Abstract
In this chapter, we present some basic results from the theory of stochastic processes and investigate the properties of some standard continuous-time stochastic processes. In Sect. 1.1, we give the definition of a stochastic process. In Sect. 1.2, we present some properties of stationary stochastic processes. In Sect. 1.3, we introduce Brownian motion and study some of its properties. Various examples of stochastic processes in continuous time are presented in Sect. 1.4. The Karhunen–Loève expansion, one of the most useful tools for representing stochastic processes and random fields, is presented in Sect. 1.5. Further discussion and bibliographical comments are presented in Sect. 1.6. Section 1.7 contains exercises.
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Notes
- 1.
The notation and basic definitions from probability theory that we will use can be found in Appendix B.
- 2.
In fact, all we need is for the stochastic process to be separable. See the discussion in Sect. 1.6.
- 3.
Note, however, that we do not know whether it is nonzero. This requires a separate argument.
- 4.
Think of R as being the inverse of the Laplacian with periodic boundary conditions. In this case, H α coincides with the standard fractional Sobolev space.
- 5.
Observe, however, that Wiener’s theorem refers to a.s. Hölder continuity, whereas the calculation presented in this section is about L 2-continuity.
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Pavliotis, G.A. (2014). Introduction to Stochastic Processes. In: Stochastic Processes and Applications. Texts in Applied Mathematics, vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1323-7_1
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