scholarly journals Upper bounds for the density of solutions to stochastic differential equations driven by fractional Brownian motions

2014 ◽  
Vol 50 (1) ◽  
pp. 111-135 ◽  
Author(s):  
Fabrice Baudoin ◽  
Cheng Ouyang ◽  
Samy Tindel
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Upper Bounds ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Density Of Solutions

scholarly journals Existence and Stability of Solutions of Fuzzy Fractional Stochastic Differential Equations with Fractional Brownian Motions

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Elhoussain Arhrrabi ◽  
M’hamed Elomari ◽  
Said Melliani ◽  
Lalla Saadia Chadli
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Stability Of Solutions ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Existence And Stability ◽  
Fractional Stochastic Differential Equations

The existence, uniqueness, and stability of solutions to fuzzy fractional stochastic differential equations (FFSDEs) driven by a fractional Brownian motion (fBm) with the Lipschitzian condition are investigated. Finally, we investigate the exponential stability of solutions.

scholarly journals Solutions to BSDEs Driven by Multidimensional Fractional Brownian Motions

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Jie Miao ◽  
Xu Yang
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Conditional Expectation ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equations ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Fixed Point Principle ◽  
Fractional Ito Formula

We study more general backward stochastic differential equations driven by multidimensional fractional Brownian motions. Introducing the concept of the multidimensional fractional (or quasi-) conditional expectation, we study some of its properties. Using the quasi-conditional expectation and multidimensional fractional Itô formula, we obtain the existence and uniqueness of the solutions to BSDEs driven by multidimensional fractional Brownian motions, where a fixed point principle is employed. Finally, solutions to linear fractional backward stochastic differential equations are investigated.

scholarly journals Optimal rate of convergence for two classes of schemes to stochastic differential equations driven by fractional Brownian motions

2020 ◽  
Author(s):  
Jialin Hong ◽  
Chuying Huang ◽  
Xu Wang
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Strong Convergence ◽  
Stochastic Differential Equations ◽  
Numerical Solutions ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Optimal Rate Of Convergence ◽  
Runge Kutta ◽  
Strong Convergence Rate

Abstract This paper investigates numerical schemes for stochastic differential equations driven by multi-dimensional fractional Brownian motions (fBms) with Hurst parameter $H\in (\frac 12,1)$. Based on the continuous dependence of numerical solutions on the driving noises, we propose the order conditions of Runge–Kutta methods for the strong convergence rate $2H-\frac 12$, which is the optimal strong convergence rate for approximating the Lévy area of fBms. We provide an alternative way to analyse the convergence rate of explicit schemes by adding ‘stage values’ such that the schemes are interpreted as Runge–Kutta methods. Taking advantage of this technique the strong convergence rate of simplified step-$N$ Euler schemes is obtained, which gives an answer to a conjecture in Deya et al. (2012) when $H\in (\frac 12,1)$. Numerical experiments verify the theoretical convergence rate.

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