Papers › Continuous stochastic processes with non-local memory
Continuous stochastic processes with non-local memory
S. S. Melnyk, V. A. Yampol'skii, O. V. Usatenko
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We study the non-Markovian random continuous processes described by the Mori-Zwanzig equation. As a starting point, we use the Markovian Gaussian Ornstein-Uhlenbeck process and introduce an integral memory term depending on the past of the process into expression for the higher-order transition probability function and stochastic differential equation. We show that the proposed processes can be considered as continuous-time interpolations of discrete-time higher-order autoregressive sequences. An equation connecting the memory function (the kernel of integral term) and the two-point correlation function is obtained. A condition for stationarity of the process is established. We suggest a method to generate stationary continuous stochastic processes with prescribed pair correlation function. As illustration, some examples of numerical simulation of the processes with non-local memory are presented.
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