Статті в журналах: "Brownian motion processes"

1

Rosen, Jay, and Jean-Dominique Deuschel. "motion, super-Brownian motion and related processes." Annals of Probability 26, no. 2 (April 1998): 602–43. http://dx.doi.org/10.1214/aop/1022855645.

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2

Takenaka, Shigeo. "Integral-geometric construction of self-similar stable processes." Nagoya Mathematical Journal 123 (September 1991): 1–12. http://dx.doi.org/10.1017/s0027763000003627.

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Recently, fractional Brownian motions are widely used to describe complex phenomena in several fields of natural science. In the terminology of probability theory the fractional Brownian motion is a Gaussian process {X(t) : t є R} with stationary increments which has a self-similar property, that is, there exists a constant H (for the Brownian motion H = 1/2, in general 0 < H < 1 for Gaussian processes) called the exponent of self-similarity of the process, such that, for any c > 0, two processes are subject to the same law (see [10]).

3

Rao, Nan, Qidi Peng, and Ran Zhao. "Cluster Analysis on Locally Asymptotically Self-Similar Processes with Known Number of Clusters." Fractal and Fractional 6, no. 4 (April 14, 2022): 222. http://dx.doi.org/10.3390/fractalfract6040222.

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We conduct cluster analysis of a class of locally asymptotically self-similar stochastic processes with finite covariance structures, which includes Brownian motion, fractional Brownian motion, and multifractional Brownian motion as paradigmatic examples. Given the true number of clusters, a new covariance-based dissimilarity measure is introduced, based on which we obtain approximately asymptotically consistent algorithms for clustering locally asymptotically self-similar stochastic processes. In the simulation study, clustering data sampled from fractional and multifractional Brownian motions with distinct Hurst parameters illustrates the approximated asymptotic consistency of the proposed algorithms. Clustering global financial markets’ equity indexes returns and sovereign CDS spreads provides a successful real world application. Implementations in MATLAB of the proposed algorithms and the simulation study are publicly shared in GitHub.

4

Andres, Sebastian, and Lisa Hartung. "Diffusion processes on branching Brownian motion." Latin American Journal of Probability and Mathematical Statistics 15, no. 2 (2018): 1377. http://dx.doi.org/10.30757/alea.v15-51.

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5

Ouknine, Y. "“Skew-Brownian Motion” and Derived Processes." Theory of Probability & Its Applications 35, no. 1 (January 1991): 163–69. http://dx.doi.org/10.1137/1135018.

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6

Katori, Makoto, and Hideki Tanemura. "Noncolliding Brownian Motion and Determinantal Processes." Journal of Statistical Physics 129, no. 5-6 (October 13, 2007): 1233–77. http://dx.doi.org/10.1007/s10955-007-9421-y.

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7

Adler, Robert J., and Ron Pyke. "Scanning Brownian Processes." Advances in Applied Probability 29, no. 2 (June 1997): 295–326. http://dx.doi.org/10.2307/1428004.

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The ‘scanning process' Z(t), t ∈ ℝk of the title is a Gaussian random field obtained by associating with Z(t) the value of a set-indexed Brownian motion on the translate t + A0 of some ‘scanning set' A0. We study the basic properties of the random field Z relating, for example, its continuity and other sample path properties to the geometrical properties of A0. We ask if the set A0 determines the scanning process, and investigate when, and how, it is possible to recover the structure of A0 from realisations of the sample paths of the random field Z.

8

Adler, Robert J., and Ron Pyke. "Scanning Brownian Processes." Advances in Applied Probability 29, no. 02 (June 1997): 295–326. http://dx.doi.org/10.1017/s0001867800028007.

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The ‘scanning process' Z(t), t ∈ ℝk of the title is a Gaussian random field obtained by associating with Z(t) the value of a set-indexed Brownian motion on the translate t + A 0 of some ‘scanning set' A 0. We study the basic properties of the random field Z relating, for example, its continuity and other sample path properties to the geometrical properties of A 0. We ask if the set A 0 determines the scanning process, and investigate when, and how, it is possible to recover the structure of A 0 from realisations of the sample paths of the random field Z.

9

SOTTINEN, TOMMI, and LAURI VIITASAARI. "CONDITIONAL-MEAN HEDGING UNDER TRANSACTION COSTS IN GAUSSIAN MODELS." International Journal of Theoretical and Applied Finance 21, no. 02 (March 2018): 1850015. http://dx.doi.org/10.1142/s0219024918500152.

