Stochastic Process Models

Part of the Mathematics for Industry book series (MFI, volume 5)


A stochastic process model describes how an objective “randomly” varies over time and is typically referred to as an infinite-dimensional random variable \(X=X(\omega )=\{X_{t}(\omega )\}_{t\in T}\) whose value is either a continuous or a càdlàg (right-continuous with left-hand limits) function of \(t\in T\subset \mathbb {R}_{+}\). The probabilistic structure of \(X\) can be wonderfully rich, ranging from a piece-wise constant type describing a low-frequency state change to a very rapidly varying type for which we cannot define \(\int f\mathrm{{d}}X\) pathwise as the Riemann-Stieltjes integral even for a smooth \(f\); typical examples are a compound-Poisson process and a Wiener process, respectively. Examples of application fields include signal processing (detection, estimation, etc.), population dynamics, finance, hydrology, radiophysics, and turbulence.


Asymptotic statistics Itô calculus Lévy process  Stochastic differential equation Stochastic process. 


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Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Institute of Mathematics for IndustryKyushu UniversityFukuokaJapan

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