stochastic process a process that evolves, as time goes by, according to a probabilistic principle rather than a deterministic principle. Such processes are also called random processes, but ‘stochastic’ does not imply complete disorderliness. The principle of evolution governing a stochastic or random process is precise, though probabilistic, in form. For example, suppose some process unfolds in discrete successive stages. And suppose that given any initial sequence of stages, S1, S2, . . . , Sn, there is a precise probability that the next stage Sn+1 will be state S, a precise probability that it will be SH, and so on for all possible continuations of the sequence of states. These probabilities are called transition probabilities. An evolving sequence of this kind is called a discrete-time stochastic process, or discrete-time random process.
A theoretically important special case occurs when transition probabilities depend only on the latest stage in the sequence of stages. When an evolving process has this property it is called a discrete-time Markov process. A simple example of a discrete-time Markov process is the behavior of a person who keeps taking either a step forward or a step back according to whether a coin falls heads or tails; the probabilistic principle of movement is always applied to the person’s most recent position. The successive stages of a stochastic process need not be discrete. If they are continuous, they constitute a ‘continuous-time’ stochastic or random process. The mathematical theory of stochastic processes has many applications in science and technology. The evolution of epidemics, the process of soil erosion, and the spread of cracks in metals have all been given plausible models as stochastic processes, to mention just a few areas of research. See also DETERMINISM , PROBABILITY , RE- GRESSION ANALYSI. T.H.