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We consider so-called regular invertible Gaussian Volterra processes and derive a formula for their prediction laws. Examples of such processes include the fractional Brownian motions and the mixed fractional Brownian motions. As an application, we consider conditional-mean hedging under transaction costs in Black–Scholes type pricing models where the Brownian motion is replaced with a more general regular invertible Gaussian Volterra process.

10

Jedidi, Wissem, and Stavros Vakeroudis. "Windings of planar processes, exponential functionals and Asian options." Advances in Applied Probability 50, no. 3 (September 2018): 726–42. http://dx.doi.org/10.1017/apr.2018.33.

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Abstract Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and present a novel simple proof of the result of DeBlassie (1987), (1988) concerning the asymptotic behavior of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, as t→∞. We use the results of the windings approach in order to obtain results for quantities associated to the pricing of Asian options.

11

Sun, Xichao, Rui Guo, and Ming Li. "Some Properties of Bifractional Bessel Processes Driven by Bifractional Brownian Motion." Mathematical Problems in Engineering 2020 (October 17, 2020): 1–13. http://dx.doi.org/10.1155/2020/7037602.

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Let B = B t 1 , … , B t d t ≥ 0 be a d -dimensional bifractional Brownian motion and R t = B t 1 2 + ⋯ + B t d 2 be the bifractional Bessel process with the index 2 HK ≥ 1 . The Itô formula for the bifractional Brownian motion leads to the equation R t = ∑ i = 1 d ∫ 0 t B s i / R s d B s i + HK d − 1 ∫ 0 t s 2 HK − 1 / R s d s . In the Brownian motion case K = 1 and H = 1 / 2 , X t ≔ ∑ i = 1 d ∫ 0 t B s i / R s d B s i , d ≥ 1 is a Brownian motion by Lévy’s characterization theorem. In this paper, we prove that process X t is not a bifractional Brownian motion unless K = 1 and H = 1 / 2 . We also study some other properties and their application of this stochastic process.

12

Didier, Kumwimba Seya, Walo Omana Rebecca, Mabela Matendo Rostin, Badibi Omak Christopher, Kankolongo Kadilu Patient, and Marcel Remon. "FUZZY ORNSTEIN-UHLENBECK AND BROWNIAN GEOMETRIC MOTION PROCESSES DRIVEN BY A FUZZY BROWNIAN MOTION." Advances in Fuzzy Sets and Systems 27, no. 1 (March 3, 2022): 95–110. http://dx.doi.org/10.17654/0973421x22005.

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13

Adler, Robert J., and Gennady Samorodnitsky. "Super Fractional Brownian Motion, Fractional Super Brownian Motion and Related Self-Similar (Super) Processes." Annals of Probability 23, no. 2 (April 1995): 743–66. http://dx.doi.org/10.1214/aop/1176988287.

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14

Dung, Nguyen Tien. "JACOBI PROCESSES DRIVEN BY FRACTIONAL BROWNIAN MOTION." Taiwanese Journal of Mathematics 18, no. 3 (May 2014): 835–48. http://dx.doi.org/10.11650/tjm.18.2014.3288.

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15

Inoue, A., and V. V. Anh. "Prediction of Fractional Brownian Motion-Type Processes." Stochastic Analysis and Applications 25, no. 3 (May 2, 2007): 641–66. http://dx.doi.org/10.1080/07362990701282971.

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16

Lim, S. C., and C. H. Eab. "Some fractional and multifractional Gaussian processes: A brief introduction." International Journal of Modern Physics: Conference Series 36 (January 2015): 1560001. http://dx.doi.org/10.1142/s2010194515600010.

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This paper gives a brief introduction to some important fractional and multifractional Gaussian processes commonly used in modelling natural phenomena and man-made systems. The processes include fractional Brownian motion (both standard and the Riemann-Liouville type), multifractional Brownian motion, fractional and multifractional Ornstein-Uhlenbeck processes, fractional and mutifractional Reisz-Bessel motion. Possible applications of these processes are briefly mentioned.

17

El-Nouty, Charles. "THE GENERALIZED BIFRACTIONAL BROWNIAN MOTION." International Journal for Computational Civil and Structural Engineering 14, no. 4 (December 21, 2018): 81–89. http://dx.doi.org/10.22337/2587-9618-2018-14-4-81-89.

