Stochastic Processes I - MIT OpenCourseWare

Lecture 5 : Stochastic Processes I1 Stochastic processA Stochastic process is a collection of random variables indexed by alternate view is that it is a probability distribution over a spaceof paths; this path often describes the evolution of some random value, orsystem, over time. In a deterministic process, there is a fixed trajectory(path) that the process follows, but in a Stochastic process, we do not knowapriori which path we will be given. One should not regard this as havingno information of the path since the information on the path is given bythe probability distribution. For example, if the probability distribution isgiven as one path having probability one, then this is equivalent to having adeterministic process. Also, it is often interpreted that the process evolvesover time. However, from the formal mathematical point of view, a betterpicture to have in mind is that we have some underlying (unknown) path,and are observing only the initial segment of this example, the functionf:R 0 Rgivenbyf(t) =tis a determin-isticprocess, but a random function f:RR 0 givenbyf(t) =twithprobability1/2 andf(t) = twith probability 1/2 is a Stochastic is a rather degernerate example and we will later see more examples ofs

Lecture 5 : Stochastic Processes I 1 Stochastic process A stochastic process is a collection of random variables indexed by time. An alternate view is that it is a probability distribution over a space of paths; this path often describes the evolution of some random value, or system, over time. In a deterministic process, there is a xed trajectory

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