Introduction to Stochastic Processes

When considering technical, economic, ecological, or other problems, in several cases the quantities \(\left \{{X}_{t},\;t \in \mathcal{T}\right \}\) being examined can be regarded as a collection of random variables. This collection describes the changes (usually in time and in space) of considered quantities. If the set \(\mathcal{T}\) is a subset of the set of real numbers, then the set \(\left \{t \in \mathcal{T}\right \}\) can be interpreted as time and we can say that the random quantities X t vary in time. In this case the collection of random variables \(\left \{{X}_{t},\;t \in \mathcal{T}\right \}\) is called a stochastic process. In mathematical modeling of randomly varying quantities in time, one might rely on the highly developed theory of stochastic processes.