The Regularity of Stochastic Convolution Driven by Tempered Fractional Brownian Motion and Its Application to Mean-field Stochastic Differential Equations

In this paper, some properties of a stochastic convolution driven by tempered fractional Brownian motion are obtained. Based on this result, we get the existence and uniqueness of stochastic mean-field equation driven by tempered fractional Brownian motion. Furthermore, combining with the Banach fixed point theorem and the properties of Mittag-Leffler functions, we study the existence and uniqueness of mild solution for a kind of time fractional mean-field stochastic differential equation driven by tempered fractional Brownian motion.     

A Set-indexed Fractional Brownian Motion

We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no “really nice” set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed fractional Brownian motion along increasing paths is analysed.      

Fractional Brownian Motion and Sheet as White Noise Functionals

In this short note, we show that it is more natural to look the fractional Brownian motion as functionals of the standard white noises, and the fractional white noise calculus developed by Hu and Фksendal follows directly from the classical white noise functional calculus. As examples we prove that the fractional Girsanov formula, the Ito type integrals and the fractional Black-Scholes formula are easy consequences of their classical counterparts. An extension to the fractional Brownian sheet is also briefly discussed.     

Professor V. Lakshmikantham,Editor Stochastic Analysis and Applications (1983–2012)

We use the stochastic calculus of variations for the fractional Brownian motion to derive formulas for the replicating portfolios for a class of contingent claims in a Bachelier and a Black–Scholes markets modulated by fractional Brownian motion. An example of such a model is the Black–Scholes process whose volatility solves a stochastic differential equation driven by a fractional Brownian motion that may depend on the underlying Brownian motion.     

On fractional tempered stable motion

Fractional tempered stable motion (fTSm) is defined and studied. FTSm has the same covariance structure as fractional Brownian motion, while having tails heavier than Gaussian ones but lighter than (non-Gaussian) stable ones. Moreover, in short time it is close to fractional stable Lévy motion, while it is approximately fractional Brownian motion in long time. A series representation of fTSm is derived and used for simulation and to study some of its sample paths properties.     

Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations

Stock exchange dynamics of fractional order are usually modeled as a non-random exponential growth process driven by a fractional Brownian motion. Here we propose to use rather a non-random fractional growth driven by a (standard) Brownian motion. The key is the Taylor’s series of fractional order where E_α(.) denotes the Mittag-Leffler function, and is the so-called modified Riemann-Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration. Various models of fractional dynamics for stock exchange are proposed, and their solutions are obtained. Mainly, the Itô’s lemma of fractional order is illustrated in the special case of a fractional growth with white noise. Prospects for the Merton’s optimal portfolio are outlined, the path probability density of fractional stock exchange dynamics is obtained, and two fractional Black-Scholes equations are derived. This approach avoids using fractional Brownian motion and thus is of some help to circumvent the mathematical difficulties so involved.     

Tempered stable Lévy motion and transient super-diffusion

The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Lévy motion, representing the accumulation of power-law jumps. The tempered stable Lévy motion uses exponential tempering to cool these jumps. A tempered fractional diffusion equation governs the transition densities, which progress from super-diffusive early-time to diffusive late-time behavior. This article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift. A temporal and spatial second-order Crank-Nicolson method is developed, based on a finite difference formula for tempered fractional derivatives. A new exponential rejection method for simulating tempered Lévy stables is presented to facilitate particle tracking codes.     

Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion

We provide an almost sure convergent expansion of fractional Brownian motion in wavelets which decorrelates the high frequencies. Our approach generalizes Lévy's midpoint displacement technique which is used to generate Brownian motion. The low-frequency terms in the expansion involve an independent fractional Brownian motion evaluated at discrete times or, alternatively, partial sums of a stationary fractional ARIMA time series. The wavelets fill in the gaps and provide the necessary high frequency corrections. We also obtain a way of constructing an arbitrary number of non-Gaussian continuous time processes whose second order properties are the same as those of fractional Brownian motion.     

分式Brown运动的局部时:白噪声分析法

本文利用白噪声分析的方法,讨论了分式布朗运动的局部时,即将其看作一个Hida分布.进一步,给出分式布朗运动的局部时的混沌分解以及局部时平方可积性.     

