Shevchenko, Existence and uniqueness of the solution of stochastic differential equation involving wiener process and fractional Brownian motion with Hurst index H > 1/2, Comm.. R˘ a¸[r]

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H-fBm is H-self-similar and has stationary increments. For H = 1 2 , **fractional** **Brownian** **motion** is standard **Brownian** **motion** and denoted by W . FBm is interesting from a theoretical point of view, since it is fairly simple, but neither a Markov process, nor a semimartingale.
Recently, the process has been studied extensively in connection to various applications, for example in finance and telecommunications. Important tools when working with fBm are its integral representations: for a fixed Hurst index K ∈ (0, 1), on the one hand, there exists a K-fBm B t K

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(ii) A portfolio is called almost simple, if there is a sequence of nondecreas- ing stopping times (τ k ) k∈N such that P (τ k = T inﬁnitely often) = 1 and the portfolio is constant on (τ k , τ k+1 ]. This means that the number of rear- ranging times is ﬁnite on almost every path, but not necessarily bounded as function on Ω. Existence of a self-ﬁnancing almost simple arbitrage has been shown by Rogers [20], making use of the history of a **fractional** Brown- ian **motion** starting at −∞ , and subsequently by Cheridito [8], taking only the history starting from 0 into account. Both constructions rely on the long memory property of the **fractional** **Brownian** **motion**.

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Itô’s semimartingale driven by a **Brownian** **motion** is typically used in modeling the asset prices, interest rates and ex- change rates, and so on. However, the assumption of **Brownian** **motion** as a driving force of the underlying asset price processes is rarely contested in practice. This naturally raises the question of whether this assumption is really appropri- ate. In the paper we propose a statistical test to answer the above question using high frequency data. The test can be used to validate the assumption of semimartingale framework and test for the existence of the long run dependence cap- tured by the **fractional** **Brownian** **motion** in a parsimonious way. Asymptotic properties of the test statistics are investi- gated. Simulations justify the performance of the test. Real data sets are also analyzed.

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1.2. Statement of problem and result
We have seen some well known examples of representations of **fractional** **Brownian** **motion**. Notice that the Molchan-Golosov representation requires integration only on a ﬁnite interval. This will be useful for extending from a Gaussian process in R to R d . In this thesis, we are interested in this kind of integral representations given by integration over compactly supported kernel functions. Roughly speaking, the extension can be achieved from R to R d by an idea of rotating vectors in R d . Indeed, let B H (t) = 0 t K(t, u)dB(u) be a **fractional** **Brownian** **motion** in R. Then we will prove the following result.

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There are many ways to tackle the **fractional** **Brownian** **motion**. In this paper, we use a convolution of a white noise by a distribution T . This distribution operates in principal value as explained in paragraphs I and II. In paragraph III anf IV is defined a Skorohod type integral with respect to the fBm. This allows to define vector valued rough paths which lead to rough paths in the sense of T.Lyons. In paragraph V, we indicate a regularization process of the fBm by convolution with some examples. We study convergence of Riemann sums in paragraph VI, this also leads to approximations by piecewise linear processes of fBm-Skorohod and fBm-Stratonovich type integrals. Paragraph VII is devoted to prove that every IR d -valued fBm defined on IR can be studied in this way.

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Abstract: In this paper, firstly, we introduce and study a self-similar Gaussian process with parameters H ∈ (0, 1) and K ∈ (0,1] that is an extension of the well known sub-**fractional** **Brownian** **motion** introduced by Bojdecki et al. [4]. Secondly, by using a decomposition in law of this process, we prove the existence and the joint continuity of its local time.

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This paper aims to give a few aspects of the recent theory and applications of the **fractional** **Brownian** **motion**. We begin by the construction of the process for which recent theoretical advances simplify the computer simulation of sample{ paths. Section 4 is devoted to the problem of the denition of a stochastic integral with respect to the **fractional** **Brownian** **motion**. In Section 5, we give several applications in dierent elds : queuing networks, ltering theory, mathematical nance. The appendix contains a very brief summary of the notion of deterministic **fractional** calculus we repeatedly use here. Throughout this paper, we will try to show how innite{dimensional processes arise naturally when studying long{range dependent processes. Figuratively, this amounts to say that we put the memory into the state of the process.

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2 Instituto de Física, Universidade de São Paulo, Rua do Matão, 1371, 05508-090 São Paulo, São Paulo, Brazil
3 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA
(Received 22 November 2017; revised manuscript received 28 December 2017; published 13 February 2018) **Fractional** **Brownian** **motion**, a stochastic process with long-time correlations between its increments, is a prototypical model for anomalous diffusion. We analyze **fractional** **Brownian** **motion** in the presence of a reflecting wall by means of Monte Carlo simulations. Whereas the mean-square displacement of the particle shows the expected anomalous diffusion behavior x 2 ∼ t α , the interplay between the geometric confinement and the long-time memory leads to a highly non-Gaussian probability density function with a power-law singularity at the barrier. In the superdiffusive case α > 1, the particles accumulate at the barrier leading to a divergence of the probability density. For subdiffusion α < 1, in contrast, the probability density is depleted close to the barrier.

