Stochastic process

Summary

• As stochastic process is a collection of random variables .
• Stochastic process on Wikipedia: “Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule”
• The index set may be the collection of all integers. The stochastic process is then written as \(\{ \ldots Y_{t-2},Y_{-1},Y_0,Y_1,Y_2 \ldots \}\) or \({\{ Y_t \}}_{t=-∞}^∞\) or simply as \(\{ Y_t \}\) if the index set is given.
• The index set may have a lower bound, for example \(\{ Y_0,Y_1,Y_2 \ldots \}\) . This is a “started process” or “a process started at \(t=0\) ”.
• A stochastic process \(\{ ε_t \}\) is called a white noise process or an innovation process if
• \(ε_t\) is independent of \(ε_s\) for \(s≠t\)
• \(E\left( ε_t|I_{t-1} \right)=0\) for all \(t\)
• \(Var\left( ε_t|I_{t-1} \right)=σ^2\) for all \(t\)
• where \(I_{t-1}\) is short for “all information known at \(t-1\) ”. \(I_{t-1}\) includes \(ε_{t-1},ε_{t-2}, \ldots \) but also all random variables with a time index \(t-1\) or lower.