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This paper deals with a class of backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1/2) with time-delayed generators. In this type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time-delay function, for instance, of the moving average type. We establish an existence and uniqueness result of solutions for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. The stochastic integral used throughout the paper is the divergence operator-type integral. [ABSTRACT FROM AUTHOR]

This paper concerns the strong convergence rate of an averaging principle for two-time-scale coupled forward–backward stochastic differential equations (CFBSDEs, for short) driven by fractional Brownian motion (fBm, for short). The fast component is a forward stochastic differential equation (FSDE, for short) driven by Brownian motion, while the slow component is a backward stochastic differential equation (BSDE, for short) driven by fBm with the Hurst index greater than 1/2. Combining Malliavin calculus theory with stochastic integral and Khasminskii's time discretization method, the rate of strong convergence for the slow component towards the solution of the averaging equation in the mean square sense is derived. The strong convergence rate of an averaging principle for fast–slow CFBSDEs driven by fBm is new. [ABSTRACT FROM AUTHOR]

This paper deals with a class of delay backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1 2 ). In this type of equation, a generator at time t can depend not only on the present but also on 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 the paper is a divergence-type integral. [ABSTRACT FROM AUTHOR]

The paper suggests a way of stochastic integration of random integrands with respect to fractional Brownian motion with the Hurst parameter H > 1∕2. The integral is defined initially on the processes that are" piecewise" predictable on a short horizon. Then the integral is extended on a wide class of square integrable adapted random processes. This class is described via a mild restriction on the growth rate of the conditional mean square error for the forecast on an arbitrarily short horizon given current observations. On the other hand, a pathwise regularity, such as Hölder condition, etc., is not required for the integrand. The suggested integration can be interpreted as foresighted integration for integrands featuring certain restrictions on the forecasting error. This integration is based on Itô's integration and does not involve Malliavin calculus or Wick products. In addition, it is shown that these stochastic integrals depend right continuously on H at H = 1/2. [ABSTRACT FROM AUTHOR]

In this paper, we study a representation for the solutions to sticky stochastic differential equations driven by a continuous process. The involved stochastic integral is interpreted in three different ways. Namely, we deal with Young integral defined by the fractional calculus, and the forward and symmetric integrals in the Russo and Vallois sense. The representation obtained in this paper depends on the amount of time spent by the solution at zero. Hence, we obtain the uniqueness for the solution among the processes that spend zero time at 0. [ABSTRACT FROM AUTHOR]

In this article, we study multiplicative SDE with fractional noise in a suitable sense, which can be regarded as a fractional Gruschin type process. Using the transfer principle and fractional integral and derivative operators, two kinds of Bismut type derivative formulae are established in this non-Markovian context under incompatible conditions. As an application, an explicit gradient estimate is derived. [ABSTRACT FROM AUTHOR]

We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter H ∈ (0 , 1). Close to a change of stability measured with a small parameter , we rely on the natural separation of time-scales and establish a simplified description of the essential dynamics. Up to an error term bounded by a power of depending on the Hurst parameter we can approximate the solution of the SPDE in first order by an SDE, the so-called amplitude equation, which describes the amplitude of the dominating pattern changing stability. In second order the approximation is given by a fast infinite-dimensional Ornstein–Uhlenbeck process. To this aim, we need to establish an explicit averaging result for stochastic integrals driven by rough fractional noise for small Hurst parameters. [ABSTRACT FROM AUTHOR]

In the last few years Ayache, Esser and Hamonier introduced a new Multifractional Process with Random Exponent (MPRE) obtained by replacing the Hurst parameter in a moving average representation of Fractional Brownian Motion through Wiener integral by an adapted Hölder continuous stochastic process indexed by the integration variable. Thus, this MPRE can be expressed as a moving average Itô integral which is a considerable advantage with respect to another MPRE introduced a long time ago by Ayache and Taqqu. Thanks to this advantage, very recently, Loboda, Mies and Steland have derived interesting results on local Hölder regularity, self-similarity and other properties of the recently introduced moving average MPRE and generalizations of it. Yet, the problem of obtaining, on a universal event of probability 1 not depending on the location, relevant lower bounds for local oscillations of such processes has remained open. We solve it in the present article under some conditions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of determining the statistical properties of the initial data from the final time data. In the case 0 < β ≤ 1 , where β is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-posedness results for the problems when β > 1. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fBm. [ABSTRACT FROM AUTHOR]

