Hurst parameterH Research Articles

Stochastic controls of fractional Brownian motion

Abstract We consider a stochastic control problem for a non-linear forward-backward stochastic differential equation driven by fractional Brownian motion, with Hurst parameter H ∈ ( 0 , 1 ) {H\in(0,1)} , in the case where the set of the control domain is convex. We provide an estimation of the solution and establish the necessary and sufficient optimality conditions in the form of the stochastic maximum principle. We apply the theory to solve a linear quadratic stochastic control problem.

Stochastic fractional differential inclusion driven by fractional Brownian motion

Abstract In this paper, we prove the existence result for a mild solution of a fractional stochastic evolution inclusion involving the Caputo derivative in the Hilbert space driven by a fractional Brownian motion with the Hurst parameter H > 1 2 {H>\frac{1}{2}} . The results are obtained by using fractional calculation, operator semigroups and the fixed point theorem for multivalued mappings.

Trajectory fitting estimation for stochastic differential equations driven by fractional Brownian motion

Abstract We consider the problem of drift parameter estimation in a stochastic differential equation driven by fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} and small diffusion. The technique that we used is the trajectory fitting method. Strong consistency and asymptotic distribution of the estimator are established as a small diffusion coefficient goes to zero.

On parameter estimation of fractional Ornstein–Uhlenbeck process

Abstract We consider a problem of parameter estimation for the fractional Ornstein–Uhlenbeck model given by the stochastic differential equation d ⁢ X t = - θ ⁢ X t ⁢ d ⁢ t + d ⁢ B t H {dX_{t}=-\theta X_{t}dt+dB_{t}^{H}} , t ≥ 0 {t\geq 0} , where θ > 0 {\theta>0} is an unknown parameter to be estimated and B H {B^{H}} is a fractional Brownian motion with Hurst parameter H ∈ ( 0 , 1 ) {H\in(0,1)} . We provide an estimator for θ, and then we study its strong consistency and asymptotic normality. The main tool in our proofs is the paper [I. Nourdin, D. Nualart and G. Peccati, The Breuer–Major theorem in total variation: Improved rates under minimal regularity, Stochastic Process. Appl. 131 2021, 1–20].

On local path behavior of Surgailis multifractional processes

Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the classical Multifractional Brownian Motion (MBM) { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90’s just by replacing the constant Hurst parameter H {\mathcal {H}} of the well-known Fractional Brownian Motion by a deterministic function H ( t ) {\mathcal {H}}(t) having some smoothness. More than 10 years later, using a different construction method, which basically relied on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two non-classical Gaussian multifactional processes denoted by { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} . In our article, under a rather weak condition on the functional parameter H ( ⋅ ) {\mathcal {H}}(\cdot ) , we show that { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} and { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} as well as { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} only differ by a part which is locally more regular than { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} itself. On one hand this result implies that the two non-classical multifractional processes { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} have exactly the same local path behavior as that of the classical MBM { M ( t ) } t ∈ R \{{\mathcal {M}}(t)\}_{t\in \mathbb {R}} . On the other hand it allows to obtain from discrete realizations of { X ( t ) } t ∈ R \{X(t)\}_{t\in \mathbb {R}} and { Y ( t ) } t ∈ R \{Y(t)\}_{t\in \mathbb {R}} strongly consistent statistical estimators for values of their functional parameter.

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.

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

Abstract 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.

Harnack-type inequality for linear fractional stochastic equations

Abstract Using the coupling method and Girsanov theorem, we prove a Harnack-type inequality for a stochastic differential equation with non-Lipschitz drift and driven by a fractional Brownian motion with Hurst parameter H < 1 2 {H<\frac{1}{2}} . We also investigate this inequality for a stochastic differential equation driven by an additive fractional Brownian sheet.

Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion

Abstract This paper deals with the existence of mild solutions for a class of non-local stochastic integro-differential equations driven by a fractional Brownian motion with Hurst parameter H ∈ 1 2 , 1 H\in \left(\tfrac{1}{2},1\right) . Discussions are based on resolvent operators in the sense of Grimmer, stochastic analysis theory and fixed-point criteria. As a final point, an example is given to illustrate the effectiveness of the obtained theory.

Stochastic partial functional integrodifferential equations driven by a sub-fractional Brownian motion, existence and asymptotic behavior

Abstract This paper deals with the existence and uniqueness of mild solutions to stochastic partial functional integro-differential equations driven by a sub-fractional Brownian motion S Q H ⁢ ( t ) {S_{Q}^{H}(t)} , with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} . By the theory of resolvent operator developed by R. Grimmer (1982) to establish the existence of mild solutions, we give sufficient conditions ensuring the existence, uniqueness and the asymptotic behavior of the mild solutions. An example is provided to illustrate the theory.

Fractional anticipated BSDEs with stochastic Lipschitz coefficients

Abstract In this paper, we deal with an anticipated backward stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H ∈ ( 1 / 2 , 1 ) {H\in(1/2,1)} . We essentially establish existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients and prove a comparison theorem in a specific case.

Stochastic Fractional Heat Equations Driven by Fractional Noises

This paper is concerned with the following stochastically fractional heat equation on(t,x)∈[0,T]×Rddriven by fractional noise:∂u(t,x)/∂t=Dδαu(t,x)+WH(t,x)⋄u(t,x), where the Hurst parameterH=(h0,h1,…,hd)and⋄denotes the Skorokhod integral. A unique solution of that equation in an appropriate Hilbert space is constructed. Moreover, the Lyapunov exponent of the solution is estimated, and the Hölder continuity of the solution on both space and time parameters is discussed. On the other hand, the absolute continuity of the solution is also obtained.