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To extend several known centered Gaussian processes, we introduce a new centered Gaussian process, named the generalized bifractional Brownian motion. This process depends on several parameters, namely α > 0 , β>0 , 0<H<1 and 0<K≤1 . When K=1, we investigate its convexity properties. Then, when 2HK≤ 1, we prove that this process is an element of the QHASI class, a class of centered Gaussian processes, which was introduced in 2015

18

Golmankhaneh, Alireza Khalili, and Renat Timergalievich Sibatov. "Fractal Stochastic Processes on Thin Cantor-Like Sets." Mathematics 9, no. 6 (March 15, 2021): 613. http://dx.doi.org/10.3390/math9060613.

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We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal.

19

Manurung, Tohap. "Hubungan Antara Brownian Motion (The Winner Process) dan Surplus Process." JURNAL ILMIAH SAINS 12, no. 1 (April 30, 2012): 47. http://dx.doi.org/10.35799/jis.12.1.2012.401.

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HUBUNGAN ANTARA BROWNIAN MOTION (THE WIENER PROCESS) DAN SURPLUS PROCESS ABSTRAK Suatu analisis model continous-time menjadi cakupan yang akan dibahas dalam tulisan ini. Dengan demikian pengenalan proses stochastic akan sangat berperan. Dua proses akan di analisis yaitu proses compound Poisson dan Brownian motion. Proses compound Poisson sudah menjadi model standard untuk Ruin analysis dalam ilmu aktuaria. Sementara Brownian motion sangat berguna dalam teori keuangan modern dan juga dapat digunakan sebagai approksimasi untuk proses compound Poisson. Hal penting dalam tulisan ini adalah menujukkan bagaimana surplus process berdasarkan proses resiko compound Poisson dihubungkan dengan Brownian motion with Drift Process. Kata kunci: Brownian motion with Drift process, proses surplus, compound Poisson RELATIONSHIP BETWEEN BROWNIAN MOTION (THE WIENER PROCESS) AND THE SURPLUS PROCESS ABSTRACT An analysis of continous-time models is covered in this paper. Thus, this requires an introduction to stochastic processes. Two processes are analyzed: the compound Poisson process and Brownian motion. The compound Poisson process has been the standard model for ruin analysis in actuarial science, while Brownian motion has found considerable use in modern financial theory and also can be used as an approximation to the compound Pisson process. The important thing is to show how the surplus process based on compound poisson risk process is related to Brownian motion with drift process. Keywords: Brownian motion with drift process, surplus process, compound Poisson

20

Le Gall, Jean-François, and Armand Riera. "Growth-fragmentation processes in Brownian motion indexed by the Brownian tree." Annals of Probability 48, no. 4 (July 2020): 1742–84. http://dx.doi.org/10.1214/19-aop1406.

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21

Abundo, Mario, and Enrica Pirozzi. "On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes." Mathematics 7, no. 10 (October 18, 2019): 991. http://dx.doi.org/10.3390/math7100991.

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We investigate the main statistical parameters of the integral over time of the fractional Brownian motion and of a kind of pseudo-fractional Gaussian process, obtained as a classical Gauss–Markov process from Doob representation by replacing Brownian motion with fractional Brownian motion. Possible applications in the context of neuronal models are highlighted. A fractional Ornstein–Uhlenbeck process is considered and relations with the integral of the pseudo-fractional Gaussian process are provided.

22

Perry, D., W. Stadje, and S. Zacks. "The first rendezvous time of Brownian motion and compound Poisson-type processes." Journal of Applied Probability 41, no. 4 (December 2004): 1059–70. http://dx.doi.org/10.1239/jap/1101840551.

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The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

23

Perry, D., W. Stadje, and S. Zacks. "The first rendezvous time of Brownian motion and compound Poisson-type processes." Journal of Applied Probability 41, no. 04 (December 2004): 1059–70. http://dx.doi.org/10.1017/s0021900200020829.

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The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

24

El-Nouty, Charles, and Darya Filatova. "ON THE QHASI CLASS AND ITS EXTENSION TO SOME GAUSSIAN SHEETS." International Journal for Computational Civil and Structural Engineering 18, no. 3 (September 27, 2022): 54–64. http://dx.doi.org/10.22337/2587-9618-2022-18-3-54-64.