A rough volatility model contains a stochastic Volterra integral with a weakly singular kernel. The classical Euler-Maruyama algorithm is not very efficient for simulating this kind of model because one needs to keep records of all the past path-values and thus the computational complexity is too large. This paper develops a fast two-step iteration algorithm using an approximation of the weakly singular kernel with a sum of exponential functions. Compared to the Euler-Maruyama algorithm, the complexity of the fast algorithm is reduced from O (N 2) to O (N log ⁡ N) or O (N log 2 ⁡ N) for simulating one path, where N is the number of time steps. Further, the fast algorithm is developed to simulate rough Heston models with (or without) regime switching, and multi-factor approximation algorithms are also studied and compared. The convergence rates of the Euler-Maruyama algorithm and the fast algorithm are proved. A number of numerical examples are carried out to confirm the high efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a Haar wavelet method is proposed for solving linear and nonlinear stochastic Itô–Volterra integral equation (SIVIE) driven by fractional Brownian motion (FBM) with Hurst parameter H ∈ (1 2 , 1). This approach reduces the solution of the problem under study to the solution of a linear and nonlinear system of algebraic equations, which is based on the operational matrix and stochastic operational matrix of the Haar wavelets basis (HWB). Moreover, the error estimate and convergence analysis of the method are established in detail. Several numerical examples are presented to test the applicability and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We consider stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H > 1 2 . We first derive supremum norm estimates for the solution and its Malliavin derivative. We then show existence and smoothness of the density under suitable nondegeneracy conditions. This extends the results in Hu and Nualart [Differential equations driven by Hölder continuous functions of order greater than 1/2, Stoch. Anal. Appl. Abel Symp. 2 (2007), pp. 399–413] and Nualart and Saussereau [Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stoch. Process. Appl. 119 (2009), pp. 391–409] where stochastic differential equations driven by fractional Brownian motion are considered. The proof uses a priori estimates for deterministic differential equations driven by a function in a suitable Sobolev space. [ABSTRACT FROM AUTHOR]

This paper deals with a class of anticipated backward stochastic differential equations driven by two mutually independent fractional Brownian motions. We essentially establish the existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral. [ABSTRACT FROM AUTHOR]

In this paper, stochastic integral equations characterized by fractional Brownian motion have been studied. The fractional stochastic integral equation has been solved by second kind Chebyshev wavelets. The convergence and error analysis have been discussed for the efficiency of the discussed method. In addition, two illustrative examples have been solved to examine the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider fuzzy stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm). These equations can be applied in hybrid real-world systems, including randomness, fuzziness and long-range dependence. Under some assumptions on the coefficients, we follow an approximation method to the fractional stochastic integral to study the existence and uniqueness of the solutions. As an example, in financial models, we obtain the solution for an equation with linear coefficients. [ABSTRACT FROM AUTHOR]

We consider the Wiener integral with respect to a d -parameter Hermite process with Hurst multi-index H = (H 1 ,.. , H d) ∈ (1 2 , 1) d and we analyze the limit behavior in distribution of this object when the components of H tend to 1 and/or 1 2. As examples, we focus on the solution to the stochastic heat equation with additive Hermite noise and to the Hermite Ornstein-Uhlenbeck process. [ABSTRACT FROM AUTHOR]

We analyze the solution to the linear stochastic heat equation driven by a multiparameter Hermite process of order q ≥ 1. This solution is an element of the q th Wiener chaos. We discuss various properties of the solution, such as the necessary and sufficient condition for its existence, self-similarity, α -variation and regularity of its sample paths. We will also focus on the probability distribution of the solution, which is non-Gaussian when q ≥ 2. [ABSTRACT FROM AUTHOR]

