Random walk approximants of mixed and time-changed Lévy processes

  • Sabir Umarov
Part of the Developments in Mathematics book series (DEVM, volume 41)


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]).


Approximate Random Walks Fractional Brownian Motion Basic Stochastic Processes Finite-dimensional Distributions Continuous Time Random Walk 
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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sabir Umarov
    • 1
  1. 1.Department of MathematicsUniversity of New HavenWest HavenUSA

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