Impact of Correlated Noises on Additive Dynamical Systems

Impact of correlated noises on dynamical systems is investigated by considering Fokker-Planck type equations under the fractional white noise measure, which correspond to stochastic differential equations driven by fractional Brownian motions with the Hurst parameterH>1/2. Firstly, by constructing the fractional white noise framework, one small noise limit theorem is proved, which provides an estimate for the deviation of random solution orbits from the corresponding deterministic orbits. Secondly, numerical experiments are conducted to examine the probability density evolutions of two special dynamical systems, as the Hurst parameterHvaries. Certain behaviors of the probability density functions are observed.

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

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.

Parameter Estimation for Long-Memory Stochastic Volatility at Discrete Observation

Ordinary least squares estimators of variogram parameters in long-memory stochastic volatility are studied in this paper. We use the discrete observations for practical purposes under the assumption that the Hurst parameterH∈(1/2,1)is known. Based on the ordinary least squares method, we obtain both the explicit estimators for drift and diffusion by minimizing the distance function between the variogram and the data periodogram. Furthermore, the resulting estimators are shown to be consistent and to have the asymptotic normality. Numerical examples are also presented to illustrate the performance of our method.

Least Squares Estimation forα-Fractional Bridge with Discrete Observations

We consider a fractional bridge defined asdXt=-α(Xt/(T-t))dt+dBtH, 0≤t<T, whereBHis a fractional Brownian motion of Hurst parameterH>1/2and parameterα>0is unknown. We are interested in the problem of estimating the unknown parameterα>0. Assume that the process is observed at discrete timeti=iΔn, i=0,…,n, andTn=nΔndenotes the length of the “observation window.” We construct a least squares estimatorα^nofαwhich is consistent; namely,α^nconverges toαin probability asn→∞.

Stochastic Volterra Equation Driven by Wiener Process and Fractional Brownian Motion

For a mixed stochastic Volterra equation driven by Wiener process and fractional Brownian motion with Hurst parameterH>1/2, we prove an existence and uniqueness result for this equation under suitable assumptions.

Fractional Black-Scholes Model and Technical Analysis of Stock Price

In the stock market, some popular technical analysis indicators (e.g., Bollinger bands, RSI, ROC, etc.) are widely used to forecast the direction of prices. The validity is shown by observed relative frequency of certain statistics, using the daily (hourly, weekly, etc.) stock prices as samples. However, those samples are not independent. In earlier research, the stationary property and the law of large numbers related to those observations under Black-Scholes stock price model and stochastic volatility model have been discussed. Since the fitness of both Black-Scholes model and short-range dependent process has been questioned, we extend the above results to fractional Black-Scholes model with Hurst parameterH>1/2, under which the stock returns follow a kind of long-range dependent process. We also obtain the rate of convergence.

Abstract Description of Internet Traffic of Generalized Cauchy Type

Self‐similar process with long‐range dependence (LRD), that is, fractional Gaussian noise (fGn) with LRD is a widely used model of Internet traffic. It is indexed by its Hurst parameter HfGn that linearly relates to its fractal dimension DfGn. Note that, on the one hand, the fractal dimension D of traffic measures local self‐similarity. On the other hand, LRD is a global property of traffic, which is characterized by its Hurst parameter H. However, by using fGn, both the self‐similarity and the LRD of traffic are measured by HfGn . Therefore, there is a limitation for fGn to accurately model traffic. Recently, the generalized Cauchy (GC) process was introduced to model traffic with the flexibility to separately measure the fractal dimension DGC and the Hurst parameter HGC of traffic. However, there is a fundamental problem whether or not there exists the generality that the GC model is more conformable with real traffic than single parameter models, such as fGn, irrelevant of traffic traces used in experimental verification. The solution to that problem remains unknown but is desired for model evaluation in traffic theory or for model selection against specific issues, such as queuing analysis relating to the autocorrelation function (ACF) of arrival traffic. The key contribution of this paper is our solution to that fundamental problem (see Theorem 3.17) with the following features in analysis. (i) Set‐valued analysis of the traffic of the fGn type. (ii) Set‐valued analysis of the traffic of the GC type. (iii) Revealing the generality previously mentioned by comparing metrics of the traffic of the fGn type to that of the GC type.

Multifractal analysis of sunspot time series: the effects of the 11-year cycle and Fouriertruncation

Multifractal theory provides an elegant statistical characterization of many complex dynamicalvariations in Nature and engineering. It is conceivable that it may enrich characterizationof the sun’s magnetic activity and its dynamical modeling. Recently, on applying Fouriertruncation to remove the 11-year cycle and carrying out multifractal detrended fluctuationanalysis of the filtered sunspot time series, Movahed et al reported that sunspot data arecharacterized by multifractal scaling laws with the exponent for the second-order moment,h(2), being 1.12. Moreover, they think the filtered sunspot data are like a fractional Brownian motionprocess with anti-persistent long-range correlations characterized by the Hurst parameterH = h(2)−1 = 0.12. By designing an adaptive detrending algorithm and critically assessing the effectiveness ofFourier truncation in eliminating the 11-year cycle, we show that the values of the fractalscaling exponents obtained by Movahed et al are artifacts of the filtering algorithm thatthey used. Instead, sunspot data with the 11-year cycle properly filtered are characterizedby a different type of multifractal with persistent long-range correlations characterized byH≈0.74.