We consider a d -parameter Hermite process with Hurst index H = (H 1 ,.. , H d) ∈ 1 2 , 1 d and we study its limit behavior in distribution when the Hurst parameters H i , i = 1 ,.. , d (or a part of them) converge to 1 2 and/or 1. The limit obtained is Gaussian (when at least one parameter tends to 1 2) and non-Gaussian (when at least one-parameter tends to 1 and none converges to 1 2). [ABSTRACT FROM AUTHOR]

Abstract We give several extensions of the sewing lemma of Feyel and de La Pradelle and show how these results generalize Young's integration theory in a simple and natural way. For illustrative purposes, we apply the lemma to integrals involving discontinuous functions of a fractional Brownian motion with the Hurst index H > 1 ∕ 2. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of the backward stochastic variational inequalities driven by m-dimensional fractional Brownian motion with Hurst parameters Hk (k = 1, . . .m) greater than 1/2. The stochastic integral used throughout the paper is the divergence type integral. [ABSTRACT FROM AUTHOR]

We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a d -dimensional fractional Brownian motion (fBm) B t with Hurst parameter H > 1 2 , where the integrands are vector fields applied to B t . It provides, for example, a direct alternative proof of Y. Hu and D. Nualart’s result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion. [ABSTRACT FROM AUTHOR]

LetQ∈C((0, 1]1 + m) and let {ξn, j: 1 ⩽j⩽n,n⩾ 1} be a triangular array of independent zero-mean random variables. In this paper, we show that for every integerm⩾ 1, the processesXmn(Q) = {Xnm(t,Q), 0 ⩽t⩽ 1},n⩾ 1 defined byconverges weakly inC[0, 1], asn→ ∞, to a multiple Wiener integral under some suitable conditions on the functionQand array {ξn, j}. This convergence includes the approximating characterization of many popular self-similar processes such as fractional Brownian motion, sub-fractional Brownian motion, Rosenblatt process, and more general Hermite processes. [ABSTRACT FROM AUTHOR]

In this paper we establish conditions that imply the conditional full support (CFS) property, introduced by Guasoni et al. (2008), for the processes St = Rt + ∫t0 ɸsdBHs, where BH is a Fractional Brownian motion, R is a continuous process, and the processes R and ɸ are either progressive or independent of BH: Moreover we build the absence of arbitrage opportunities without calculating the risk-neutral probability. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the stochastic averaging principle for stochastic differential equations (SDEs) with non-Lipschitz coefficients driven by fractional Brownian motion (fBm) of the Hurst parameter . We define the stochastic integrals with respect to the fBm in the integral formulation of the SDEs as pathwise integrals and we adopt the non-Lipschitz condition proposed by Taniguchi (1992) which is a much weaker condition with wider range of applications. The averaged SDEs are established. We then use their corresponding solutions to approximate the solutions of the original SDEs both in the sense of mean square and of probability. One can find that the similar asymptotic results are suitable for those non-Lipschitz SDEs with fBm under different types of stochastic integrals. [ABSTRACT FROM AUTHOR]

We prove existence and uniqueness for stochastic functional differential equations driven by a one-dimensional fractional Brownian motion with Hurst parameter H > 1/2. The stochastic integral with respect to fractional Brownian motion is defined as a pathwise Riemann-Stieltjes integral. [ABSTRACT FROM AUTHOR]

We show that if a random variable is a final value of an adapted Hölder continuous process, then it can be represented as a stochastic integral with respect to fractional Brownian motion, and the integrand is an adapted process, continuous up to the final point. [ABSTRACT FROM PUBLISHER]

We construct an iterated stochastic integral with respect to fractional Brownian motion (fBm) withH>1/2. The first integrand is a deterministic function, and each successive integral is with respect to an independent fBm. We show that this symmetric stochastic integral is equal to the Malliavin divergence integral. By a version of the Fourth Moment Theorem of Nualart and Peccati [10], we show that a family of such integrals converges in distribution to a scaled Brownian motion. An application is an approximation to the windings for a planar fBm, previously studied by Baudoin and Nualart [2]. [ABSTRACT FROM AUTHOR]

