scholarly journals Numerical Implementation of Stochastic Operational Matrix Driven by a Fractional Brownian Motion for Solving a Stochastic Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
R. Ezzati ◽  
M. Khodabin ◽  
Z. Sadati
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Fractional Brownian Motion ◽  
Operational Matrix ◽  
Standard Brownian Motion ◽  
One Dimensional ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Stochastic Linear System

An efficient method to determine a numerical solution of a stochastic differential equation (SDE) driven by fractional Brownian motion (FBM) with Hurst parameterH∈(1/2,1)andnindependent one-dimensional standard Brownian motion (SBM) is proposed. The method is stated via a stochastic operational matrix based on the block pulse functions (BPFs). With using this approach, the SDE is reduced to a stochastic linear system ofmequations andmunknowns. Then, the error analysis is demonstrated by some theorems and defnitions. Finally, the numerical examples demonstrate applicability and accuracy of this method.

scholarly journals Numerical Solution of Nonlinear Stochastic Itô–Volterra Integral Equations Driven by Fractional Brownian Motion Using Block Pulse Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mengting Deng ◽  
Guo Jiang ◽  
Ting Ke
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Integral Equations ◽  
Stochastic Integral ◽  
Operational Matrix ◽  
Volterra Integral Equations ◽  
Hurst Parameter ◽  
Stochastic Integral Equation ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix

This paper presents a valid numerical method to solve nonlinear stochastic Itô–Volterra integral equations (SIVIEs) driven by fractional Brownian motion (FBM) with Hurst parameter H ∈ 1 / 2 , 1 . On the basis of FBM and block pulse functions (BPFs), a new stochastic operational matrix is proposed. The nonlinear stochastic integral equation is converted into a nonlinear algebraic equation by this method. Furthermore, error analysis is given by the pathwise approach. Finally, two numerical examples exhibit the validity and accuracy of the approach.

Large deviations and Berry–Esseen inequalities for estimators in nonhomogeneous diffusion driven by fractional Brownian motion

2020 ◽  
Vol 28 (3) ◽  
pp. 183-196
Author(s):  
Kouacou Tanoh ◽  
Modeste N’zi ◽  
Armel Fabrice Yodé
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Maximum Likelihood ◽  
Large Deviations ◽  
Stochastic Differential Equation ◽  
Fractional Brownian Motion ◽  
Maximum Likelihood Estimator ◽  
Bayes Estimator ◽  
Likelihood Estimator

AbstractWe are interested in bounds on the large deviations probability and Berry–Esseen type inequalities for maximum likelihood estimator and Bayes estimator of the parameter appearing linearly in the drift of nonhomogeneous stochastic differential equation driven by fractional Brownian motion.

Smooth first-passage densities for one-dimensional diffusions

1987 ◽  
Vol 24 (02) ◽  
pp. 370-377 ◽  
Author(s):  
E. J. Pauwels
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Time Scale ◽  
First Passage ◽  
One Dimensional ◽  
Drift Coefficient

The purpose of this paper is to show that smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic differential equation imply the existence and smoothness of a first-passage density. In order to be able to prove this, we shall show that Brownian motion conditioned to first hit a point at a specified time has the same distribution as a Bessel (3)-process with changed time scale.

Admission Control for Multidimensional Workload input with Heavy Tails and Fractional Ornstein-Uhlenbeck Process

2015 ◽  
Vol 47 (2) ◽  
pp. 476-505
Author(s):  
Amarjit Budhiraja ◽  
Vladas Pipiras ◽  
Xiaoming Song
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Fractional Brownian Motion ◽  
Admission Control ◽  
Control Policy ◽  
Uhlenbeck Process ◽  
Control Policies ◽  
Ornstein Uhlenbeck Process ◽  
Admission Control Policy

The infinite source Poisson arrival model with heavy-tailed workload distributions has attracted much attention, especially in the modeling of data packet traffic in communication networks. In particular, it is well known that under suitable assumptions on the source arrival rate, the centered and scaled cumulative workload input process for the underlying processing system can be approximated by fractional Brownian motion. In many applications one is interested in the stabilization of the work inflow to the system by modifying the net input rate, using an appropriate admission control policy. In this paper we study a natural family of admission control policies which keep the associated scaled cumulative workload input asymptotically close to a prespecified linear trajectory, uniformly over time. Under such admission control policies and with natural assumptions on arrival distributions, suitably scaled and centered cumulative workload input processes are shown to converge weakly in the path space to the solution of a d-dimensional stochastic differential equation driven by a Gaussian process. It is shown that the admission control policy achieves moment stabilization in that the second moment of the solution to the stochastic differential equation (averaged over the d-stations) is bounded uniformly for all times. In one special case of control policies, as time approaches ∞, we obtain a fractional version of a stationary Ornstein-Uhlenbeck process that is driven by fractional Brownian motion with Hurst parameter H > ½.

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