Introduction to Stochastic Processes

Any collection of observations \(x_t \in \mathbb {R}\) recorded at fixed times t is called a times series. Times series For example, \(x_t\) may represent a stock price at end of the day t or a log-return. In this case the time series is indexed by discrete time indices and we can call it historical prices or returns. Since nowadays electronic trading is almost continuous in time, we can have a continuous-time series over some time interval \([0,T]\) . Our aim is to build a model to describe the time evolution of such quantities in the future, trying to fit some probabilistic framework to the past behavior of a time series. From now on, we assume \((\Omega ,\mathcal {F},\mathsf {P})\) is a fixed probability space.