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Introduced in 2018 the generalized bifractional Brownian motion is considered as an element of the quasi-helix with approximately stationary increment class of real centered Gaussian processes conditioning by parameters. This paper proves that the generalized bifractional Brownian motion is an element of the above mentioned class with no condition on parameters. The quasi-helix with approximately stationary increment class of real centered Gaussian processes is extended to two-dimensional processes as the fractional Brownian sheet, the sub-fractional Brownian sheet, and the bifractional Brownian sheet. This generalized presentation of the class of stochastic processes is used to augment the training samples for generative adversarial networks in computer vision problem.

25

ENGELKE, SEBASTIAN, and JEANNETTE H. C. WOERNER. "A UNIFYING APPROACH TO FRACTIONAL LÉVY PROCESSES." Stochastics and Dynamics 13, no. 02 (March 4, 2013): 1250017. http://dx.doi.org/10.1142/s0219493712500177.

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Starting from the moving average representation of fractional Brownian motion, there are two different approaches to constructing fractional Lévy processes in the literature. Applying L2-integration theory, one can keep the same moving average kernel and replace the driving Brownian motion by a pure jump Lévy process with finite second moments. Alternatively, in the framework of alpha-stable random measures, the Brownian motion is replaced by an alpha-stable Lévy process and the exponent in the kernel is reparametrized by H - 1/α. We now provide a unified approach taking kernels of the form [Formula: see text], where γ can be chosen according to the existing moments and the Blumenthal–Getoor index of the underlying Lévy process. These processes may exhibit both long and short range dependence. In addition we will examine further properties of the processes, e.g., regularity of the sample paths and the semimartingale property.

26

Pagnini, Gianni, Antonio Mura, and Francesco Mainardi. "Generalized Fractional Master Equation for Self-Similar Stochastic Processes Modelling Anomalous Diffusion." International Journal of Stochastic Analysis 2012 (October 16, 2012): 1–14. http://dx.doi.org/10.1155/2012/427383.

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The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function) emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

27

Munis, Rafaele Almeida, Diego Aparecido Camargo, Richardson Barbosa Gomes da Silva, Miriam Harumi Tsunemi, Siti Nur Iqmal Ibrahim, and Danilo Simões. "Price Modeling of Eucalyptus Wood under Different Silvicultural Management for Real Options Approach." Forests 13, no. 3 (March 18, 2022): 478. http://dx.doi.org/10.3390/f13030478.

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Choosing the ideal number of rotations of planted forests under a silvicultural management regime results in uncertainties in the cash flows of forest investment projects. We verified if there is parity in the Eucalyptus wood price modeling through fractional Brownian motion and geometric Brownian motion to incorporate managerial flexibilities into investment projects in planted forests. We use empirical data from three production cycles of forests planted with Eucalyptus grandis × E. urophylla in the projection of discounted cash flows. The Eucalyptus wood price, assumed as uncertainty, was modeled using fractional and geometric Brownian motion. The discrete-time pricing of European options was obtained using the Monte Carlo method. The root mean square error of fractional and geometric Brownian motions was USD 1.4 and USD 2.2, respectively. The real options approach gave the investment projects, with fractional and geometric Brownian motion, an expanded present value of USD 8,157,706 and USD 9,162,202, respectively. Furthermore, in both models, the optimal harvest ages execution was three rotations. Thus, with an indication of overvaluation of 4.9% when assimilating the geometric Brownian motion, there is no parity between stochastic processes, and three production cycles of Eucalyptus planted forests are economically viable.

28

Fu, James C., and Tung-Lung Wu. "Linear and Nonlinear Boundary Crossing Probabilities for Brownian Motion and Related Processes." Journal of Applied Probability 47, no. 4 (December 2010): 1058–71. http://dx.doi.org/10.1239/jap/1294170519.

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We propose a new method to obtain the boundary crossing probabilities or the first passage time distribution for linear and nonlinear boundaries for Brownian motion. The method also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a finite Markov chain, and the boundary crossing probability of Brownian motion is cast as the limiting probability of the finite Markov chain entering a set of absorbing states induced by the boundaries. Error bounds are obtained. Numerical results for various types of boundary studied in the literature are provided in order to illustrate our method.

