A stochastic process is a collection of random variables <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi><mi>t</mi></msub><mo>:</mo><mi>t</mi><mo>∈</mo><mi>T</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{X_t : t \in T\}</annotation></semantics></math>{Xt​:t∈T} defined on a common probability space and indexed by a set <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>T. The index <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>t usually stands for time. We use stochastic processes to describe random systems that change over time.
Examples of important processes:
<ul>
<li>Brownian motion</li>
<li>Markov chains</li>
<li>Gaussian processes</li>
<li>Poisson processes</li>
</ul>
We will learn how to select the right type of process to model a given situation.
We will then study:
<ul>
<li>ergodicity</li>
<li>stationarity</li>
<li>stochastic integration</li>
</ul>

Stochastic Processes: Introduction