scholarly journals On Comparison of the Estimators of the Hurst Index of the Solutions of Stochastic Differential Equations Driven by the Fractional Brownian Motion

Informatica ◽  
2011 ◽  
Vol 22 (1) ◽  
pp. 97-114 ◽  
Author(s):  
Kęstutis Kubilius ◽  
Dmitrij Melichov
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Hurst Index

scholarly journals ɛ-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations

2021 ◽  
Author(s):  
Yi Chen ◽  
Jing Dong ◽  
Hao Ni
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Probability Space ◽  
Index Formula ◽  
Regularity Conditions ◽  
Hurst Index ◽  
Simulation Techniques ◽  
And Diffusion

Consider a fractional Brownian motion (fBM) [Formula: see text] with Hurst index [Formula: see text]. We construct a probability space supporting both BH and a fully simulatable process [Formula: see text] such that[Formula: see text] with probability one for any user-specified error bound [Formula: see text]. When [Formula: see text], we further enhance our error guarantee to the α-Hölder norm for any [Formula: see text]. This enables us to extend our algorithm to the simulation of fBM-driven stochastic differential equations [Formula: see text]. Under mild regularity conditions on the drift and diffusion coefficients of Y, we construct a probability space supporting both Y and a fully simulatable process [Formula: see text] such that[Formula: see text] with probability one. Our algorithms enjoy the tolerance-enforcement feature, under which the error bounds can be updated sequentially in an efficient way. Thus, the algorithms can be readily combined with other advanced simulation techniques to estimate the expectations of functionals of fBMs efficiently.

scholarly journals Fuzzy stochastic differential equations driven by fractional Brownian motion

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Marek T. Malinowski ◽  
M. J. Ebadi
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Approximation Method ◽  
Existence And Uniqueness ◽  
Financial Models ◽  
Stochastic Integral ◽  
Long Range Dependence ◽  
World Systems

AbstractIn this paper, we consider fuzzy stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm). These equations can be applied in hybrid real-world systems, including randomness, fuzziness and long-range dependence. Under some assumptions on the coefficients, we follow an approximation method to the fractional stochastic integral to study the existence and uniqueness of the solutions. As an example, in financial models, we obtain the solution for an equation with linear coefficients.

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