Abstract
In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Then, we use the variation of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. Finally, we give three examples to demonstrate the applicability of our obtained results.
Highlights
In the last few years, the interest of the scientific community towards fractional calculus has experienced an exceptional boost, and so its applications can be found in a great variety of scientific fields—for example, anomalous diffusion [1,2,3], medicine [4], solute transport [5], random and disordered media [6,7,8], information theory [9], electrical circuits [10], and so on
We investigate the analytical solution of the following multi-time scale fractional stochastic differential equation:
We gave analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions
Summary
In the last few years, the interest of the scientific community towards fractional calculus has experienced an exceptional boost, and so its applications can be found in a great variety of scientific fields—for example, anomalous diffusion [1,2,3], medicine [4], solute transport [5], random and disordered media [6,7,8], information theory [9], electrical circuits [10], and so on. Fractional permutation entropy, sample entropy, and fractional sample entropy play important roles It is well-known that entropy is used to quantify the complexity and uncertainty in financial time series and others. For stochastic differential systems, analytical solutions may provide a useful tool for assessing the influence of some parameters on statistical properties, permutation entropy, fractional permutation entropy, sample entropy, and fractional sample entropy. It is well-known that entropy theory is an important issue because it enables hydraulic and control engineers to quantify uncertainties, determine risk and reliability, estimate parameters, model processes, and design more robust and reliable hydraulic canals control systems.
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