Deplay BSDEs driven by fractional Brownian motion

This paper deals with a class of deplay backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1 2 {\frac{1}{2}} ). In this type of equation, a generator at time t can depend not only on the present but also the past solutions. We essentially establish existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout this paper is the divergence-type integral.

CONSISTENCY OF THE DRIFT PARAMETERS ESTIMATES IN THE FRACTIONAL BROWNIAN DIFFUSION MODEL AND ESTIMATION OF THE HURST PARAMETER BY MAXIMUM LIKELIHOOD METHOD

In this paper, we study the problem of estimating the unknow parameters in a long memory process based on the maximum likelihood method. We consider again a diffusion model involving fractional Brownian motion. Our goal is to study the consistency of the drift parameter estimates depending on the form of the model.

Stability of Hybrid SDEs Driven by fBm

In this paper, the exponential stability of stochastic differential equations driven by multiplicative fractional Brownian motion (fBm) with Markovian switching is investigated. The quasi-linear cases with the Hurst parameter H ∈ (1/2, 1) and linear cases with H ∈ (0, 1/2) and H ∈ (1/2, 1) are all studied in this work. An example is presented as a demonstration.

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

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.

Approximation of SDEs: a stochastic sewing approach

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (Electron J Probab 25:55, 2020. 10.1214/20-EJP442). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler–Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is $$H\in (0,1)$$ H ∈ ( 0 , 1 ) and the drift is $$\mathcal {C}^\alpha $$ C α , $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] and $$\alpha >1-1/(2H)$$ α > 1 - 1 / ( 2 H ) , we show the strong $$L_p$$ L p and almost sure rates of convergence to be $$((1/2+\alpha H)\wedge 1) -\varepsilon $$ ( ( 1 / 2 + α H ) ∧ 1 ) - ε , for any $$\varepsilon >0$$ ε > 0 . Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016. 10.1016/j.spa.2016.02.002). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence $$1/2-\varepsilon $$ 1 / 2 - ε of the Euler–Maruyama scheme for $$\mathcal {C}^\alpha $$ C α drift, for any $$\varepsilon ,\alpha >0$$ ε , α > 0 .

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

Proposal of an estimator of the Hurst parameter for a self-similar process representative of the degree of randomness of the recorded traffic in IEEE 802.3-2005 networks

An hypothesis for the existence of a process with long term memory structure, that represents the independence between the degree of randomness of the traffic generated by the sources and the pattern of traffic stream exhibited by the network is presented, discussed and developed. This methodology is offered as a new and alternative way of approaching the estimation of performance and the design of computer networks ruled by the standard IEEE 802.3-2005.