29

Fu, James C., and Tung-Lung Wu. "Linear and Nonlinear Boundary Crossing Probabilities for Brownian Motion and Related Processes." Journal of Applied Probability 47, no. 04 (December 2010): 1058–71. http://dx.doi.org/10.1017/s0021900200007361.

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We propose a new method to obtain the boundary crossing probabilities or the first passage time distribution for linear and nonlinear boundaries for Brownian motion. The method also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a finite Markov chain, and the boundary crossing probability of Brownian motion is cast as the limiting probability of the finite Markov chain entering a set of absorbing states induced by the boundaries. Error bounds are obtained. Numerical results for various types of boundary studied in the literature are provided in order to illustrate our method.

30

Kleptsyna, M. L., P. E. Kloeden, and V. V. Anh. "Linear filtering with fractional Brownian motion in the signal and observation processes." Journal of Applied Mathematics and Stochastic Analysis 12, no. 1 (January 1, 1999): 85–90. http://dx.doi.org/10.1155/s1048953399000076.

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Integral equations for the mean-square estimate are obtained for the linear filtering problem, in which the noise generating the signal is a fractional Brownian motion with Hurst index h∈(3/4,1) and the noise in the observation process includes a fractional Brownian motion as well as a Wiener process.

31

López, Sergio I. "Convergence of tandem Brownian queues." Journal of Applied Probability 53, no. 2 (June 2016): 585–92. http://dx.doi.org/10.1017/jpr.2016.22.

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AbstractIt is known that in a stationary Brownian queue with both arrival and service processes equal in law to Brownian motion, the departure process is a Brownian motion, identical in law to the arrival process: this is the analogue of Burke's theorem in this context. In this paper we prove convergence in law to this Brownian motion in a tandem network of Brownian queues: if we have an arbitrary continuous process, satisfying some mild conditions, as an initial arrival process and pass it through an infinite tandem network of queues, the resulting process weakly converges to a Brownian motion. We assume independent and exponential initial workloads for all queues.

32

Araman, Victor F., and Peter W. Glynn. "Fractional Brownian Motion with H < 1/2 as a Limit of Scheduled Traffic." Journal of Applied Probability 49, no. 3 (September 2012): 710–18. http://dx.doi.org/10.1239/jap/1346955328.

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In this paper we show that fractional Brownian motion with H < ½ can arise as a limit of a simple class of traffic processes that we call ‘scheduled traffic models’. To our knowledge, this paper provides the first simple traffic model leading to fractional Brownnian motion with H < ½. We also discuss some immediate implications of this result for queues fed by scheduled traffic, including a heavy-traffic limit theorem.

33

Marouby, Matthieu. "Micropulses and Different Types of Brownian Motion." Journal of Applied Probability 48, no. 3 (September 2011): 792–810. http://dx.doi.org/10.1239/jap/1316796915.

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In this paper we study sums of micropulses that generate different kinds of processes. Fractional Brownian motion and bifractional Brownian motion are obtained as limit processes. Moreover, we not only prove the convergence of finite-dimensional laws but also, in some cases, convergence in distribution in the space of right-continuous functions with left limits. Finally, we obtain generalizations with multidimensional indices.

34

Marouby, Matthieu. "Micropulses and Different Types of Brownian Motion." Journal of Applied Probability 48, no. 03 (September 2011): 792–810. http://dx.doi.org/10.1017/s0021900200008329.

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In this paper we study sums of micropulses that generate different kinds of processes. Fractional Brownian motion and bifractional Brownian motion are obtained as limit processes. Moreover, we not only prove the convergence of finite-dimensional laws but also, in some cases, convergence in distribution in the space of right-continuous functions with left limits. Finally, we obtain generalizations with multidimensional indices.

35

Kobryn, Hayashi, and Arimitsu. "QUANTUM STOCHASTIC PROCESSES: BOSON AND FERMION BROWNIAN MOTION." Condensed Matter Physics 6, no. 4 (2003): 637. http://dx.doi.org/10.5488/cmp.6.4.637.

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36

MANCINO, MARIA ELVIRA. "DIFFUSION PROCESSES WITH RESPECT TO FREE BROWNIAN MOTION." Infinite Dimensional Analysis, Quantum Probability and Related Topics 03, no. 03 (September 2000): 435–43. http://dx.doi.org/10.1142/s0219025700000273.

