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2015 | OriginalPaper | Buchkapitel

8. Random walk approximants of mixed and time-changed Lévy processes

verfasst von : Sabir Umarov

Erschienen in: Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Verlag: Springer International Publishing

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Abstract

Random walks are used to model various random processes in different fields. In this chapter we are only interested in random walks as approximating processes of some basic driving processes of stochastic differential equations discussed in the previous chapter. There is a vast literature (see, e.g., [GK54, Don52, Bil99, Taq75, GM98-1, GM01, MS01]) devoted to approximation of various basic stochastic processes like Brownian motion, fractional Brownian motion, Lévy processes, and their time-changed counterparts. In the context of approximation, the question in what sense a random walk approximates (or converges to) an associated stochastic process becomes important. We will be interested only in the convergence in the sense of finite-dimensional distributions, which is equivalent to the locally uniform convergence of corresponding characteristic functions (see, e.g., [Bil99]).

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Vorheriges Kapitel Fractional order Fokker-Planck-Kolmogorov equations and associated stochastic processes

Nächstes Kapitel Complex ΨDOSS and systems of complex differential equations

For the dimension in this chapter we use the letter d, since n is overloaded.

The Gaussian density function with mean 0 and correlation matrix I evolving in time.

The set (0, 2) can be replaced by (0, 2], but this requires an additional care (see [US06]).

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Titel: Random walk approximants of mixed and time-changed Lévy processes
verfasst von: Sabir Umarov
Verlag: Springer International Publishing
Buch: Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols
Print ISBN: 978-3-319-20770-4

Electronic ISBN: 978-3-319-20771-1

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-3-319-20771-1_8

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