We introduce a Multifractal Random Walk (MRW) defined as a stochastic integral of an infinitely divisible noise with respect to a dependent fractional Brownian motion. Using the techniques of the Malliavin calculus, we study the existence of this object and its properties. We then propose a continuous time model for financial modeling that captures the main properties observed in the empirical data, including the leverage effect. We illustrate our result by numerical simulations. [ABSTRACT FROM AUTHOR]

Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida's white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we use a definition of the fractional stochastic integral given by Carmona et al. (2003) in [19] and develop a simple approximation method to study quasi-linear stochastic differential equations by fractional Brownian motion. We also propose a stochastic process, namely fractional semimartingale, to model for the noise driving in some financial models. [Copyright &y& Elsevier]

Abstract: We give a strong approximation of Rosenblatt process via transport processes and we give the rate of convergence. [Copyright &y& Elsevier]

By using multiple Wiener-Itô stochastic integrals, we study the cubic variation of a class of self-similar stochastic processes with stationary increments (the Rosenblatt process with self-similarity order H∊(½, 1)). This study is motivated by statistical purposes. We prove that this renormalized cubic variation satisfies a noncentral limit theorem, and its limit is (in the L²(Ω) sense) still the Rosenblatt process. [ABSTRACT FROM AUTHOR]

Abstract: We use a method developed in Carmona et al. (2003) to study the fractional geometric mean-reversion processes. Our obtained results hold for any . [Copyright &y& Elsevier]

We consider a stochastic volatility model where the volatility process is a fractional Brownian motion. We estimate the memory parameter of the volatility from discrete observations of the price process. We use criteria based on Malliavin calculus in order to characterize the asymptotic normality of the estimators. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to provide a semimartingale approximation of a fractional stochastic integration. This result leads us to approximate the fractional Black–Scholes model by a model driven by semimartingales, and a European option pricing formula is found. [Copyright &y& Elsevier]

Abstract: We construct a family of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener–Itô integral with respect to the fractional Brownian motion. We assume that and we prove our approximation result for the integrands f in a rather general class. [Copyright &y& Elsevier]

Abstract: In this paper, we study almost sure central limit theorems for sequences of functionals of general Gaussian fields. We apply our result to non-linear functions of stationary Gaussian sequences. We obtain almost sure central limit theorems for these non-linear functions when they converge in law to a normal distribution. [Copyright &y& Elsevier]

Abstract: Fourier normal ordering (Unterberger, 2009)  is a new algorithm to construct explicit rough paths over arbitrary Hölder-continuous multidimensional paths. We apply in this article the Fourier normal ordering algorithm to the construction of an explicit rough path over multi-dimensional fractional Brownian motion with arbitrary Hurst index (in particular, for , which was till now an open problem) by regularizing the iterated integrals of the analytic approximation of defined in Unterberger (2009) . The regularization procedure is applied to ‘Fourier normal ordered’ iterated integrals obtained by permuting the order of integration so that innermost integrals have highest Fourier modes. The algebraic properties of this rough path are best understood using two Hopf algebras: the Hopf algebra of decorated rooted trees (Connes and Kreimer, 1998)  for the multiplicative or Chen property, and the shuffle algebra for the geometric or shuffle property. The rough path lives in Gaussian chaos of integer orders and is shown to have finite moments. As well-known, the construction of a rough path is the key to defining a stochastic calculus and solving stochastic differential equations driven by . The article Unterberger gives a quick overview of the method. [Copyright &y& Elsevier]

Abstract: By using Malliavin calculus and multiple Wiener–Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in the Watanabe spaces. [Copyright &y& Elsevier]

Abstract: We construct a sequence of processes that converges strongly to fractional Brownian motion uniformly on bounded intervals for any Hurst parameter , and we derive a rate of convergence, which becomes better when approaches . The construction is based on the Mandelbrot–van Ness stochastic integral representation of fractional Brownian motion and on a strong transport process approximation of Brownian motion. The objective of this method is to facilitate simulation. [Copyright &y& Elsevier]

We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener–Itô integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry–Esséen bounds in the Breuer–Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler’s formula for Ornstein–Uhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors. [ABSTRACT FROM AUTHOR]