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37

Macedo-Junior, A. F., and A. M. S. Macêdo. "Brownian-motion ensembles: correlation functions of determinantal processes." Journal of Physics A: Mathematical and Theoretical 41, no. 1 (December 12, 2007): 015004. http://dx.doi.org/10.1088/1751-8113/41/1/015004.

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38

McCauley, Joseph L., Gemunu H. Gunaratne, and Kevin E. Bassler. "Hurst exponents, Markov processes, and fractional Brownian motion." Physica A: Statistical Mechanics and its Applications 379, no. 1 (June 2007): 1–9. http://dx.doi.org/10.1016/j.physa.2006.12.028.

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39

Abramson, Joshua, and Steven N. Evans. "Lipschitz minorants of Brownian motion and Lévy processes." Probability Theory and Related Fields 158, no. 3-4 (March 30, 2013): 809–57. http://dx.doi.org/10.1007/s00440-013-0497-9.

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40

Herzog, Bodo. "Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion." Mathematics 10, no. 3 (January 23, 2022): 340. http://dx.doi.org/10.3390/math10030340.

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The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions {BtH,t≥0} and sub-fractional Brownian motions {ξtH,t≥0} with Hurst parameter H∈(12,1). We start by establishing the connection between a fPDE and SDE via the Feynman–Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional- and sub-fractional Brownian motions and prove the generalized Feynman–Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory.

41

Caramellino, Lucia, and Barbara Pacchiarotti. "Large deviation estimates of the crossing probability for pinned Gaussian processes." Advances in Applied Probability 40, no. 2 (June 2008): 424–53. http://dx.doi.org/10.1239/aap/1214950211.

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The paper deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay in n fixed points at n fixed past instants. In particular, functional large deviation results are stated for small time. Several examples are considered: integrated or not fractional Brownian motions and m-fold integrated Brownian motion. As an application, the asymptotic behavior of the exit probability is studied and used for the practical purpose of the numerical computation, via Monte Carlo methods, of the hitting probability up to a given time of the unpinned process.

42

Caramellino, Lucia, and Barbara Pacchiarotti. "Large deviation estimates of the crossing probability for pinned Gaussian processes." Advances in Applied Probability 40, no. 02 (June 2008): 424–53. http://dx.doi.org/10.1017/s0001867800002597.

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The paper deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay in n fixed points at n fixed past instants. In particular, functional large deviation results are stated for small time. Several examples are considered: integrated or not fractional Brownian motions and m-fold integrated Brownian motion. As an application, the asymptotic behavior of the exit probability is studied and used for the practical purpose of the numerical computation, via Monte Carlo methods, of the hitting probability up to a given time of the unpinned process.

43

Vasylyk, O. I., та I. I. Lovytska. "Simulation of a strictly φ-sub-Gaussian generalized fractional Brownian motion". Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, № 1 (2021): 11–19. http://dx.doi.org/10.17721/1812-5409.2021/1.1.

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In the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fractional Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly φ-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when φ(x) = (|x|^p)/p, |x| ≥ 1, p > 1. In order to obtain these results, we use some results from the theory of φ-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment.

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Vasylyk, O. I., I. V. Rozora, T. O. Ianevych та I. I. Lovytska. "On some method on model construction for strictly φ-sub-Gaussian generalized fractional Brownian motion". Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, № 2 (2021): 18–25. http://dx.doi.org/10.17721/1812-5409.2021/2.3.

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In the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fracti-onal Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly φ-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when φ(x) = exp{|x|} − |x| − 1, x ∈ R. In order to obtain these results, we use some results from the theory of φ-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment.

45

Araman, Victor F., and Peter W. Glynn. "Fractional Brownian Motion with H < 1/2 as a Limit of Scheduled Traffic." Journal of Applied Probability 49, no. 03 (September 2012): 710–18. http://dx.doi.org/10.1017/s0021900200009487.

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In this paper we show that fractional Brownian motion with H < ½ can arise as a limit of a simple class of traffic processes that we call ‘scheduled traffic models’. To our knowledge, this paper provides the first simple traffic model leading to fractional Brownnian motion with H < ½. We also discuss some immediate implications of this result for queues fed by scheduled traffic, including a heavy-traffic limit theorem.

