Approximation of Multiple Stratonovich Fractional Integrals

Abstract

We study multiple Riemann-Stieltjes integral approximations to multiple Stratonovich fractional integrals. Two standard approximations (Wong-Zakai and Mollifier approximations) are considered and we show the convergence in the mean square sense and uniformly on compact time intervals of these approximations to the multiple Stratonovich fractional integral.     

A strong approximation for double subfractional integrals

A Riemann–Stieltjes integral strong approximation to double Stratonovich integrals with respect to odd and even fractional Brownian motions is considered. We prove the convergence in quadratic mean, uniformly on compact time intervals, of the ordinary double integral process obtained by linear interpolation of the odd and even fractional Brownian motions, to the double Stratonovich integral. The deterministic integrands are continuous or are given by bimeasures.     

Stratonovich Calculus with Respect to Fractional Brownian Sheet

Abstract

We introduce two types of Stratonovich stochastic integrals for two-parameter process. The relationship of Stratonovich integrals to Skorohod integrals will be investigated. By using this relationship, we prove that a differentiation formula for fractional Brownian sheet in Stratonovich form can be expressed as the sum of Stratonovich integrals of two types introduced in this article.     

Differentiation formula in Stratonovich version for fractional Brownian sheet

We introduce two types of the Stratonovich stochastic integrals for two-parameter processes, and investigate the relationship of these Stratonovich integrals and various types of Skorohod integrals with respect to a fractional Brownian sheet. By using this relationship, we derive a differentiation formula in the Stratonovich sense for fractional Brownian sheet through Itô formula. Also the relationship between the two types of the Stratonovich integrals will be obtained and used to derive a differentiation formula in the Stratonovich sense. In this case, our proof is based on the repeated applications of differentiation formulas in the Stratonovich form for one-parameter Gaussian processes.     

Wick-Itô Formula for Gaussian Processes

Abstract

A Wick-Itô formula for Gaussian processes is obtained. This is a change of variables formula, which is to Wick-Itô integrals what the usual Itô formula is to Itô integrals. The conditions are weak enough to allow processes with infinite quadratic variation. They are satisfied by fractional Brownian motion with parameter 1/4 < H < 1.     

On a Multiple Stratonovich-type Integral for Some Gaussian Processes

We construct a multiple Stratonovich-type integral with respect to Gaussian processes with covariance function of bounded variation. This construction is based on the previous definition of the multiple Itô-type integral given by Huang and Cambanis [Ann. Propab. 6(4), 585–614] and on a Hu–Meyer formula (that is, an expression of the multiple Stratonovich integral as a sum of Itô-type integrals of inferior or equal order) for the elementary functions. We also apply our results to the fractional Brownian motion with Hurst parameter $H > \frac{1}{2}We construct a multiple Stratonovich-type integral with respect to Gaussian processes with covariance function of bounded variation. This construction is based on the previous definition of the multiple It?-type integral given by Huang and Cambanis [Ann. Propab. 6(4), 585–614] and on a Hu–Meyer formula (that is, an expression of the multiple Stratonovich integral as a sum of It?-type integrals of inferior or equal order) for the elementary functions. We also apply our results to the fractional Brownian motion with Hurst parameter .     

On the Expectations of Multiple Stratonovich Integrals

Abstract

In the construction of numerical methods for solving stochastic differential equations it becomes necessary to calculate the expectations of products of multiple stochastic integrals. In the Itô case, explicit formulae for the expectation of a multiple integral with integrand identically equal to 1 and for the product of two such integrals are known. In this paper formulae for the expectation of any multiple Stratonovich integral as well as for the product of a broad class of two Stratonovich integrals have been derived.     

Bahadur–Kiefer representations for time dependent quantile processes

We define a time dependent empirical process based on n independent fractional Brownian motions and describe strong approximations to it by Gaussian processes. They lead to strong approximations and functional laws of the iterated logarithm for the quantile or inverse of this empirical process. They are obtained via time dependent Bahadur–Kiefer representations.     