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FBm is interesting from a theoretical point of view, since it is fairly simple but neither Markov nor semimartingale. The latter fact, for example, makes integration with respect to it challenging.
FBm was first mentioned and studied by Kolmogorov in 1940 under the name of Wiener spiral (see [6]). The modern name **fractional** **Brownian** **motion** was proposed by Mandelbrot and Van Ness in 1968, when they described fBm by a Wiener integral process of a **fractional** integral kernel, namely

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LAURI VIITASAARI
Department of Mathematics and System Analysis, Aalto University School of Science, Helsinki, P.O. Box 11100, FIN-00076 Aalto, FINLAND
Abstract. We calculate the regular conditional future law of the **fractional** **Brownian** **motion** with index H ∈ (0, 1) conditioned on its past. We show that the conditional law is continuous with respect to the conditioning path. We investigate the path properties of the conditional process and the asymptotic behavior of the conditional covariance.

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t , ..., B t H,d ); t ∈ [0, T ]
o
be a d-dimensional **fractional** **Brownian** **motion** (fBm) defined on the probability space (Ω, F, P ). That is, B H is a zero mean Gaussian vector whose components are independent one- dimensional **fractional** **Brownian** motions with Hurst parameter H ∈ (0, 1), i.e., for every i = 1, ..., d B H,i is a Gaussian process and covariance function given by

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CYLINDRICAL **FRACTIONAL** **BROWNIAN** **MOTION** IN BANACH SPACES
ELENA ISSOGLIO AND MARKUS RIEDLE
Abstract. In this article we introduce cylindrical **fractional** **Brownian** motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical **fractional** **Brownian** **motion** is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical **fractional** **Brownian** **motion**, which is analogous to the Karhunen-Lo`eve expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical **fractional** **Brownian** **motion**.

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9. Cheﬁdito, P: Regularizing **Fractional** **Brownian** **Motion** with a Miew towards Stock Price Modeling. Dissertation, Zurich University, Zurich (2002)
10. El-Nouty, C: The **fractional** mixed **fractional** **Brownian** **motion**. Stat. Probab. Lett. 65, 111-120 (2003)
11. He, X, Chen, W: The pricing of credit default swaps under a **fractional** mixed **fractional** **Brownian** **motion**. Phys. A 404, 26-33 (2014)

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Inspired by all the above works, the purpose of this paper is to research a weak approx- imation of a complex **fractional** **Brownian** **motion** from a standard Poisson process and from a Lévy process, respectively, by the method in Delgado and Jolis [11].
Let {M t , t ≥ 0} be a Poisson process of parameter 2. We deﬁne {N t , t ≥ 0} and {N t , t ≥ 0}

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The purpose of our paper is to develop a stochastic calculus with respect to the **fractional** **Brownian** **motion** B with Hurst parameter H > 2 1 using the techniques of the Malliavin calculus. Unlike some previous works (see, for instance, [3]) we will not use the integral representation of B as a stochastic integral with respect to a Wiener process. Instead of this we will rely on the intrinsic Malliavin calculus with respect to B.

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Abstract
This research aims to investigate a model for pricing of currency options in which value governed by the **fractional** **Brownian** **motion** model (FBM). The **fractional** partial differ- ential equation and some Greeks are also obtained. In addition, some properties of our pricing formula and simulation studies are presented, which demonstrate that the FBM model is easy to use.

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Keywords: Stock Prices; Black-Scholes Model; Brown Movement; **Fractional**
**Brownian** **Motion**; Options; Volatility
1. Introduction
Brown **Motion** theory is usually used for researching the change of asset price in previous option pricing theory, while comparatively speaking, **Fractional** Brown **Motion** theory has related nature between incremental quantities, and using it to research asset price can more reflect some characteristics of stock yield with more extensive significance. In this text, **Fractional** Brown **Motion** theory during random process is applied to research option pricing problem. [1-4].

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The Hurst parameter H accounts not only for the sign of the correlation of the increments, but also for the regularity of the sample paths. Indeed, for H > 1 2 , the increments are positively correlated, and for H < 1 2 they are negatively correlated. Furthermore, for every β ∈ (0, H), its sample paths are almost surely H¨older continuous with index β. Finally, it is worthy of note that for H > 1 2 , according to Beran’s definition [3], it is a long memory process: the covariance of increments at distance u decrease as u 2H −2 . These significant properties make **fractional** **Brownian** **motion** a natural can- didate as a model of noise in mathematical finance (see Comte and Re- nault [5], Rogers [26]), and in communication networks (see, for instance, Leland, Taqqu and Willinger [16]).

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This follows exactly from the same arguments as in the proof of Theorem 1.2 in [4]. 5. Synthesis of the mfBm
5.1. Introduction. The exact simulation of the **fractional** **Brownian** **motion** has been a question of great interest in the nineties. This may be done by generating a sample path of a **fractional** Gaussian noise. An important step towards efficient simulation was obtained after the work of Wood and Chan [32] about the simulation of arbitrary stationary Gaussian sequences with prescribed covariance function. The technique relies upon the embedding of the covariance matrix into a circulant matrix, a square root of which is easily calculated using the discrete Fourier transform. This leads to a very efficient algorithm, both in terms of computation time and storage needs. Wood and Chan method is an exact simulation method provided that the circulant matrix is semidefinite positive, a property that is not always satisfied. However, for the **fractional** Gaussian noise, it can be proved that the circulant matrix is definite positive for all H ∈ (0, 1), see [9, 13].

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