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Self-exciting multifractional processes. (English) Zbl 1464.60035

Summary: We propose a new multifractional stochastic process which allows for self-exciting behavior, similar to what can be seen for example in earthquakes and other self-organizing phenomena. The process can be seen as an extension of a multifractional Brownian motion, where the Hurst function is dependent on the past of the process. We define this by means of a stochastic Volterra equation, and we prove existence and uniqueness of this equation, as well as giving bounds on the \(p\)-order moments, for all \(p\geq 1\). We show convergence of an Euler-Maruyama scheme for the process, and also give the rate of convergence, which is dependent on the self-exciting dynamics of the process. Moreover, we discuss various applications of this process, and give examples of different functions to model self-exciting behavior.

MSC:

60G22 Fractional processes, including fractional Brownian motion
60H20 Stochastic integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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