On drift parameter estimation in models with fractional Brownian motion

We consider a stochastic differential equation involving standard and fractional Brownian motion with unknown drift parameter to be estimated. We investigate the standard maximum likelihood estimate of the drift parameter, two non-standard estimates and three estimates for the sequential estimation. Model strong consistency and some other properties are proved. The linear model and Ornstein–Uhlenbeck model are studied in detail. As an auxiliary result, an asymptotic behaviour of the fractional derivative of the fractional Brownian motion is established.     

Brownian Bridge and Other Path-dependent Gaussian Processes Vectorial Simulation

The iterative simulation of the Brownian bridge is well known. In this article, we present a vectorial simulation alternative based on Gaussian processes for machine learning regression that is suitable for interpreted programming languages implementations. We extend the vectorial simulation of path-dependent trajectories to other Gaussian processes, namely, sequences of Brownian bridges, geometric Brownian motion, fractional Brownian motion, and Ornstein–Ulenbeck mean reversion process.     

The local fractional bootstrap

We introduce a bootstrap procedure for high‐frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high‐frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first‐order validity of the bootstrap method, and in simulations, we observe that the bootstrap‐based hypothesis test provides considerable finite‐sample improvements over an existing test that is based on a central limit theorem. This is important when studying the roughness properties of time series data. We illustrate this by applying the bootstrap method to two empirical data sets: We assess the roughness of a time series of high‐frequency asset prices and we test the validity of Kolmogorov's scaling law in atmospheric turbulence data.     

Parameter identification for the discretely observed geometric fractional Brownian motion

This paper deals with the problem of estimating all the unknown parameters of geometric fractional Brownian processes from discrete observations. The estimation procedure is built upon the marriage of the quadratic variation and the maximum likelihood approach. The asymptotic properties of the estimators are provided. Moveover, we compare our derived method with the approach proposed by Misiran et al. [Fractional Black-Scholes models: complete MLE with application to fractional option pricing. In International conference on optimization and control; Guiyang, China; 2010. p. 573–586.], namely the complete maximum likelihood estimation. Simulation studies confirm theoretical findings and illustrate that our methodology is efficient and reliable. To show how to apply our approach in realistic contexts, an empirical study of Chinese financial market is also presented.     

Approximation of fractional Brownian sheet by Wiener integral

In this paper, we consider an approximation of the fractional Brownian sheet by two parameter wiener integral. We obtain that there exists an unique two parameter wiener integral closest to the fractional Brownian sheet.     

Ruin problem of a two-dimensional fractional Brownian motion risk process

This paper investigates ruin probability and ruin time of a two-dimensional fractional Brownian motion risk process. The net loss process of an insurance company is modeled by a fractional Brownian motion. The two-dimensional fractional Brownian motion risk process models the surplus processes of an insurance and a reinsurance company, where the net loss is divided between them in some specified proportions. The ruin problem considered is that of the two-dimensional risk process first entering the negative quadrant, that is, the simultaneous ruin problem. We derive both asymptotics of the ruin probability and approximations of the scaled conditional ruin time as the initial capital tends to infinity.     

Parameter Identification for Drift Fractional Brownian Motions with Application to the Chinese Stock Markets

This article deals with the problem of estimating all the unknown parameters in the drift fractional Brownian motion with discretely sampled data. The estimation procedure is built upon the marriage of the variation method and the ergodic theory. The strong consistencies of these estimators are provided. Moreover, our method and two existing approaches are compared based on the computational running time and the accuracy of estimation via simulation studies. We also apply the proposed method to the real high-frequency financial data within a window of 4 h in the trading day from the Chinese mainland stock market.     

Stochastically Curtailed Tests Under Fractional Brownian Motion

Stochastic curtailment has been considered for the interim monitoring of group sequential trials (Davis and Hardy, 1994). Statistical boundaries in Davis and Hardy (1994) were derived using theory of Brownian motion. In some clinical trials, the conditions of forming a Brownian motion may not be satisfied. In this paper, we extend the computations of Brownian motion based boundaries, expected stopping times, and type I and type II error rates to fractional Brownian motion (FBM). FBM includes Brownian motion as a special case. Designs under FBM are compared to those under Brownian motion and to those of O’Brien–Fleming type tests. One- and two-sided boundaries for efficacy and futility monitoring are also discussed. Results show that boundary values decrease and error rates deviate from design levels when the Hurst parameter increases from 0.1 to 0.9, these changes should be considered when designing a study under FBM.     

Pricing American put option on zero-coupon bond under fractional CIR model with transaction cost

The motivation of this study is to evaluate the American put option on zero-coupon bond, when the interest rate model is governed by a fractional CIR (FCIR) interest rate model. Since the existence of fractional Brownian motion, leading to create the arbitrage, we employ the transaction cost for eliminating the arbitrage. We first of all apply the Leland's hedging strategy for a self-financing portfolio that contains an American option and zero-coupon bond and derive a formula for the transaction cost. We perform the least-square Monte Carlo (LSM) method for pricing American option under the proposed interest rate model.     