The Feynman-Stratonovich semigroup and stratonovich integral expansions in nonlinear filtering

Representations for the solution of the Zakai equation in terms of multiple Stratonovich integrals are derived. A new semigroup (the Feynman-Stratonovich semigroup) associated with the Zakai equation is introduced and using the relationship between multiple Stratonovich integrals and iterated Stratonovich integrals, a representation for the unnormalized conditional density,u(t,x), solely in terms of the initial density and the semigroup, is obtained. In addition, a Fourier seriestype representation foru(t,x) is given, where the coefficients in this representation uniquely solve an infinite system of partial differential equations. This representation is then used to obtain approximations foru(t,x). An explicit error bound for this approximation, which is of the same order as for the case of multiple Wiener integral representations, is obtained. Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL03-92-G0008.     

Chaos decomposition of multiple fractional integrals and applications

Chaos decomposition of multiple integrals with respect to fractional Brownian motion (with H > 1/2) is given. Conversely the chaos components are expressed in terms of the multiple fractional integrals. Tensor product integrals are introduced and series expansions in those are considered. Strong laws for fractional Brownian motion are proved as an application of multiple fractional integrals. Received: 22 September 1998 / Revised version: 20 April 1999     

Convergence rate of multiple fractional Stratonovich type integral for Hurst parameter less than 1/2

In this paper, we have investigated the problem of the convergence rate of the multiple integralwhere f ∈ Cn+1([0, T ]n) is a given function, π is a partition of the interval [0, T ] and {BtHi ,π} is a family of interpolation approximation of fractional Brownian motion BtH with Hurst parameter H < 1/2. The limit process is the multiple Stratonovich integral of the function f . In view of known results, the convergence rate is different for different multiplicity n. Under some mild conditions, we obtain that the uniform convergence rate is 2H in the mean square sense, where is the norm of the partition generating the approximations.     

Stochastic heat equation driven by fractional noise and local time

The aim of this paper is to study the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with Hurst parameter H ∈ (0,1) in time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos expansion for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion.     

Computational scheme for solving nonlinear fractional stochastic differential equations with delay

Abstract

This paper studies the numerical solution of fractional stochastic delay differential equations driven by Brownian motion. The proposed algorithm is based on linear B-spline interpolation. The convergence and the numerical performance of the method are analyzed. The technique is adopted for determining the statistical indicators of stochastic responses of fractional Langevin and Mackey-Glass models with stochastic excitations.     

Wong–Zakai approximations with convergence rate for stochastic partial differential equations

ABSTRACT

The goal of this paper is to prove a convergence rate for Wong–Zakai approximations of semilinear stochastic partial differential equations driven by a finite-dimensional Brownian motion. Several examples, including the HJMM equation from mathematical finance, illustrate our result.     

Couplings and Strong Approximations to Time-Dependent Empirical Processes Based on I.I.D. Fractional Brownian Motions

We define a time-dependent empirical process based on n i.i.d. fractional Brownian motions and establish Gaussian couplings and strong approximations to it by Gaussian processes. They lead to functional laws of the iterated logarithm for this process.     

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.     

Instrumental Variable Estimation for Linear Stochastic Differential Equations Driven by Fractional Brownian Motion

Abstract

We investigate the asymptotic properties of instrumental variable estimators of the drift parameter for stochastic processes satisfying linear stochastic differential equations driven by fractional Brownian motion.     

Fractional Brownian motion approximation based on fractional integration of a white noise

We study simple approximations to fractional Gaussian noise and fractional Brownian motion. The approximations are based on spectral properties of the noise. They allow one to consider the noise as the result of fractional integration/differentiation of a white Gaussian noise. We consider correlation properties of the approximation to fractional Gaussian noise and point to the peculiarities of persistent and anti-persistent behaviors. We also investigate self-similarity properties of the approximation to fractional Brownian motion, namely, `τ^H laws' for the structure function and the range. We conclude that the models proposed serve as a convenient tool for modelling of natural processes and testing and improvement of methods aimed at analysis and interpretation of experimental data.     

Parametric estimation for linear stochastic differential equations driven by mixed fractional Brownian motion

ABSTRACT

We investigate the asymptotic properties of the maximum likelihood estimator and Bayes estimator of the drift parameter for stochastic processes satisfying linear stochastic differential equations driven by a mixed fractional Brownian motion. We obtain a Bernstein–von Mises-type theorem also for such a class of processes.     

Strassen-type Laws of Iterated Logarithm for a Fractional Brownian Sheet

Abstract

We study Strassen-type laws of iterated logarithm for a fractional Brownian sheet including that for small time, which imply most of the former laws of the iterated logarithm and Strassen's laws for one-parameter and two-parameter Wiener processes.