46

Chronopoulou, Alexandra, and Georgios Fellouris. "Optimal Sequential Change Detection for Fractional Diffusion-Type Processes." Journal of Applied Probability 50, no. 1 (March 2013): 29–41. http://dx.doi.org/10.1239/jap/1363784422.

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The problem of detecting an abrupt change in the distribution of an arbitrary, sequentially observed, continuous-path stochastic process is considered and the optimality of the CUSUM test is established with respect to a modified version of Lorden's criterion. We apply this result to the case that a random drift emerges in a fractional Brownian motion and we show that the CUSUM test optimizes Lorden's original criterion when a fractional Brownian motion with Hurst index H adopts a polynomial drift term with exponent H+1/2.

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Chronopoulou, Alexandra, and Georgios Fellouris. "Optimal Sequential Change Detection for Fractional Diffusion-Type Processes." Journal of Applied Probability 50, no. 01 (March 2013): 29–41. http://dx.doi.org/10.1017/s0021900200013097.

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The problem of detecting an abrupt change in the distribution of an arbitrary, sequentially observed, continuous-path stochastic process is considered and the optimality of the CUSUM test is established with respect to a modified version of Lorden's criterion. We apply this result to the case that a random drift emerges in a fractional Brownian motion and we show that the CUSUM test optimizes Lorden's original criterion when a fractional Brownian motion with Hurst index H adopts a polynomial drift term with exponent H+1/2.

48

Xie, Huantian, and Nenghui Kuang. "Least squares type estimations for discretely observed nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind." AIMS Mathematics 7, no. 1 (2021): 1095–114. http://dx.doi.org/10.3934/math.2022065.

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<abstract><p>We consider the nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind defined by $ dX_t = \theta X_tdt+dY_t^{(1)}, t\geq 0, X_0 = 0 $ with an unknown parameter $ \theta > 0, $ where $ dY_t^{(1)} = e^{-t}dG_{a_{t}} $ and $ \{G_t, t\geq 0\} $ is a mean zero Gaussian process with the self-similar index $ \gamma\in (\frac{1}{2}, 1) $ and $ a_t = \gamma e^{\frac{t}{\gamma}} $. Based on the discrete observations $ \{X_{t_i}:t_i = i\Delta_n, i = 0, 1, \cdots, n\} $, two least squares type estimators $ \hat{\theta}_n $ and $ \tilde{\theta}_n $ of $ \theta $ are constructed and proved to be strongly consistent and rate consistent. We apply our results to the cases such as fractional Brownian motion, sub-fractional Brownian motion, bifractional Brownian motion and sub-bifractional Brownian motion. Moreover, the numerical simulations confirm the theoretical results.</p></abstract>

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Buchmann, Boris, and Claudia Klüppelberg. "Maxima of stochastic processes driven by fractional Brownian motion." Advances in Applied Probability 37, no. 3 (September 2005): 743–64. http://dx.doi.org/10.1239/aap/1127483745.

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We study stationary processes given as solutions to stochastic differential equations driven by fractional Brownian motion. This rich class includes the fractional Ornstein-Uhlenbeck process and those processes that can be obtained from it by state space transformations. An explicit formula in terms of Euler's Γ-function describes the asymptotic behaviour of the covariance function of the fractional Ornstein-Uhlenbeck process near zero, which, by an application of Berman's condition, guarantees that this process is in the maximum domain of attraction of the Gumbel distribution. Necessary and sufficient conditions on the state space transforms are stated to classify the maximum domain of attraction of solutions to stochastic differential equations driven by fractional Brownian motion.

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Buchmann, Boris, and Claudia Klüppelberg. "Maxima of stochastic processes driven by fractional Brownian motion." Advances in Applied Probability 37, no. 03 (September 2005): 743–64. http://dx.doi.org/10.1017/s0001867800000458.

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We study stationary processes given as solutions to stochastic differential equations driven by fractional Brownian motion. This rich class includes the fractional Ornstein-Uhlenbeck process and those processes that can be obtained from it by state space transformations. An explicit formula in terms of Euler's Γ-function describes the asymptotic behaviour of the covariance function of the fractional Ornstein-Uhlenbeck process near zero, which, by an application of Berman's condition, guarantees that this process is in the maximum domain of attraction of the Gumbel distribution. Necessary and sufficient conditions on the state space transforms are stated to classify the maximum domain of attraction of solutions to stochastic differential equations driven by fractional Brownian motion.

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