Long-range dependence and approximate Bayesian computation

In this work, we propose a method for estimating the Hurst index, or memory parameter, of a stationary process with long memory in a Bayesian fashion. Such approach provides an approximation for the posterior distribution for the memory parameter and it is based on a simple application of the so-called approximate Bayesian computation (ABC), also known as likelihood-free method. Some popular existing estimators are reviewed and compared to this method for the fractional Brownian motion, for a long-range binary process and for the Rosenblatt process. The performance of our proposal is remarkably efficient.     

Group sequential tests under fractional Brownian motion in monitoring clinical trials

Group sequential tests have been effective tools in monitoring long term clinical trials. There have been several popular discrete sequential boundaries proposed for modeling interim analysis of clinical trials under the assumption of Brownian motion for the stochastic processes generated from test statistics. In this paper, we study the five sequential boundaries in Lan and DeMets (Biometrika 70:659–663, 1983) under the fractional Brownian motion. The fractional Brownian includes the classic Brownian motion as a special case. An example from a real data set is used to illustrate the applications of the boundaries.     

Conditional moving linear regression: Modeling the recruitment process for ALLHAT

Effective recruitment is a prerequisite for successful execution of a clinical trial. ALLHAT, a large hypertension treatment trial (N = 42,418), provided an opportunity to evaluate adaptive modeling of recruitment processes using conditional moving linear regression. Our statistical modeling of recruitment, comparing Brownian and fractional Brownian motion, indicates that fractional Brownian motion combined with moving linear regression is better than classic Brownian motion in terms of higher conditional probability of achieving a global recruitment goal in 4-week ahead projections. Further research is needed to evaluate how recruitment modeling can assist clinical trialists in planning and executing clinical trials.     

The effect of long-range dependence on change-point estimators

We study the asymptotic behaviour of a class of estimators of the time of change in the mean of Gaussian observations having long-range dependence. We prove that after a suitable normalization the estimators converge in distribution to functionals of fractional Brownian motion.     

Fractional Brownian motion and long term clinical trial recruitment

Prediction of recruitment in clinical trials has been a challenging task. Many methods have been studied, including models based on Poisson process and its large sample approximation by Brownian motion (BM); however, when the independent incremental structure is violated for BM model, we could use fractional Brownian motion to model and approximate the underlying Poisson processes with random rates. In this paper, fractional Brownian motion (FBM) is considered for such conditions and compared to BM model with illustrated examples from different trials and simulations.     

A note on the differentiation formula in Stratonovich type for fractional Brownian sheet

We consider a new proof on the differentiation formula in Stratonovich type for fractional Brownian sheet. Our proof is based on the repeated applications of differentiation formulas in Stratonovich form for a one-parameter Gaussian process.     

The Modelling of Ethernet Data and of Signals that are Heavy‐tailed with Infinite Variance*

We provide an overview of some of the research of the last ten years involving computer network data traffic. We describe the original Ethernet data study which suggested that computer traffic is inherently different from telephone traffic and that in the context of computer networks, self‐similar models such as fractional Brownian motion, should be used. We show that the on–off model can physically explain the presence of self‐similarity. While the on–off model involves bounded signals, it is also possible to consider arbitrary unbounded finite‐variance signals or even infinite‐variance signals whose distributions have heavy tails. We show that, in the latter case, one can still obtain self‐similar processes with dependent increments, but these are not the infinite‐variance fractional stable Lévy motions which have been commonly considered in the literature. The adequate model, in fact, can either have dependent or independent increments, and this depends on the respective size of two parameters, namely, the number of workstations in the network and the time scale under consideration. We indicate what happens when these two parameters become jointly asymptotically large. We conclude with some comments about high frequency behaviour and multifractals.     

Approximation to two independent Gaussian processes from a unique Lévy process and applications

Abstract

In this article, we construct two families of processes, from a unique Lévy process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes. As applications of this result, we obtain families of processes that converge in law towards fractional Brownian motion, sub-fractional Brownian motion and bifractional Brownian motion, respectively.     

Fractional Brownian motion and clinical trials

The purpose of this paper is to extend the widely used classical Brownian motion technique for monitoring clinical trial data to a larger class of stochastic processes, i.e. fractional Brownian motion, and compare these results. The beta-blocker heart attack trial is presented as an example to illustrate both methods.     

Fisher Information for Fractional Brownian Motion Under High-Frequency Discrete Sampling

We investigate the issue of the validation of the local asymptotic normality property of three characterizing parameters of the fractional Brownian motion under high-frequency discrete sampling. We prove that the local asymptotic normality property holds true for the likelihood only when at least one of the volatility parameter and the Hurst exponent is known. We provide optimal rates of convergence of the three parameters and Fisher information matrix in closed form.