scholarly journals Selection Properties and Set-Valued Young Integrals of Set-Valued Functions

2020 ◽  
Vol 75 (4) ◽  
Author(s):  
Mariusz Michta ◽  
Jerzy Motyl
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Differential Inclusions ◽  
Stochastic Integrals ◽  
Hölder Continuous ◽  
Stochastic Differential Inclusions ◽  
Different Types ◽  
Convexity Assumptions

AbstractThe paper deals with some selection properties of set-valued functions and different types of set-valued integrals of a Young type. Such integrals are considered for classes of Hölder continuous or with bounded Young p-variation set-valued functions. Two different cases are considered, namely set-valued functions with convex values and without convexity assumptions. The integrals contain as a particular case set-valued stochastic integrals with respect to a fractional Brownian motion, and therefore, their properties are crucial for the investigation of solutions to stochastic differential inclusions driven by a fractional Brownian motion.

scholarly journals Set-Valued Functions of Bounded Generalized Variation and Set-Valued Young Integrals

2020 ◽  
Author(s):  
Mariusz Michta ◽  
Jerzy Motyl
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Differential Inclusions ◽  
Stochastic Integrals ◽  
Stochastic Differential Inclusions ◽  
Generalized Variation

AbstractThe paper deals with some properties of set-valued functions having bounded Riesz p-variation. Set-valued integrals of Young type for such multifunctions are introduced. Selection results and properties of such set-valued integrals are discussed. These integrals contain as a particular case set-valued stochastic integrals with respect to a fractional Brownian motion, and therefore, their properties are crucial for the investigation of solutions to stochastic differential inclusions driven by a fractional Brownian motion.

Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion

2017 ◽  
Vol 17 (02) ◽  
pp. 1750013 ◽  
Author(s):  
Yong Xu ◽  
Bin Pei ◽  
Jiang-Lun Wu
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Stochastic Averaging ◽  
Hurst Parameter ◽  
Stochastic Integrals ◽  
Averaging Principle ◽  
Mean Square ◽  
Asymptotic Results ◽  
Different Types

In this paper, we are concerned with the stochastic averaging principle for stochastic differential equations (SDEs) with non-Lipschitz coefficients driven by fractional Brownian motion (fBm) of the Hurst parameter [Formula: see text]. We define the stochastic integrals with respect to the fBm in the integral formulation of the SDEs as pathwise integrals and we adopt the non-Lipschitz condition proposed by Taniguchi (1992) which is a much weaker condition with wider range of applications. The averaged SDEs are established. We then use their corresponding solutions to approximate the solutions of the original SDEs both in the sense of mean square and of probability. One can find that the similar asymptotic results are suitable for those non-Lipschitz SDEs with fBm under different types of stochastic integrals.

scholarly journals Adapted integral representations of random variables

2015 ◽  
Vol 36 ◽  
pp. 1560004 ◽  
Author(s):  
Georgiy Shevchenko ◽  
Lauri Viitasaari
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stochastic Integral ◽  
Continuous Process ◽  
Random Variables ◽  
Random Variable ◽  
Integral Representations ◽  
Hölder Continuous ◽  
Continuous Processes ◽  
Mixed Fractional Brownian Motion

We study integral representations of random variables with respect to general Hölder continuous processes and with respect to two particular cases; fractional Brownian motion and mixed fractional Brownian motion. We prove that an arbitrary random variable can be represented as an improper integral, and that the stochastic integral can have any distribution. If in addition the random variable is a final value of an adapted Hölder continuous process, then it can be represented as a proper integral. It is also shown that in the particular case of mixed fractional Brownian motion, any adapted random variable can be represented as a proper integral.

scholarly journals Micropulses and Different Types of Brownian Motion

2011 ◽  
Vol 48 (03) ◽  
pp. 792-810 ◽  
Author(s):  
Matthieu Marouby
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Continuous Functions ◽  
Convergence In Distribution ◽  
Bifractional Brownian Motion ◽  
Finite Dimensional ◽  
Multidimensional Indices ◽  
Different Types

In this paper we study sums of micropulses that generate different kinds of processes. Fractional Brownian motion and bifractional Brownian motion are obtained as limit processes. Moreover, we not only prove the convergence of finite-dimensional laws but also, in some cases, convergence in distribution in the space of right-continuous functions with left limits. Finally, we obtain generalizations with multidimensional indices.

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