined in various ways. A definition frequently given is in terms of a physical system or other entity which depends on the parameter / (time) and whose state is specified by the position of a point Q = Q (t) varying in some space in accordance with a given probability law. The probability that Q will be in given point sets at given times is specified, or, if the process is a Markoff process, the probability that Q will be in a given point set at time t+8 (5>0) if it was in a given position at time i is specified. Bachelier (I, II, III) f was the first to study these processes. His work was pioneering, and without any attempt at rigor. Most investigations have studied the particular case of Markoff processes^ or else what are called below differential processes; that is, those in which the changes of Q (t) in non-overlapping intervals are independent in the usual probability sense. If Q represents the state of a physical system, the combination of the physical system and the probability relations is to be taken as the stochastic process. This, however, is not a mathematical definition, but a concretization of an unstated mathematical definition. Sometimes Q (T) is simply described as a function varying in accordance with a given law of probability. ||Khintchine (II) defined a stochastic process as a one-parameter family of chance variables. If Q varies on the x-axis, the probability relations of such a process are determined by specifying the probability of any set of inequalities of the form * Presented to the Society, September 1, 1936; received by the editors August 23, 1936. f Roman numerals refer to the bibliography at the end of the paper. This bibliography does not pretend to completeness. It refers only to those papers on probability which are fairly closely related to this one. t Cf. Hostinsky (I), Khintchine (III, pp. 24-59), Kolmogoroff (IV). Hostinsky has an extended bibliography, including references to papers on the diffusion problem which leads to a study of Markoff processes. Many papers have also been written studying Markoff processes in which Q-Q (t) can have only a finite number of positions. These have not studied the specific difficulties of definition, many of which do not arise in this special case, so specific references will not be given. Bachelier (I, II, III) ; Khintchine (III, pp. 68-75) ; Kolmogoroff (I, II) ; Levy (II, III, and several papers in the Paris Comptes Rendus whose results are given in II) ; Wiener (I, Chapters 9 and 10, and several earlier papers whose results are given in I). || Finetti (I, II). 107