Recent research suggests that fractional Brownian motion can be used to model the long-range dependence structure of the stock market. Fractional Brownian motion is not a semi-martingale and arbitrage opportunities do exist, however. Hu and Øksendal [Infin. Dimens. Anal., Quant. Probab. Relat. Top., 2003, 6, 1-32] and Elliott and van der Hoek [Math. Finan., 2003, 13, 301-330] propose the use of the white noise calculus approach to circumvent this difficulty. Under such a setting, they argue that arbitrage does not exist in the fractional market. To unravel this discrepancy, we examine the definition of self-financing strategies used by these authors. By refining their definitions, a new notion of continuously rebalanced self-financing strategies, which is compatible with simple buy and hold strategies, is given. Under this definition, arbitrage opportunities do exist in fractional markets. [ABSTRACT FROM AUTHOR]

The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of the chaos space of the random field, combined with the Wick product, leads to the Itô-Skorokhod integral, and provides an efficient tool to study the integral, both analytically and numerically. For a Gaussian process, a natural definition of the integral follows from a canonical correspondence between random processes and a special class of random fields. Also considered are the corresponding linear stochastic evolution equations. [ABSTRACT FROM AUTHOR]

We study the pathwise regularity of the map ... where φ is a vector function on R[supd] belonging to some Banach space V,X is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A continous version of this map, seen as a random element of the topological dual of V will be called stochastic current. We give sufficient conditions for the current to live in some Sobolev space of distributions and we provide elements to conjecture that those are also necessary. Next we verify the sufficient conditions when the process X is a d-dimensional fractional Brownian motion (fBm); we identify regularity in Sobolev spaces fBm with Hurst index H ∊ (1/4,1). Next we provide some results about general Sobolev regularity of currents when W is a standard Weiner process. Finally we discuss applications to a model of random vortex filaments in turbulent fluids. [ABSTRACT FROM AUTHOR]

We study several properties of the sub-fractional Brownian motion (fBm) introduced by Bojdecki et al. related to those of the fBm. This process is a self-similar Gaussian process depending on a parameter H ∈ (0, 2) with non stationary increments and is a generalization of the Brownian motion (Bm). The strong variation of the indefinite stochastic integral with respect to sub-fBm is also discussed. [ABSTRACT FROM AUTHOR]

Abstract: Methods are presented for calculating the evolution in time of the second moment properties of the output of linear systems subjected to fractional Brownian motion and fractional Gaussian noise, defined as the formal derivative of fractional Brownian motion. The study also examines whether the output of linear systems to fractional Brownian motion and fractional Gaussian noise exhibits long range dependence. Numerical examples are presented to illustrate the calculation of output statistics for some linear systems with fractional Brownian motion and fractional Gaussian noise input, and show that output of linear systems to these input processes may not have long memory. [Copyright &y& Elsevier]

Abstract: Multiple stochastic fractional integral expansions are applied to the problem of non-linear filtering of a signal observed in the presence of an additive noise, where the noise is modelled by a fractional Brownian motion with Hurst index greater than ½. It is shown that the best mean-square estimate of the signal can be represented as a ratio of two multiple integral series, where the stochastic integrals are defined in either the Itô or Stratonovich sense and taken with respect to the observation process, which is a persistent fractional Brownian motion under a suitable probability measure. Finally, motivated by practical considerations, finite expansion approximations to the optimal filter are studied. [Copyright &y& Elsevier]

Using the multiple stochastic integrals, we prove an existence and uniqueness result for a linear stochastic equation driven by the fractional Brownian motion with any Hurst parameter. We study both the one- and two-parameter cases. When the drift is zero. we show that in the one-parameter case the solution is an exponential--thus positive--function while in the two-parameter setting the solution is negative on a non negligible set. [ABSTRACT FROM AUTHOR]

In this paper we study necessary and sufficient conditions for the equivalence of Volterra Gaussian processes. Though this topic has already been studied in the literature, we provide new proofs, precisions and new theorems. We also give some examples of equivalent Volterra processes all related to the extensively studied fractional Brownian motion. Finally, we give an extension to general Gaussian processes of a recent regularization theorem by P. Cheridito. [Copyright &y& Elsevier]