1. Random Walk and Brownian Motion
Society for Industrial and Applied Mathematics eBooks · 2009
1 WHAT IS A STOCHASTIC PROCESS? Denoting by the value of a stock at an nth unit of time, one may represent its (erratic) evolution by a family of random variables indexed by the discrete-time parameter . The number of car accidents in a city during the time interval [0, t] gives rise to a collection of random variables indexed by the continuous-time parameter t. The velocity at a point u in a turbulent wind field provides a family of random variables indexed by a multidimensional spatial parameter u. More generally we make the following definition. Definition 1.1. Given an index set I, a stochastic process indexed by I is a collection of random variables on a probability space (Ω, ℱ, P) taking values in a set S. The set S is called the state space of the process. In the above, one may take, respectively: (i) , ; (ii) , ; (iii) , . For the most part we shall study stochastic processes indexed by a one-dimensional set of real numbers (e.g., time). Here the natural ordering of numbers coincides with the sense of evolution of the process. This order is lost for stochastic processes indexed by a multidimensional parameter; such processes are usually referred to as random fields. The state space S will often be a set of real numbers, finite, countable, (i.e., discrete) or uncountable. However, we also allow for the possibility of vector-valued variables. As a matter of convenience in notation the index set is often suppressed when the context makes it clear. In particular, we often write in place of and in place of .