The fractional multivariate normal tempered stable process

In this paper, the multivariate process having long-range dependency is presented. The process is defined by the time-changed fractional Brownian motion whose subordinator is given by the fractional tempered stable subordinator. The fractional tempered stable subordinator is a generalization of the non-decreasing tempered stable process with long-range dependence. The multivariate process allows for (1) the long-range dependence in the endogenous noise, (2) the long-range dependence in time or the volatility, (3) the fat-tailed marginal distribution, and (4) an asymmetric dependence structure between elements. Numerical methods to generating sample paths for the process are discussed.     

多重分数斯特拉托诺维奇积分的一个注记

本文研究了多重分数斯特拉托诺维奇积分,通过卷积逼近技巧和分数布朗运动的随机积分的性质,构造了当Hurst参数小于二分之一时的多重随机积分.这种方法是新的不同于文[8]中的构造方法.     

Large deviations for optimal filtering with fractional Brownian motion

We establish large deviation estimates for the optimal filter where the observation process is corrupted by a fractional Brownian motion. The observation process is transformed to an equivalent model which is driven by a standard Brownian motion. The large deviations in turn are established by proving qualitative properties of perturbations of the equivalent observation process.     

On the Weighted Local Time and the Tanaka Formula for the Multidimensional Fractional Brownian Motion

Abstract

We determine the weighted local time for the multidimensional fractional Brownian motion from the occupation time formula. We also discuss on the Itô and Tanaka formula for the multidimensional fractional Brownian motion. In these formulas the Skorohod integral is applicable if the Hurst parameter of fractional Brownian motion is greater than 1/2. If the Hurst parameter is less than 1/2, then we use the Skorohod type integral introduced by Nualart and Zakai for the stochastic integral and establish the Itô and Tanaka formulas.     

Fractional stochastic differential equation with discontinuous diffusion

In this article, we study a class of stochastic differential equations driven by a fractional Brownian motion with H > 1/2 and a discontinuous coefficient in the diffusion. We prove existence and uniqueness for the solution of these equations. This is a first step to define a fractional version of the skew Brownian motion.     

分数随机利率模型中的期权定价

本文研究分数随机利率模型中的期权定价问题.通过选取不同的资产作为计价单位及相应的测度交换,将经典模型中的测度变换方法推广到分数布朗运动市场环境,既丰富了分数期权定价的拟鞅方法,也得到了股票价格与利率分别服从几何分数布朗运动时的期权定价公式.     

Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion

Abstract

We prove an existence and uniqueness theorem for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter H > 1/2 and a multidimensional standard Brownian motion. The proof relies on some a priori estimates, which are obtained using the methods of fractional integration and the classical Itô stochastic calculus. The existence result is based on the Yamada–Watanabe theorem.     

Random Dynamical Systems and Stationary Solutions of Differential Equations Driven by the Fractional Brownian Motion

Abstract

Linear and semilinear stochastic evolution equations with additive noise, where the forcing term is an infinite dimensional fractional Brownian motion are studied. Under usual dissipativity conditions the equations are shown to define random dynamical systems which have unique, exponentially attracting fixed points. The results are applied to stochastic parabolic PDE's. They are also applicable to standard finite-dimensional dissipative stochastic equation driven by fractional Brownian motion.     

Parameter estimation for stochastic equations with additive fractional Brownian sheet

We study the maximum likelihood estimator for stochastic equations with additive fractional Brownian sheet. We use the Girsanov transform for the the two-parameter fractional Brownian motion, as well as the Malliavin calculus and Gaussian regularity theory.      

Example of a Gaussian Self-Similar Field With Stationary Rectangular Increments That Is Not a Fractional Brownian Sheet

We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet (fBs). This Gaussian field is an extension of fractional Brownian motion. It is well known that the fractional Brownian motion is a unique Gaussian self-similar process with stationary increments. The main result of this article is an example of a Gaussian self-similar field with stationary rectangular increments that is not an fBs. So we proved that the structure of self-similar Gaussian fields can be substantially more involved then the structure of self-similar Gaussian processes. In order to establish the main result, we prove some properties of covariance function for self-similar fields with rectangular increments. Also, using Lamperti transformation, we obtain properties of covariance function for the corresponding stationary fields.     

Stochastic Analysis of the Fractional Brownian Motion

Since the fractional Brownian motion is not a semi-martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Itô formula, the Itô–Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations.