· 2019. 1. 23. · V. S. Koroliuk, I. V. Samoilenko Estimation of the remainder in asymptotic expansion of a functional of semi-Markov random evolution 205 M. Sreehari Order of

ТЕОРІЯ ЙМОВІРНОСТЕЙ ТА МАТЕМАТИЧНА СТАТИСТИКА

ІSSN 0868-6904

1(98)/2018

Засновано 1970 року

Публікуються оригінальні статті з актуальних питань теорії ймовірнос-тей, теорії випадкових процесів та полів, математичної статистики та їх засто-сувань. Усі матеріали, що надходять до редколегії, рецензуються. Після виходу у світ усі матеріали реферуються у «Mathematical Reviews» (AMS), «Zentralblatt für Mathematik» (Springer) та «Statistical Methods and Applications» (ISI). Журнал включено до науково-метричної бази Scopus (Elsevier) із 2004 р.

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ГОЛОВНИЙ РЕДАКТОР В. С. Королюк (Україна)

ЗАСТУПНИКИ ГОЛОВНОГО РЕДАКТОРА

Ю. В. Козаченко (Україна), Ю. С. Мішура (Україна)

ВІДПОВІДАЛЬНИЙ СЕКРЕТАР Л. М. Сахно (Україна)

РЕДАКЦІЙНА КОЛЕГІЯ В. В. Анісімов (Велика Британія), Т. Боднар (Швеція), О. Ю. Веретенніков (Велика Британія, Російська Федерація), О. О. Гущін (Російська Федерація), М. Доцці (Франція), О. К. Закусило (Україна), О. В. Іванов (Україна), М. В. Карташов (Україна), О. Г. Кукуш (Україна), М. М. Леоненко (Велика Британія), І. К. Мацак (Україна), М. П. Моклячук (Україна), І. Молчанов (Швейцарія), О. О. Новіков (Австралія), А. Я. Оленко (Австралія), Е. Орсінгер (Італія), М. І. Портенко (Україна), Д. С. Сільвестров (Швеція), Г. М. Шевченко (Україна)

Адреса редколегії Кафедра теорії ймовірностей, статистики та актуарної математики, механіко-математичний факультет, Київський національний університет імені Тараса Шевченка, вул. Володимирська, 64/13, м. Київ, Україна, 01601; ( і факс: (38 044) 259 03 92, e-mail: [email protected] web-page: http://probability.univ.kiev.ua/tims/ http://www.ams.org/journals/tpms/

Затверджено Вченою радою механіко-математичного факультету 16.04.18 (протокол № 8)

Засновник Київський національний університет імені Тараса Шевченка

Зареєстровано Міністерством юстиції України. Свідоцтво про державну реєстрацію КВ № 22437-12337ПР від 24.11.16

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Адреса видавця ВПЦ "Київський університет" (кімн. 43), б-р Т. Шевченка, 14, м. Київ, Україна, 01601 ( (38 044) 239 32 22; факс 239 31 28

© Київський національний університет імені Тараса Шевченка, Видавничо-поліграфічний центр "Київський університет", 2018

TEORIYA ĬMOVIRNOSTEĬ TA MATEMATYCHNA STATYSTYKA

ІSSN 0868-6904

1(98)/2018

Founded in 1970

The journal publishes original research papers devoted to advanced prob-lems in probability theory, theory of random processes and fields, mathematical sta-tistics and its applications. All articles submitted to the Editorial board are peer-reviewed. After publication, articles are indexed in “Mathematical Reviews” (AMS), “Zentralblatt für Mathematik” (Springer) and “Statistical Methods and Applications” (ISI). The journal is indexed in Scopus (Elsevier) since 2004.

For scientists, professors, students.

EDITOR-IN-CHIEF V. S. Korolyuk (Ukraine)

DEPUTY EDITORS-IN-CHIEF Yu. V. Kozachenko (Ukraine), Yu. S. Mishura (Ukraine)

EXECUTIVE SECRETARY L. M. Sakhno (Ukraine)

EDITORIAL BOARD V. V. Anisimov (UK), T. Bodnar (Sweden), M. Dozzi (France), A. A. Gushchin (Russia), A. V. Ivanov (Ukraine), M. V. Kartashov (Ukraine), A. G. Kukush (Ukraine), M. M. Leonenko (UK), I. K. Matsak (Ukraine), M. P. Moklyachuk (Ukraine), I. Molchanov (Switzerland), A. A. Novikov (Australia), A. Ya. Olenko (Australia), E. Orsingher (Italy), M. I. Portenko (Ukraine), D. S. Silvestrov (Sweden), G. M. Shevchenko (Ukraine), A. Y. Veretennikov (UK & Russia), O. K. Zakusylo (Ukraine)

Address of Editorial board Department of Probability Theory, Statistics and Actuar-ial Mathematics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 64/13 Volodymyrska str., Kyiv, Ukraine, 01601; (/fax: (38 044) 259 03 92, e-mail: [email protected] web-page: http://probability.univ.kiev.ua/tims/ http://www.ams.org/journals/tpms/

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© Taras Shevchenko National University of Kyiv,

Kyiv University publishing and printing center, 2018

3

ÇÌIÑÒ

Þ. Ìiøóðà, Ë. Ñàõíî, Ã. Øåâ÷åíêî

Âiä ðåäàêöi¨ 5

Â. Â. Àí, Ì. Ì. Ëåîíåíêî

Äðîáîâå ðiâíÿííÿ Ñòîêñà �Áócñiíåñêà �Ëàíæåâåíà òà ìiòòàã-ëåôôëåðiâñüêå ñïàäàííÿ êîðåëÿöi¨

8

À. Àéÿø, ß. Åñìiëi

Âåéâëåò-àíàëiç ìóëüòèäðîáîâîãî ïðîöåñó ó äîâiëüíîìó âiíåðiâñüêîìóõàîñi

29

Å. Àçìóäåõ, Ä. Ãàñáàððà

Ïðî íîâèé êëàñ Øåôôåðà ïîëiíîìiâ, ïîâ'ÿçàíèõ iç ðîçïîäiëîìíîðìàëüíîãî äîáóòêó

51

I. Ì. Áîäíàð÷óê, Â. Ì. Ðàä÷åíêî

Õâèëüîâå ðiâíÿííÿ íà ïëîùèíi, êåðîâàíå çàãàëüíîþ ñòîõàñòè÷íîþìiðîþ

70

Õ. Â. Áó÷àê, Ë. Ì. Ñàõíî

Ïðî êåðóþ÷i ðiâíÿííÿ äëÿ ïðîöåñiâ Ïóàññîíà òà Ñêåëëàìà çâèïàäêîâîþ çàìiíîþ ÷àñó, çàäàíîþ îáåðíåíèìè ñóáîðäèíàòîðàìè

87

Ð. Ãîðåíôëî, Ô. Ìàéíàðäi

Ôóíêöiÿ Ìiòòàã-Ëåôôëåðà â òåîði¨ ðîçðiäæåííÿ ïðîöåñiâ âiäíîâëåííÿ 100

ß. Õó

Ðiâíÿííÿ Øðåäiíãåðà ç ãàóññiâñüêèì ïîòåíöiàëîì 109

Þ. Ëó÷êî

Ïðèíöèïè ñóáîðäèíàöi¨ äëÿ áàãàòîâèìiðíîãî äðîáîâîãî çà ïðîñòîðîìi ÷àñîì äèôóçiéíî-õâèëüîâîãî ðiâíÿííÿ

121

Þ. Ìiøóðà, Ê. Ðàëü÷åíêî, Ã. Øåâ÷åíêî

Iñíóâàííÿ òà ¹äèíiñòü ì'ÿêîãî ðîçâ'ÿçêó ñòîõàñòè÷íîãî ðiâíÿííÿòåïëîïðîâiäíîñòi ç áiëèì òà äðîáîâèì øóìàìè

142

Ë. I. Ðóñàíþê, Ã. Ì. Øåâ÷åíêî

Ðiâíÿííÿ êîëèâàíü îäíîðiäíî¨ ñòðóíè iç çàêðiïëåíèìè êiíöÿìè,âèìóøåíèõ âèïàäêîâèì ñòiéêèì øóìîì

163

Ì. Ñëàói, ×. À. Òóäîð

Ãðàíè÷íà ïîâåäiíêà ðîçåíáëàòòiâñüêîãî ïðîöåñó Îðíøòåéíà �Óëåíáåêà âiäíîcíî iíäåêñó Õþðñòà

173

Ò. Ñîòòiíåí, Ë. Âiiòàñààði

Ïðèíöèï ïåðåíåñåííÿ äëÿ äðîáîâîãî áðîóíiâñüêîãî ðóõó n-ãî ïîðÿäêó iéîãî çàñòîñóâàííÿ äëÿ ïðîãíîçóâàííÿ òà åêâiâàëåíòíîñòi çà ðîçïîäiëîì

188

Â. Ñ. Êîðîëþê, I. Â. Ñàìîéëåíêî

Îöiíêà çàëèøêîâîãî ÷ëåíà â àñèìïòîòè÷íîìó ðîçêëàäi ôóíêöiîíàëàâiä íàïiâìàðêîâñüêî¨ âèïàäêîâî¨ åâîëþöi¨

205

Ì. Ñðiãàði

Ïîðÿäîê àïðîêñèìàöi¨ â öåíòðàëüíié ãðàíè÷íié òåîðåìi äëÿàñîöiéîâàíèõ âèïàäêîâèõ âåëè÷èí òà ðåçóëüòàò ïðî ïîìiðíi âiäõèëåííÿ

215

Ì. Â. Êàðòàøîâ , Â. Â. Ãîëîìîçèé

Íåðiâíîñòi äëÿ ôóíêöi¨ ðèçèêó ó íåîäíîðiäíié çà ÷àñîì òà ïðîñòîðîììîäåëi Êðàìåðà �Ëóíäáåðãà

228

Ìèêîëà Âàëåíòèíîâè÷ Êàðòàøîâ (18.10.1952 � 1.01.2018) 239

4

CONTENTS

Yu. Mishura, L. Sakhno, G. ShevchenkoEditorial 5

V. V. Anh, N. N. LeonenkoFractional Stokes–Boussinesq–Langevin equation and Mittag-Lefflercorrelation decay

8

A. Ayache, Y. EsmiliWavelet analysis of a multifractional process in an arbitrary Wiener chaos 29

E. Azmoodeh, D. GasbarraOn a new Sheffer class of polynomials related to normal productdistribution

51

I. M. Bodnarchuk, V. M. RadchenkoWave equation in a plane driven by a general stochastic measure 70

K. V. Buchak, L. M. SakhnoOn the governing equations for Poisson and Skellam processes time-changed by inverse subordinators

87

R. Gorenflo, F. MainardiThe Mittag-Leffler function in the thinning theory for renewal processes 100

Y. HuSchrödinger equation with Gaussian potential 109

Yu. LuchkoSubordination principles for the multi-dimensional space-time-fractionaldiffusion-wave equation

121

Yu. Mishura, K. Ralchenko, G. ShevchenkoExistence and uniqueness of mild solution to stochastic heat equation withwhite and fractional noises

142

L. I. Rusaniuk, G. M. ShevchenkoEquation for vibrations of a string with fixed ends, forced by a stablerandom noise

163

M. Slaoui, C. A. TudorLimit behavior of the Rosenblatt Ornstein–Uhlenbeck process with respectto the Hurst index

173

T. Sottinen, L. ViitasaariTransfer principle for nth order fractional Brownian motion withapplications to prediction and equivalence in law

188

V. S. Koroliuk, I. V. SamoilenkoEstimation of the remainder in asymptotic expansion of a functional ofsemi-Markov random evolution

205

M. SreehariOrder of approximation in the central limit theorem for associated randomvariables and a moderate deviation result

215

M. V. Kartashov , V. V. GolomoziyInequalities for a risk function in the time and space inhomogeneousCramer–Lundberg risk model

228

Mykola Valentynovych Kartashov (18.10.1952 — 1.01.2018) 239

5

EDITORIAL

Y. MISHURA, L. SAKHNO, G. SHEVCHENKO

Fractality and fractionality became a dynamic research topic during the last decades.The reason is that these notions, referring to self-similarity and irregularity, are encoun-tered everywhere both in applied areas, such as geophysics, fluid mechanics, crystallog-raphy, astronomy, biology, chemistry, medicine, electronics, and in various mathematicalareas: number theory, geometry, dynamical systems, probability theory etc. The notion“fractality” is usually employed when the fractal behavior manifests itself statically, asspatial self-similarity property of the objects of systems involved in the research. Theword “fractionality” is a dynamical counterpart of this notion, referring to the frac-tal evolution of individual agents in some macroscopic collections. Multifractality andmultifractionality, which further extend these two notions, are used to describe objects,dynamical systems, and phenomena, whose self-similarity properties are present onlylocally and varying with respect to time, space, or scale.

At the present time, there are different fields of mathematics studying fractality andfractionality: fractional calculus, fractal analysis, fractional dynamical systems, stochas-tic fractional analysis, multifractal analysis etc. Motivated by the numerous applications,the variety of problems occur, and the techniques are constantly developing. The dedi-cated conferences and journals help to keep up with the rapid development of the fieldsand to disseminate the results and ideas quickly.

Given all of the above, we decided to dedicate the most part of the current issue ofour journal to fractionality and related topics and invited specialists in this area. Weare thankful to all contributors for submitting their newest research results. In the listbelow we introduce the authors of the of “fractional” part of the present issue, togetherwith short descriptions of their research interests.

Vo V. Anh is currently a Visiting Distinguished Professor in the School of Mathemat-ics and Computational Science, Xiangtan University, Hunan, China. Research interests:fractional processes and random fields, stochastic analysis and modelling, financial math-ematics. Author of more than 250 research papers and 1 edited book.

Antoine Ayache is a Full Professor in Applied Mathematics at the laboratory PaulPainlevé (UMR CNRS 8524) of the “Université de Lille” in France. Research interests:sample path behavior of multifractional and anisotropic fields, statistical inference formultifractional and anisotropic fields, random series, local times, fractal dimensions.Author of about 50 research papers.

Ehsan Azmoodeh is an adjunct professor at University of Helsinki and researcherat Ruhr University Bochum. His research interests are stochastic analysis, mathematicalfinance and Malliavin–Stein approach.

Iryna Bodnarchuk is a senior engineer-mathematician at Taras Shevchenko NationalUniversity of Kyiv, candidate of sciences in physics and mathematics (Ph. D.). Researchinterests: stochastic partial differential equations, general stochastic measures. Authorof 6 research papers.

Khrystyna Buchak is Ph. D. student under the supervision of Lyudmila Sakhno,research topic: time-changed Lévy processes. Author of 5 research papers.

Yassine Esmili is a first year Ph. D. student at the laboratory Paul Painlevé (UMRCNRS 8524) of the “Université de Lille” in France; he is under the supervision of ProfessorAntoine Ayache.

6 Y. MISHURA, L. SAKHNO, G. SHEVCHENKO

Dario Gasbarra is lecturer at the Department of Mathematics and Statistics of theUniversity of Helsinki (Finland). He has authored 36 publications in the fields stochasticanalysis, mathematical finance and statistics.

Yaozhong Hu is a Full Professor of the Department of Mathematical and StatisticalSciences, University of Alberta at Edmonton, Canada. Research interests: probabilityand applications, statistics of stochastic processes, mathematical finance. He is authorof more than 120 research papers and 3 books.

Nikolai/Mykola Leonenko is a Full Professor at School of Mathematics of CardiffUniversity, UK. Research interests: limit theorems for random fields with short-and longrange dependence, stochastic analysis and financial mathematics, actuarial mathemat-ics, fractality and multifractality, stochastic processes and fractional calculus, statisticalinference for stochastic processes and random fields, random PDE. Author of more than240 research papers, 2 monographs and 1 textbook.

Yuri Luchko is a Full Professor at Department of Mathematics, Physics, and Chem-istry of the Beuth Technical University of Applied Sciences Berlin. Research interests:fractional calculus, ordinary and partial fractional differential equations, integral trans-forms, and special functions. Author of more than 150 scientific publications.

Francesco Mainardi is a retired professor of Mathematical Physics from the Uni-versity of Bologna (since November 2013 at the age of 70) where he has taught thiscourse for 40 years. Even being retired, he continues to carry out teaching and researchactivity. His fields of research concern several topics of applied mathematics, includingdiffusion and wave problems, asymptotic methods, integral transforms, special functions,fractional calculus and non-Gaussian stochastic processes. At present his h-index is morethan 50.

Yuliya Mishura is a Full Professor and Head of the Department of Probability,Statistics and Actuarial Mathematics at Taras Shevchenko National University of Kyiv.Research interests: fractional processes, short- and long-memory phenomena, stochasticprocesses and stochastic analysis, financial mathematics. Author of more than 250 re-search papers and 9 books.

Vadym Radchenko is a Full Professor of the Department of Mathematical Analysisat Taras Shevchenko National University of Kyiv. Research interests: stochastic analysis,stochastic partial differential equations. Author of more than 50 research papers and3 books.

Kostiantyn Ralchenko is an Associate Professor at the Department of Probabil-ity, Statistics and Actuarial Mathematics at Taras Shevchenko National University ofKyiv. Research interests: fractional and multifractional processes and fields, stochasticdifferential equations, statistical inference for stochastic processes. Author of 26 researchpapers and 1 book.

Larysa Rusanyuk is Ph. D. student in stochastic analysis and stochastic differentialequations, under supervision of G. Shevchenko. Author of 6 research papers.

Lyudmyla Sakhno is Head of Scientific Research Laboratory at the Mechanics andMathematics Faculty of Taras Shevchenko National University of Kyiv. Research in-terests: statistical problems for random processes and fields with short and long-rangedependence, spectral analysis of random fields, differential equations and related stochas-tic processes. Author of about 50 research papers.

Georgiy Shevchenko is a Full Professor at the Department of Probability, Statis-tics and Actuarial Mathematics at Taras Shevchenko National University of Kyiv. Re-search interests: stochastic analysis, stochastic differential equations, stochastic partialdifferential equations, fractional and multifractional processes, long-range dependence,heavy-tailed distributions. Author of about 50 research papers.

Meryem Slaoui is Ph. D. student in stochastic analysis since 2016.

EDITORIAL 7

Tommi Sottinen is a Full Professor of Business Mathematics at the University ofVaasa. Research interests: fractional, Gaussian, self-similar, and quadratic variationprocesses; stochastic analysis; statistics for stochastic processes; stochastic simulation;mathematical finance; financial engineering. Author of more than 30 research papers.

Ciprian Tudor is Full Professor at University of Lille, France. His research interestsare related to fractional processes, Malliavin calculus, stochastic differential equation andtheir applications. Author of more than 110 papers and 2 books.

Lauri Viitasaari is a postdoctoral researcher at the University of Helsinki, Finland,Department of Mathematics and Statistics. Research interests: Gaussian processes, sto-chastic analysis, stochastic (backward) differential equations, mathematical finance, sta-tistical inference for stochastic processes, time series analysis, functional data analysis.Author of 20 research papers.

Òåîðiÿ éìîâiðíîñòåé Teoriya �Imovirnoste��òà ìàòåìàòè÷íà ñòàòèñòèêà ta Matematychna StatystykaÂèï. 1(98)/2018, ñ. 8�28 No. 1(98)/2018, pp. 8�28

UDC 519.21

FRACTIONAL STOKES–BOUSSINESQ–LANGEVIN EQUATION

AND MITTAG-LEFFLER CORRELATION DECAY

V. V. ANH, N. N. LEONENKO

This contribution is dedicated to the 85th anniversary of Professor Mykhailo Iosipovych Yadrenko

Abstract. This paper presents some stationary processes which are solutions of the fractional Stokes–

Boussinesq–Langevin equation. These processes have reflection positivity and their correlation func-tions, which may exhibit the Alder–Wainwright effect or long-range dependence, are expressed in terms

of the Mittag-Leffler functions. These properties are established rigorously via the theory of KMO–

Langevin equation and a combination of Mittag-Leffler functions and fractional derivatives. A rela-tionship to fractional Riesz–Bessel motion is also investigated. This relationship permits to study the

effects of long-range dependence and second-order intermittency simultaneously.

Key words and phrases. Anomalous diffusion, Stokes–Boussinesq–Langevin equation, Langevin equa-

tion with delay, long-range dependence, Mittag-Leffler function.

2010 Mathematics Subject Classification. Primary 60G10; Secondary 60M20, 82C31.

1. Introduction

In a computer experiment of molecular dynamics, Alder and Wainwright [2, 3] foundthe tail behavior t−3/2 as t→∞ for the autocorrelation function of a stationary process ofnon-Markovian type. This behavior is known as the Alder–Wainwright effect. The usualLangevin equation for the velocity ξ(t) of a Brownian particle of mass m at position x(t)in a fluid, which neglects the effect of the fluid flow around the particle, is not adequate tocapture this behavior. Taking into account the hydrodynamic drag force, which is due tothe acceleration of the particle, the Langevin equation becomes the Stokes–Boussinesq–Langevin equation:

dξ(t)

dt= − 1

σ∗ξ(t)− a

σ∗√πν

∫ t

−∞

1√t− s

dξ(s)

dsds +

1

m∗W (t), (1.1)

where m∗ = m(

1 + ρ2ρ0

)is the effective mass, ρ being the density of the fluid, ρ0 being

the density of the particle, σ∗ = m∗µ is the modified relaxation time, µ is the mobilitycoefficient, ν is the kinematic viscosity of the fluid, and W (t) denotes the random forcearising from rapid thermal fluctuations (see Appendix A for the derivation of equation(1.1)). It was shown in Widom [64], for example, that the autocorrelation function ofthe random process ξ(t) defined by equation (1.1) has the tail t−3/2 as t → ∞, whichagrees with the result of the Alder–Wainwright experiment.

Many works on physical models of anomalous diffusion reported a Mittag-Leffler decayfor the autocorrelation function:

ρ(t) = Eα(−b|t|α

), t ∈ R, 0 < α ≤ 1, b ≥ 0, (1.2)

Partially supported by the Australian Research Council grant DP160101366. We wish to thank

Professor Akihiko Inoue and a referee for many suggestions to rectify some earlier results and improvethe paper.

FRACTIONAL STOKES–BOUSSINESQ–LANGEVIN EQUATION 9

where Eα is the one-parameter Mittag-Leffler function (defined in Section 2). Thisformula covers a complete range from the exponential decay of Ornstein–Uhlenbeck pro-cesses to the hyperbolic decay of strongly dependent processes, and includes the Alder–Wainwright effect. Metzler et al. [44] introduced a fractional Fokker–Planck equationusing fractional derivatives to describe subdiffusive behavior of a system close to thermalequilibrium. Based on this equation, they showed that the mean square displacement of aparticle has a Mittag-Leffler decay as t→∞, hence implying the long-range dependence(LRD) for its velocity. Metzler and Klafter [43] and Barkai and Silbey [9] investigated afractional Klein–Kramers equation, from which the fractional Fokker–Planck equation isdeduced, and again established the Mittag-Leffler relaxation.

Lutz [40] described another pathway to anomalous diffusion using random matrixtheory. This approach considers a system coupled to a fractal heat bath with a random-matrix interaction. In the limit of weak coupling, the following fractional Langevinequation is obtained:

mẍ(t) + m

∫ t

0

γ(t− s)ẋ(s)ds = W (t), (1.3)

where W (t) is a Gaussian random force with mean zero and covariance function

RW (t) = E(W (t)W (0)) ∼ 2A0Γ(α) cos(απ

2

)t−α, 0 < α < 2,

in the limit of large bandwidth, A0 being the strength of the coupling, and γ(t) isa response kernel that obeys the second fluctuation–dissipation theorem mκTγ(t) =RW (t), κ ≡ Boltzmann constant, T ≡ absolute temperature (Kubo [36]). Using thisequation, the Mittag-Leffler decay of the autocorrelation function is obtained. Kou andXie [35] and Min et al. [46] used the fractional Langevin equation (1.3) to investigatesubdiffusion (0 < α < 1) within a single protein molecule. Fa [25], Lim and Teo [39],Eab and Lim [23] extended the fractional Langevin equation (1.3) to the case where theresponse kernel γ(t) is given in terms of a Mittag-Leffler function and the time derivativeof ẋ(t) is replaced by a Caputo fractional time derivative. The resulting equation is calleda fractional generalized Langevin equation (FGLE). Camargo et al. [12] considered a two-parameter Mittag-Leffler function in the response kernel γ(t) for the FGLE, while Sandevet al. [58, 59] considered a three-parameter Mittag-Leffler function for γ(t). The paper[59] provides a review of works in this direction. It should be noted that these worksused the fluctuation–dissipation theorem and followed the Laplace transform methodapplied to the FGLE, which is a random equation, to obtain a formal expression for thedisplacement x(t).

From a different angle, Okabe [50, 51] introduced and gave a rigorous treatment ofthe linear stochastic delay equation

Ẋ(t) = −βX(t)−∫ t

0

γ(t− s)Ẋ(s)ds + αI(t), (1.4)

in which the solution X(t) is defined as a random tempered distribution, and Ẋ(t) is itsderivative. Here, α and β are positive numbers, the delay kernel γ : (0,∞)→ [0,∞) hasthe representation

γ(t) =

∫ ∞

0

e−tλρ(dλ), t > 0, (1.5)

ρ being a Borel measure on (0,∞) such that∫ ∞

0

(λ−1 + λ

)ρ(dλ)

10 V. V. ANH, N. N. LEONENKO

and I(t) is a stationary Gaussian random tempered distribution associated with X(t),called the Kubo noise of the process X(t) (this concept comes from Kubo’s linear re-sponse theory detailed in Kubo [36], Kubo et al. [37]). The Kubo noise I(t) is neededfor a fluctuation–dissipation theorem to hold. Equation (1.4) has a physical meaningby considering X(t) to be the x-component of the velocity of a particle as describedin Appendix A. A key feature of equation (1.4) is that it describes the time evolutionof a stationary Gaussian process with reflection positivity (defined in the next section);this concept arises from an axiom of the quantum field theory. Under the conditions onthe measure ρ(dλ), the diffusion coefficient D =

∫∞0 RX(t)dt is finite for equation (1.4).

Inoue [33] extended Okabe’s work by considering the case D = ∞. In this latter work,equation (1.4) is also established, but with β = 0. A key result obtained is that the solu-tion of equation (1.4) (with β = 0) possesses both long-range dependence and reflectionpositivity. A causality condition (defined in (2.18) of Section 2) is needed for uniquenessof the solution.

In Section 2, we apply the theory of Okabe [50, 51] and Inoue [33] to a fractionalgeneralization of the Stokes–Boussinesq–Langevin equation:

Ẋ(t) = −λX(t)− bD1−αX(t) + W (t), t ∈ R, λ ≥ 0, b ≥ 0, (1.6)

where the fractional derivative D1−α, 0 ≤ α ≤ 1, is defined in (2.15) below, and W (t) isKubo noise with a certain spectral density. In this application, the delay kernel γ(t) of(1.5) takes the specific form of the fractional derivative D1−α and the Kubo noise hastwo specifications in Theorems 2.1 and 2.2 respectively. In Theorem 2.1 we confirm ana-lytically the Mittag-Leffler decay in the autocorrelation function of the solution process,while in Theorem 2.2, for α = 1/2 the asymptotic behaviour of the correlation functionis ρX(t) = O

(t−3/2

), t → ∞, which is the Alder–Wainwright effect. A new aspect of

Theorems 2.1 and 2.2 is that the results are given in an explicit form using the Mittag-Leffler functions, rather than asymptotic results as given in Inoue [33]. These exactresults highlight the important role played by a combination of Mittag-Leffler functionsand fractional derivatives, which takes advantage of the availability of nice formulae ofthe Laplace transform in terms of Mittag-Leffler functions.

In Section 3, we consider the fractional Stokes–Boussinesq–Langevin equation (1.6) in

the context of fractional Gaussian noise ḂH(t). Equation (1.6) will then take the form

Ẋ(t) = −λX(t)− bD1−2HX(t) + ḂH(t), t ∈ R, λ ≥ 0, b ≥ 0. (1.7)

We will look at two cases of interest: λ = 0 and then b = 0 separately. For λ = 0 in (1.7),existence and uniqueness of a stationary solution for (1.7) with long-range dependenceis confirmed for 0 < H < 1/2. For the case 1/2 < H < 1, we will consider equation (1.7)with b = 0 in the Itô approach. The corresponding equation is the Ornstein–Uhlenbeckequation driven by fractional Brownian motion BH(t), 1/2 < H < 1. Existence anduniqueness of a stationary solution with long-range dependence is also obtained, as wellas an explicit form for its spectral density.

In the approach of Okabe [50, 51] and Inoue [33], the noise term (Kubo noise) is associ-ated with the underlying process via its spectral decomposition. If we are able to obtainlong-range dependence or the Alder–Wainwright effect in the solution of the fractionalStokes–Boussinesq–Langevin equation under the scenario of system-independent noise,we may use the noise term to represent other effects such as intermittency (see Frisch[27] for example). There may also be more flexibility in defining the response functionγ(t). We demonstrate these possibilities in Section 4, where a stationary process relatedto fractional Riesz–Bessel motion (Anh et al. [5, 7]) is derived. This permits to study theeffects of long-range dependence and second-order intermittency simultaneously. Theseeffects are known to be important features of data in geophysics, turbulence and finance.


2. Stationary processes governed by the fractionalStokes–Boussinesq–Langevin equation

2.1. Reflection positivity. Let X = {X(t), t ∈ R} be a real-valued, measurable, mean-square continuous, stationary (in the wide sense) random process with meanEX(t) = const, covariance function RX(t) = Cov(X(t), X(0)), t ∈ R, and spectraldensity fX(ω), ω ∈ R, that is,

RX(t− s) =∫

Rcos{ω(t− s)}fX(ω)dω. (2.1)

The concept of reflection positivity arises in the axiomatic quantum field theory (Os-terwalder and Schrader [53], Nelson [49], Hegerfeldt [30], Glimm and Jaffe [28, p. 90–92]).Following Osterwalder and Schrader [53], we say that the process X has reflection posi-tivity if its covariance function (2.1) satisfies

n∑

j,k=1

zjRX(tj + tk)z̄k ≥ 0, tj ∈ [0,∞), j = 1, . . . , n,

for any n ≥ 1, zj ∈ C, j = 1, . . . , n. Hida and Streit [31] showed that a Gaussian processX has reflection positivity if and only if there exists uniquely a bounded non-negativeBorel measure σ on [0,∞) such that

RX(t)

RX(0)= ρX(t) = σ({0}) +

∫

(0,∞)ρλ(t)σ(dλ), (2.2)

where

ρλ(t) = e−|t|λ, t ∈ R, λ > 0, (2.3)

is the correlation function of the stationary Gaussian Ornstein–Uhlenbeck (OU) processξ(t) defined by the equation

dξ(t) = −λξ(t) dt + γ dB(t), t ∈ R, λ > 0, γ > 0. (2.4)Here, B = {B(t), t ∈ R} is a one-dimensional Brownian motion or Wiener process suchthat

EB(t) = 0, VarB(t) = |t|. (2.5)The stationary Gaussian solution of (2.4) has the following covariance function and spec-tral density:

Rξ(t) =γ2

2λe−λ|t|, t ∈ R; fξ(ω) =

A

ω2 + λ2, A =

γ2

2π, ω ∈ R. (2.6)

By Bernstein’s theorem (see Feller [26, p. 426]) we obtain that the condition (2.2) isequivalent to the complete monotonicity of the function ρX(t) on (0,∞), that is,

(−1)k dk

dtkρX(t) ≥ 0, t > 0, k = 0, 1, 2, . . . (2.7)

The following functions on (0,∞) are known to be completely monotone:exp{−ctγ}, c > 0, 0 < γ ≤ 1;(2v−1Γ(ν)

)−1(c√t)ν

Kν(c√t), c > 0, ν > 0;

(1 + ctγ)ν, c > 0, 0 < γ ≤ 1, ν > 0;

2ν(ec√t + e−c

√t)−ν

, c > 0, ν > 0;

Eα,β(−t), (0 < α < 1,β = 1) or (0 < α ≤ 1,β ≥ α);Eα,1(−tγ), 0 < α < 1, 0 < γ < 1.


In this list, Kν is the modified Bessel function and Eα,β(−x), x > 0 is the Mittag-Lefflerfunction of the negative real argument (see formula (2.32) below).

If we assume that

σ({0}) = 0, 0 < σ([0,∞)) 0, (2.11)

which is also known as the probability density of the generalized Linnik distribution(Erdogǎn and Ostrovskii [24]). Analytic and asymptotic properties of the function (2.11)has been studied by Erdogǎn and Ostrovskii [24]. In particular, for 0 < α < 1, ω ↓ 0,

fX(ω) =1

2Γ(α) cos{απ2

} 1|ω|1−α(1− θ(ω)

), θ(ω)→ 0, (2.12)

while for α = 1, ω ↓ 0,

fX(ω) =1

πlog

1

|ω| −γ

π+

1

2|ω|2 log 1|ω| + O

(ω2),

where γ = Γ′(1) is Euler constant. Thus, for α ∈ (0, 1] the process X with covariancefunction (2.10) and spectral density (2.11) displays long-range dependence.

2.2. Fractional Stokes–Boussinesq–Langevin equation. Let us recall some defini-tions of fractional derivatives (see Caputo [13], Caputo and Mainardi [14], Miller andRoss [45], Samko et al. [56], Djrbashian [17], Podlubny [55] among others).

Under certain natural conditions on the real-valued function f(t), the Caputo frac-tional derivative of order β ∈ [n− 1, n), n = 1, 2, . . . , is defined as

aDβCf(t) =1

Γ(n− β)

∫ t

a

(dn/dτn)f(τ)

(t− τ)β+1−n dτ, (2.13)

while the Riemann–Liouville fractional derivative of order β ∈ [n− 1, n), n = 1, 2, . . . , isdefined as

aDβRLf(t) =1

Γ(n− β)dn

dtn

∫ t

a

f(τ)

(t− τ)β+1−n dτ. (2.14)

The main advantage of Caputo’s definition is that the fractional derivative of a constant

C is equal to zero: 0DβCC = 0, while in the Riemann–Liouville definition we have

0DβRLC = Ct−β/

Γ(1− β), 0 ≤ β < 1.


Putting a = −∞ in both definitions (2.13) and (2.14) and requiring reasonable behaviorof f(t) and its derivatives for t→ −∞, we arrive at the same formula

Dβf(t) = −∞DβCf(t) = −∞DβRLf(t) =1

Γ(n− β)

∫ t

−∞

(dn/dτn)f(τ)

(t− τ)β+1−n dτ, (2.15)

where n − 1 ≤ β < n, n = 1, 2, . . . The fractional derivative (2.14) is also called Weyl’sfractional derivative (see Samko et al. [56, p. 356]).

In order to obtain exact formulae for equation (1.6) instead of asymptotic expressions,we will widely use the one-parameter and two-parameter Mittag-Leffler functions (see,for example, Djrbashian [17]). In particular, the entire function of order 1/α of type 1

Eα(z) =

∞∑

j=0

zj

Γ(αj + 1), z ∈ C, α > 0,

is known as the one-parameter Mittag-Leffler function. For real x ≥ 0 the function

Eα(−x) =∞∑

j=0

(−1)jxjΓ(αj + 1)

, x ≥ 0, 0 < α ≤ 1, (2.16)

is infinitely differentiable and completely monotone. It follows from the definition that

E1(−x) = e−x, E1/2(−x) = ex2

(1− 2√

π

∫ x

0

e−t2

dt

), x ≥ 0.

From Djrbashian [17, p. 5], we obtain the following asymptotic formula:

Eα(−x) = −N∑

k=1

(−1)kx−kΓ(1− αk) + O

(|x|−N−1

), 0 < α < 1, (2.17)

as x→∞.The next theorem is concerned with the fractional Stokes–Boussinesq–Langevin equa-

tion (1.6) for the case λ = 0. The uniqueness of its stationary solution is obtained underthe causality condition

Σt(X) = Σt(W ) (2.18)

for any t ∈ R, where Σt(Y ) denotes the closed linear hull of{Y (φ), φ ∈ D(R), supp{φ} ⊂ (−∞, t]

}

in L2(Ω,F ,P), (Ω,F ,P) being the underlying complete probability space and D(R) beingthe space of all φ ∈ C∞(R) with compact support (see Appendix B for further details).It should be noted that equation (1.6) is not an Itô stochastic differential equation ingeneral because the fractional operator (2.15) is not local. For λ ≥ 0, b > 0, equation(1.6) is a particular case of the second KMO–Langevin equation (Okabe [51], Inoue [33]).

Theorem 2.1. There exists a unique stationary solution X (in the sense of randomdistributions) of the fractional Stokes–Boussinesq–Langevin equation (1.6) with λ = 0and spectral density of the Kubo noise of the form

fW (ω) = γb

√2

πcos

{(1− α)π

2

}|ω|1−α, 0 < α < 1, γ > 0, b > 0, (2.19)

under the causality condition (2.18). The solution X is a purely nondeterministic zero-mean stationary Gaussian process having the following properties:

RX(0) = γ√

2π ; (2.20)

ρX(t) = RX(t)/RX(0) = Eα(−b|t|α), t ∈ R, 0 < α < 1; (2.21)


for α ∈ (0, 1) and large t > 0

ρX(t) =1

|t|αbΓ(1− α) + O(

1

|t|2α)

; (2.22)

for α ∈ (0, 1), X has reflection positivity, that is, (2.2) holds with

σ(dλ) =sin{πα}π

λα−1

1 + 2 cos{απ}λα + λ2α dλ; (2.23)

its spectral density is

fX(ω) =sin{πω}π2

∫ ∞

0

uα du

(ω2 + u2)(1 + 2 cos{απ}uα + u2α) , ω > 0, 0 < α < 1; (2.24)

the correlation function (2.21) is the unique solution of the Cauchy problem for thefractional differential equation

0DαCρX(t) + bρX(t) = 0, b > 0, ρX(0) = 1; (2.25)the process X has the time-domain representation as a random distribution:

X(t) = limM→∞

1(0,M)Eα(−b | · |α) ∗W (t) =

=

∫ t

−∞Eα(−b|t− s|α)W (s) ds, 0 < α < 1. (2.26)

Proof. The existence and uniqueness of the stationary solution of equation (1.6) withλ = 0 follows from Theorems 1.1 and 1.2 of Inoue [33]. The expression (2.19) of the Kubonoise is given in Example 5.11 of Inoue [33]. The result (2.20) is part of Theorem 5.10of Inoue [33]. Moreover, from (ii) of Theorem 1.2 of Inoue [33] we obtain

∫ ∞

0

eiζt(RX(t)/RX(0)

)dt =

(−iζ− iζ∫ ∞

0

eiζtγ(t) dt

)−1,

Im ζ > 0, which, with

γ(t) = btα−1/

Γ(α), 0 < α < 1, (2.27)

and ζ = ip, reduces to the following equation:∫ ∞

0

e−ptρX(t) dt =1

p + bp1−α, p > 0. (2.28)

It is known (see, for example, Samko et al. [56, p. 21]) that the Laplace transform of theMittag-Leffler function (2.21) is

∫ ∞

0

e−ptEα(−b|t|α)dt =pα−1

pα + b, Re p > |b|1/α, (2.29)

0 < α < 1. Thus, (2.21) follows from (2.27) and (2.28). The formula (2.22) then followsfrom (2.17), and (2.23) is a particular case of Theorems 1.3–1.5 of Djrbashian [17]. Theformula (2.24) follows from (2.8) and (2.23). The fractional differential equation (2.25)is solved by (2.21) in Djrbashian and Nersesian [18], which uses the same definitionof fractional derivatives as Caputo’s. The representation (2.26) is a particular case ofTheorem 1.2 of Inoue [33], while (2.19) follows from (5.17) of Inoue [33] with appropriatechoice of γ(t) according to (2.27). �

Remark 2.1. If α = 1, the corresponding equation is (2.4) and the covariance function ofits stationary solution is given by (2.6). The correlation function (2.21) formally reducesto (2.6) (up to constants) in this case. For α ∈ (0, 1) the process X given in Theorem 2.1displays LRD, that is, ∫ ∞

0

ρX(τ) dτ =∞. (2.30)


The exact formula (2.21) in terms of the Mittag-Leffler function thus gives a completeinterpolation between the exponential covariance function of OU processes and the hy-perbolic covariance function of LRD processes. The representation (2.26) of the processitself also interpolates the moving-average representations of OU processes and LRDprocesses.

In what follows we need the two-parameter Mittag-Leffler function (see again Djr-bashian [17, p. 1–6]), which can be defined by the series expansion

Eα,β(z) =∞∑

k=0

zk

Γ(αk + β), α > 0, β > 0, z ∈ C. (2.31)

It is clear that Eα,1(z) = Eα(z), and E1,1(z) = ez, E2,1(z) = cosh

√z, E2,2(z) =

= (sinh√z )/√z, E1,2(z) = (e

z − 1)/z, E1,3(z) = (ez − 1− z)/z2. If α < 1, β ≥ α thefunction

Eα,β(−u) =∞∑

k=0

(−1)kukΓ(αk + β)

, u ≥ 0, (2.32)

is completely monotone, that is,

Eα,β(−u) =∫ ∞

0

e−uτqα,β(τ) dτ, (2.33)

where

qα,β(τ) = −1

π

∞∑

k=0

(−1)kk!

Γ(1− β+ α(k + 1)

)sin{π(α(k + 1)− β)

}τk ≥ 0.

Theorem 2.2. There exists a unique stationary solution X (in the sense of distributions)of the fractional Stokes–Boussinesq–Langevin equation (1.6) with λ > 0, b > 0 and thespectral density of the Kubo noise of the form

fW (ω) = γ

√2

π

(λ+ b

∫ ∞

0

ω2

u2 +ω2du

uα

), ω ∈ R− {0}, 0 < α < 1. (2.34)

The solution X is a stationary Gaussian process with reflection positivity and correlationfunction of the form

ρX(t) = RX(t)/RX(0) =∞∑

k=0

(−1)kk!

(λt)kE(k)α,1+k−ακ(−btα), (2.35)

where E(0)α,β(x) = Eα,β(x) is defined by (2.32) and

E(k)α,β(x) =

dk

dxkEα,β(x) =

∞∑

j=0

xj(j + k)!

j!Γ(αj + αk + β), k = 1, 2, . . . (2.36)

Proof. The existence and uniqueness of a stationary solution of (1.6) with λ > 0, b > 0is given in Okabe [51]. The causality condition (2.18) follows in this solution. Theexpression (2.34) is the spectral density for Kubo noise of model (6.6) of Inoue [33]corresponding to the response function γ(t) of (2.27). With this choice of γ(t), we obtainfrom (6.1) of Inoue [33] that

∫ ∞

0

e−ptρX(t) dt =(λ+ p + bp1−α

)−1, p > 0. (2.37)

From Djrbashian [17], Podlubny [55], we can obtain the following expression for theLaplace transform of the function (2.36):∫ ∞

0

e−pttαk+β−1E(k)α,β(±btα)dt =k!pα−β

(pα ∓ b)k+1, Re p > |b|1/α, k = 0, 1, . . . , (2.38)


and for 0 < α < 1 the function (2.37) can be written as

1

λ

λpα−1

p1−α + b

[1 +

λpα−1

p1−α + b

]=

1

λ

∞∑

k=0

(−1)kλk+1pα−1+k(α−1)/(

p1−α + b)k+1

. (2.39)

Term-by-term inversion of (2.39) based on the general expansion theorem for the Laplacetransform using (2.38) produces (2.35). �

Remark 2.2. Putting b = 0 formally in (2.35) we obtain

ρX(t) =∞∑

k=0

(−1)kk!

(λt)k = e−λt, t ≥ 0,

which is the correlation function (2.3) of the OU process (2.4). Moreover, putting λ = 0formally in (2.35) with b > 0 we obtain

ρX(t) = Eα,1(−btα), t ≥ 0,which coincides with the correlation function (2.21) of the stationary solution to (1.6)with λ = 0, α ∈ (0, 1].Remark 2.3. The correlation function (2.35) is the inverse Laplace transform of thefunction

gα(p) =1

λ+ p + bp1−α, p > 0, 0 < α < 1.

The behaviour of gα(p) as p → 0 is c(1−O

(p1−α

)), c being a constant. This yields,

via Watson’s lemma, the behaviour ρX(t) = O(t2−α

)as t → ∞. Thus, for α = 1/2 the

asymptotic behaviour of the correlation function is ρX(t) = O(t−3/2

), t → ∞, which is

the Alder–Wainwright effect; but the exact expression (2.35) for the correlation functionis more informative.

3. Fractional Stokes–Boussinesq–Langevin equation driven by fractionalGaussian noise

3.1. Fractional Stokes–Boussinesq–Langevin equation. Fractional Brownian mo-tion (FBM), BH = {BH(t), t ∈ R}, with Hurst parameter H ∈ (0, 1), is a Gaussian,mean-zero and H-self-similar process with BH(0) = 0 and stationary increments. By

H-self-similarity we mean that, for a > 0, {BH(at), t ∈ R} d= {aHBH(t), t ∈ R}, whered= stands for equality in finite-dimensional distributions. The FBM BH with H =

12 is

the usual Brownian motion B = {B(t), t ∈ R}.Samorodnitsky and Taqqu [57] provided an introduction to FBM. For a detailed

treatment of FBM we refer to Mishura [47]. We note that the covariance function ofFBM is

Cov(BH(t), BH(s)

)=

c

2{|t|2H + |s|2H − |t− s|2H}, t, s ∈ R, (3.1)

where c = VarBH(1), while the covariance function of an increment of FBM is given by

Cov(BH(t + h)−BH(h), BH(t + s + h)−BH(s + h)

)=

= Cov(BH(t), BH(t + s)−BH(s)

)=

= c∞∑

n=1

t2n

(2n)!

(2n−1∏

k=0

(2H − k))s2H−2n =

= cN∑

n=1

t2n

(2n)!

(2n−1∏

k=0

(2H − k))s2H−2n + O

(s2H−2N−2

), s→∞,


for every N ∈ {1, 2, . . .} and all 0 < t < s, h ∈ R. If we denoteRH(n) = Cov

(BH(t), BH((n + 1)t)−BH(nt)

),

then, for H ∈(0, 12),∑∞

n=−∞RH(n) = 0 (negative correlation property) and, forH ∈

(12 , 1),∑∞

n=−∞ |RH(n)| =∞ (long-range dependence).FBM admits a time-domain representation in the form of Itô stochastic integral with

respect to standard Brownian motion B(t):

BH(t) =1

Γ(H + 12 )

∫

R

[gH

(t− s)− gH

(−s)]dB(s), t ∈ R,

where gH

(s) = sH−12 1(0,∞)(s). We consider the spectral representation

B(t) =

∫

R

e−itω − 1−iω Z(dω),

where Z(dω) is a complex Gaussian random measure with

E|Z(dω)|2 = σ2 dω.Then the spectral representation of FBM is

BH(t) =

∫

R

e−itω − 1−iω

1

(−iω)H− 12Z(dω), (3.2)

from which we get a formal representation of the derivative process, which exists only inthe sense of random distributions:

d

dtBH(t) = ḂH(t) =

∫

Re−itω(−iω) 12−HZ(dω), (3.3)

where(−iω) 12−H = lim

η↓0(−iζ) 12−H , ζ = ω+ iη

and we choose the branch of (−iζ) 12−H such that (−iζ) 12−H∣∣∣ζ=i

= 1. The above formulae

show that we can consider the fractional noise ḂH = {ḂH(t), t ∈ R} as a randomdistribution with spectral density

σ2|ω|1−2H , H ∈ (0, 1). (3.4)Remark 3.1. The formulae (3.2)–(3.3) correspond to the definition of the outer functionh(ξ) as the boundary value of an analytic function h(ζ) in the upper half-plane Im ζ > 0(see Appendix C). More often (see, for instance, Samorodnitsky and Taqqu [57] or Igloiand Terdik [32]) the expressions such as

B(t) =

∫

R

eitω − 1iω

Z(dω)

are used. These expressions correspond to considering the Hardy functions in the lowerhalf-plane Im ζ < 0. Thus, to use these latter expressions, we must change the definitionof the outer function (C.1) such that this function becomes analytical in Im ζ < 0.

Remark 3.2. The variance of BH(t) has the spectral representation

VarBH(t) = 4

∫ ∞

0

(1− cosωt)fBH (ω)dω,

where, in view of (3.4), fBH (ω) = σ2|ω|−(2H+1), ω ∈ R, is the spectral density of BH(t).

We have denoted above that VarBH(1) = c; thus the connection between σ2 and c is

σ2 =c

4∫∞0 (1− cosω)|ω|−(2H+1)dω

=cΓ(2H + 1) sin(πH)

2π.


For a nonrandom function f , integration with respect to FBM BH can be based onformal calculations:∫

Rf(t) dBH(t) =

∫

Rf(t)ḂH(t) dt =

=

∫

Re−itω∫

Rf(t)(−iω) 12−HZ(dω) dt =

=

∫

R

{∫

Re−itωf(t)dt

}(−iω) 12−HZ(dω).

A precise meaning is given by the following definition.

Definition. Let f ∈ L2(R) be a non-random real-valued function and∫

R

∣∣∣∣∫

Re−itωf(t) dt

∣∣∣∣2

|ω|1−2Hdω


Then, by Eqs. (3.6), (3.10) and (3.11) we get∫

Re−itω

{−iω+ b(−iω)1−2H

}h

X(ω)Z(dω) =

∫

Re−itωh

Ḃ(ω)Z(dω),

or [−iω+ b(−iω)1−2H

]h

X(ω) = h

Ḃ(ω),

which yields ∣∣−iω+ b(−iω)1−2H∣∣2f

X(ω) = f

Ḃ(ω).

The spectral density of X(t) is then given by

fX

(ω) =fḂ

(ω)

|−iω+ b(−iω)1−2H |2 .

Using (3.4), we obtain for H ∈ (0, 12 ] that

fX

(ω) =σ2ω1−2H

ω2 + b2ω2(1−2H) + 2bω2(1−H) sin(π(12 −H

)) , ω ∈ R. (3.12)

3.2. Ornstein–Uhlenbeck equation driven by fractional Brownian motion. Wehave seen in the subsection above that the linear response theory with Kubo noise worksfor the case 0 < H < 1/2. In this subsection we pay attention to the case 1/2 < H < 1 ofstrongly correlated noise. We consider the linear stochastic differential equation drivenby FBM:

dX(t) = −λX(t)dt + dBH(t), λ > 0, t ∈ R. (3.13)Note that equation (3.13) was discussed by a number of authors including Comte andRenault [16], Igloi and Terdik [32], Cheridito et al. [15]. One can show that there existsa unique continuous solution of equation (3.13) in the form

X(t) =

∫ t

−∞e−λ(t−s) dBH(s) (3.14)

or, in the frequency domain,

X(t) =

∫

Re−itω

1

−iω+ λ (−iω)−H+ 12Z(dω), t ∈ R, (3.15)

for a complex Gaussian random measure Z(dω) with

E|Z(dω)|2 = σ2dω.Thus, the stationary process (3.14) has spectral density

fX(ω) =σ2

ω2 + λ2|ω|1−2H , ω ∈ R, (3.16)

and covariance function

RX(t) =1

2σ2

N∑

n=1

λ−2n(

2n−1∏

k=0

(2H − k))t2H−2n + O

(t2H−2N−2

)

as t → ∞, for any N = 1, 2, . . . and H 6= 12 . In particular, for H ∈(12 , 1)

the process(3.14) is stationary and exhibits long-range dependence, that is,

∫

RRX(s) ds =∞,

while, for H ∈(0, 12), it has the negative correlation property

∫

RRX(s) ds = 0.


We assume for simplicity that σ2 = 1, λ = 1. In the former case, the outer function andcanonical representation kernel can be obtained explicitly (see Inoue and Kasahara [34]).By applying Exercises 2.3.4 and 2.7.2 of Dym and McKean [21] to the rational functions

1

1− iζ = exp{

1

2πi

∫

R

1 +ωζ

ω− ζlog(1 +ω2

)−1

1 +ω2dω

}, Im ζ > 0,

−iζ = exp{

1

2πi

∫

R

1 +ωζ

ω− ζlogω2

1 +ω2dω

}, Im ζ > 0

(noting that both 11−iζ and −iζ are positive on the upper imaginary axis), we obtain forthe outer function (C.1) an explicit formula:

h(ζ) =(−iζ) 12−H

1− iζ , Im ζ > 0. (3.17)

For the function (3.17) we have

h(ζ) =1√2π

∫ ∞

0

eiζtE(t) dt (3.18)

with

E(t) =

√π

Γ(H − 12

)∫ t

0

es−tsH−32 ds, t > 0. (3.19)

Thus, the covariance function of the process (3.14) with σ2 = 1, λ = 1 has the remarkablerepresentation

R(t) =1

2π

∫ ∞

0

E(|t|+ s)E(s) ds (3.20)

with the function E given by (3.19). To our knowledge, this is the only case wherethe canonical representation (C.3) can be written explicitly, unless the process is anOrnstein–Uhlenbeck process.

4. Stationary processes related to fractional Riesz–Bessel motion

Fractional Riesz–Bessel motion (FRBM) was introduced in Anh et al. [5] and further

investigated in Anh et al. [7]. Its model is governed by the operator −(I −∆)β2 (−∆)γ2 ,where ∆ is the Laplace operator, the fractional operators (I−∆)β/2 and (−∆)γ/2 are theinverses of the Bessel and Riesz potentials respectively. Formally, a real-valued Gaussianprocess X(t) which has (i) X(0) = 0 a. s., (ii) stationary increments, and (iii) spectraldensity of the form

fX(ω) =const

|ω|2γ(1 +ω2)β , 1/2 < γ < 3/2, β ≥ 0, ω ∈ R,

is called a fractional Riesz–Bessel motion (FRBM). Fractional Brownian motion canbe deduced from fractional Riesz–Bessel motion by putting β = 0, γ = H + 1/2. Ifγ ∈ (1, 3/2), the process displays long-range dependence (as |ω| → 0). The sum γ + βdescribes clustering of extreme values (as |ω| → ∞) of FRBM (see Anh et al. [6]).

The spectral density of the stationary increments of FRBM has the form

f(ω) =const

|ω|2δ(1 +ω2)β+1 , ω ∈ R, (4.1)

where δ = γ− 1 ∈ (−1/2, 1/2), β > −1. In this section we introduce a model related tostationary fractional Riesz–Bessel motion which has spectral density similar to (4.1). Thismodel is based on the fractional Stokes–Boussinesq–Langevin equation with stationaryrandom noise.


Consider again a fractional Stokes–Boussinesq–Langevin equation of the followingform

Ẋ(t) = −λX(t)− bDδX(t) + W (t), t ∈ R, (4.2)where λ ≥ 0, b ≥ 0, δ ∈ (0, 1) and W (t), t ∈ R, is stationary random noise withspectral distribution FW (ω) and spectral density fW (ω). The fractional derivative Dδis defined in (2.15). As we cannot interpret equation (4.2) in the Itô sense, we will usethe approach proposed by Wong and Hajek [66, pp. 78–116]. In particular, every mean-square continuous second-order stationary process η(t), t ∈ R, with mean zero can berepresented as

η(t) =

∫

Re−iωt dη̂(ω), (4.3)

where η̂(ω) is the spectral process with orthogonal increments such that

E[η̂(a)− η̂(b)][η̂(c)− η̂(d)] = Fη([a, b) ∩ [c, d)

).

Then ∫

Rg(ω) dη̂(ω) =

∫

Rĝ(t)η(t) dt, g ∈ D(R),

where ĝ denotes the Fourier transform of g, D(R) is the space of all functions in C∞(R)with compact support (see Appendix B). In particular, η(t) can be (generalized) whitenoise. The integral

∫R ĝ(t)η(t)dt can be handled by replacing it with the second-order sto-

chastic integral∫R g(ω)Zη(dω), where Zη is a random measure such that E|Zη([a, b))|2 =

= Fη([a, b)). Then

Dδη(t) =∫

R(−iω)δe−itω dη̂(ω) =

∫

R(−iω)δe−itω Zη(dω). (4.4)

We will write ZX , ZW for the random measures corresponding to X(t) and W (t) of (4.2)in this approach. Since (4.2) can be rewritten as

(D+bDδ + λ

)X(t) = W (t), (4.5)

we have

ZX(dω) =[−iω+ b(−iω)δ + λ

]−1ZW (dω).

Thus, there exists a stationary solution of equation (4.2) with λ > 0, b > 0 (Gaussian ifW (t) is Gaussian) of the form

X(t) =

∫

Re−itω

1

−iω+ b(−iω)δ + λ ZW (dω), 0 < δ < 1, (4.6)

where

E|ZW (dω)|2 = fW (ω) dω.The spectral density of the process (4.6) is of the form

fX(ω) =fW (ω)

|g(ω)|2 , ω ∈ R, (4.7)

where

g(ω) = −iω+ b(−iω)δ + λ = λ+ |ω|e− iπ2 + b|ω|δe− iπδ2 .A direct calculation yields

|g(ω)|2 = λ2+|ω|2+2b|ω|1+δ cos π(1− δ)2

+2λb|ω|δ cos πδ2

+b2|ω|2δ, 0 < δ < 1,ω ∈ R.(4.8)

Suppose now that the correlation function of the random noise W (t) is of the form

ρW (t) = c1(β)|t|β+1/2Kβ+1/2(|t|), t ∈ R, β > −1/2, (4.9)


where

c1(β) =[Γ(β+ 1/2)2β−1/2

]−1.

Here, Kµ(z) is the modified Bessel function of the third kind of order µ (Abramowitzand Stegun [1]). Note that

Kµ(z) = K−µ(z), z > 0; Kµ(z) ∼ Γ(µ)2µ−1z−µ, z ↓ 0, µ > 0. (4.10)From (4.9) and (4.10) we obtain ρW (0) = 1.

It is known (see Donoghue [19, p. 293]) that the spectral density which correspondsto (4.9) has the form

fW (ω) =c2(β)

(1 +ω2)β+1, ω ∈ R, β > −1/2, (4.11)

where

c2(β) =Γ(β+ 1)√πΓ(β+ 1/2)

. (4.12)

From (4.7)–(4.11) we obtain the following theorem.

Theorem 4.1. There exists a stationary solution of the fractional Stokes–Boussinesq–Langevin equation (4.2) with δ ∈ (0, 1), λ ≥ 0, b > 0 and stationary random noise W (t),t ∈ R, with correlation function (4.9). The spectral density of this solution is given by

fX(ω) =c2(β)/2π

(1 +ω2)β+1[λ2 +ω2 + 2b|ω|1+δ cos π(1−δ)2 + 2λb|ω|δ cos δπ2 + b2|ω|2δ

] ,

(4.13)ω ∈ R, where δ ∈ (0, 1), β > −1/2 and c2(β) is defined by (4.12). The process X(t) isGaussian if W (t) is Gaussian.

Remark 4.1. If the parameter λ = 0 but b > 0, the spectral density (4.13) reduces to

fX(ω) =c2(β)/2π

|ω|2δ(1 +ω2)β+1(b2 + |ω|2(1−δ) + 2b|ω|1−δ cos π(1−δ)2

) , (4.14)

where δ ∈ (0, 1/2), β > −1/2. This spectral density displays LRD for δ ∈ (0, 1/2).The asymptotic properties of the spectral density (4.14) is similar to those of (4.1). Wetherefore conclude that the stationary process described in Theorem 4.1 with λ = 0provides a dynamic model of fractional Riesz–Bessel motion in the same way as the OUprocess providing a dynamic model of Brownian motion.

Remark 4.2. A general form of equation (4.5) is(AnDβny(t) + . . . + A1Dβ1y(t) + A0Dβ0

)X(t) = W (t) (4.15)

with constant coefficients An, . . . , A1, A0 and βn > β1 > . . . > β1 > β0, n ≥ 1. Thespectral density of the stationary solution X(t) then takes the form (4.7) with

|g(ω)|2 = A20 +n∑

j=1

A2j |ω|2βj + 2∑

1≤i 1/2 or βi + βj > 1, i, j ∈ {1, . . . , n}.If A0 = 0, this spectral density displays LRD of the form O

(|ω|−2β1

)as |ω| → 0 for

β1 ∈ (0, 1/2) and second-order intermittency of the form O(|ω|−2βn

)as |ω| → ∞.


Appendix A. Dynamic models of Brownian motion and related processes

This appendix is based on Kubo [36], Nelson [48], Hauge and Martin-Löf [29], Kuboet al. [37], Okabe [52], Mainardi and Pironi [41]. Our aim is to recall a few historicalfacts which clarify our considerations in Sections 2 and 3. The term “Brownian particle”refers to a body of microscopically visible size suspended in a fluid. Its motion is causedby a molecular bombardment of the fluid and is called Brownian motion because it wasfirst described by Robert Brown, a botanist, in 1827 (see Nelson [48] for some interestinghistorical facts).

The first mathematical theory of Brownian motion was proposed by Einstein [22] andSmoluchowski [61] based on the kinetic theory of heat. Einstein derived the diffusionequation or heat equation for the transition probability density of the position of aBrownian particle as

∂p

∂t= D ∂

2p

∂x2,

where D is a positive constant, called the diffusion coefficient. The second part of Ein-stein’s argument relates D to other physical quantities (see Einstein’s relation (A.7)below). A more rigorous theory was developed by Wiener [65]. Therefore Brownianmotion is also known as the Wiener process.

Langevin [38] initiated, and Uhlenbeck and Ornstein [63] developed the equation ofthe motion of a Brownian particle of mass m. This theory is derived from Newton’ssecond law: F = ma, which in this special case reads

md2x(t)

dt2= F (t) + W (t), (A.1)

where x(t) is the position of the particle, F (t) is the frictional force (due to the fluid)and W (t) denotes the random force arising from rapid thermal fluctuations. Equation(A.1) can be equivalently rewritten as the following system of two equations:

dx(t)

dt= ξ(t), m

dξ(t)

dt= F (t) + W (t), (A.2)

where ξ(t) is the velocity of the particle. Assuming for the frictional force the Stokesexpression for the drag of a spherical particle of radius a, it is given by

F = − 1µξ(t),

1

µ= 6πaρν,

where µ denotes the mobility coefficient and ρ and ν are the density and the kinematicviscosity of the fluid, respectively. The constant σ = mµ is called the friction character-istic time. Thus, the Langevin equation (A.1) reads

dξ(t)

dt= − 1

σξ(t) +

1

mW (t). (A.3)

We assume that the Brownian particle has been kept for a sufficiently long time in thefluid at absolute temperature T. Then, for any time t, the equilibrium law for the energydistribution requires that

mE(ξ2(t)

)= κT,

where κ is the Boltzmann constant (a knowledge of κ is equivalent to a knowledge of Avo-gadro’s number and hence of molecular sizes). If we assume that there exists a Gaussianstationary solution to the Langevin equation (A.3), then the previous assumptions leadto the following expressions for the covariance functions of the velocity of the Brownianparticle and the noise term:

E(ξ(t)ξ(s)

)= E

(ξ2(0)

)e−|t−s|/σ =

κT

me−|t−s|/σ, (A.4)


E(W (t)W (s)

)=

m2

σE(ξ2(0)

)δ(t− s) = mκT

σδ(t− s), (A.5)

where t, s ∈ R and δ(τ) denotes the Dirac distribution. The constant (finite of infinite)

D = σE(ξ2(0)

)=

∫ ∞

0

E(ξ(τ)ξ(0)

)dτ = lim

t→∞EX2(t)

2t(A.6)

is known as the diffusion coefficient and the Einstein relation

D = σκTm

= µκT (A.7)

holds.The Langevin equation (A.3) and Einstein relation (A.7) have been extremely useful

in statistical physics and financial mathematics (see Shiryaev [60], for example). It isinteresting to note that Bachelier [8] made the first attempt towards a mathematical de-scription of the evolution of stock prices (on the Paris market) on the basis of probabilisticconcepts analogous to Brownian motion.

In the theory of hydrodynamics, the Langevin equation (A.3) needs to be modified,since it ignores the effect of the added mass and retarded viscous force, which are due tothe acceleration of the particle (see Hauge and Martin-Löf [29], for example). The addedmass effect requires to substitute the mass of the particle m with the so-called effectivemass given by

m∗ = m[1 + ρ/(2ρ0)],

where ρ0 denotes the density of the particle. Keeping the Stokes drag law unmodified,the relaxation time changes from σ = mµ to σ∗ = m∗µ. The corresponding Langevinequation then has the form (A.3) with m replaced by m∗ and σ by σ∗. Consequently,the diffusion coefficient is unmodified and turns out to be

D = σ∗E(ξ2(0)

)= µκT,

so the Einstein relation (A.7) still holds.The retarded viscous force effect is due to an additional term to the Stokes drag, which

is related to the history of the particle acceleration. This additional drag force, proposedby Stokes [62], Boussinesq [11] and Basset [10], is referred to as the Basset history force(see Hauge and Martin-Löf [29] or Maxey and Riley [42], for example). In our notation,

F (t) = − 1µ

a√πν

∫ t

b

dξ(τ)/dτ√t− τ dτ = −

a

µ√ν

bD1/2C ξ(t), (A.8)

where bD1/2C ξ(t) is the fractional derivative (2.13) of order 1/2 in the Caputo sense (seeCaputo [13], Caputo and Mainardi [14], Mainardi and Pironi [41]). Then using (A.3),(A.8), the generalised Langevin equation or Stokes–Boussinesq–Langevin equation turnsout to be

dξ(t)

dt= − 1

σ∗ξ(t)− a

σ∗√ν

bD1/2C ξ(t) +1

m∗W (t). (A.9)

It is worth noting that if the process is in thermodynamic equilibrium (at b = 0), wewould account for the long memory of hydrodynamic interaction, and thus it is correctto integrate equation (A.9) from b = −∞. On the other hand Dufty [20] proposed in thecase b = 0 to modify the random force by replacing W (t) by

W ∗(t) = W (t)− aµ√πν

∫ 0

−∞

dξ(τ)/dτ√t− τ dτ.

In any case, the fluctuation–dissipation theory of Kubo [36] proposes to introduce a mem-ory function, and one of the possible memory functions gives us the Stokes–Boussinesq–Langevin equation (A.9) or more general equation (1.6) or (4.2). This is the reasonwhy we study in this paper the fractional version of the Stokes–Boussinesq–Langevin


equation (1.6) or (4.2), which can also be called Langevin equation with Basset historyforce. Our consideration suggests useful models for financial mathematics in view ofsignificant analogies (see again Shiryaev [60]) between Newtonian mechanics and stockprice motions.

Appendix B. Random distributions

We denote by H the Hilbert space of C-valued random variables, defined on a proba-bility space (Ω,F ,P), with zero expectation and finite variance:

(f, g) = E[fḡ], ‖f‖ = (f, f) 12 .By D(R) we denote the space of all φ ∈ C∞(R) with compact support, endowed withthe usual topology. A random distribution is a linear and continuous map from D(R) toH. A random distribution X is stationary if

(X(τhφ), X(τhψ)

)=(X(φ), X(ψ)

)

for all φ,ψ ∈ D(R) and h ∈ R, where τhφ(t) = φ(t + h). We then denote by µX itsspectral measure:

(X(φ), X(ψ)

)=

∫ ∞

−∞φ̂(ξ)ψ̂(ξ)µX(dξ),

where φ̂ is the Fourier transform of φ, namely φ̂(ξ) =∫∞−∞ e

−itξφ(t) dt. Any stationaryrandom distribution X has the following spectral representation:

X(φ) =

∫ ∞

−∞φ̂(ξ)ZX(dξ),

where ZX is the orthogonal measure corresponding to the spectral measure µX(dξ):

E|Z(dξ)|2 = µX(dξ). We write Ẋ for the derivative of a random distribution X:Ẋ(φ) = −X

(φ̇).

Let X and Y be random distributions. Then X is said to be stationarily correlatedwith Y if

(X(τhφ), Y (τhψ)

)=(X(φ), Y (ψ)

)for all φ,ψ ∈ D(R) and h ∈ R; this

is equivalent to(X(t + s), Y (s)

)=(X(t), Y (0)

)for all t, s ∈ R if X and Y are both

processes. We denote by M(Y ) the closed linear hull of {Y (φ) : φ ∈ D(R)} in H. Thenwe have M(Y ) =

{∫∞−∞ g(ξ)dZY (ξ), g ∈ L2(µY )

}. A stationary random distribution Y

is said to be purely non-deterministic if⋂t∈R

Mt(Y ) = {0}, that is, the remote past doesnot contain any information at all.

Appendix C. Canonical representation

Suppose that X is a purely non-deterministic process, then X has a spectral densityf = {f(ω),ω ∈ R} of the Hardy class:

(1 +ω2)−1 log f(ω) ∈ L1(R).Following Dym and McKean [21] we write h for the outer function of X :

h(ζ) = exp

{1

2πi

∫

R

1 +ωζ

ω− ζlog f(ω)

1 +ω2dω

}, Im ζ > 0, (C.1)

and E for the canonical representation kernel of X, that is, E = ĥ, where ĥ is the Fouriertransform of

h(·) = limη↓0

h(·+ iη) ∈ L2(R),i. e.,

ĥ(t) = limM→∞

∫ M

−Me−itξh(ξ) dξ.


We have

h(ζ) =1

2π

∫ ∞

0

eiζtE(t) dt, Im ζ > 0, (C.2)

and

X(t) =1√2π

∫ t

−∞E(t− s) dB(s), E ∈ L2(R), (C.3)

or

R(t) =1√2π

∫ ∞

0

E(|t|+ s)E(s) ds (C.4)

is called the canonical representation of X, where B = {B(t), t ∈ R} is the standardBrownian motion. Note that the representations (C.3) and (C.4) play an important rolein the prediction theory of stationary processes.

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School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434,

Brisbane QLD 4001, Australia, School of Mathematics and Computational Science, Xiangtan

University, Hunan 411105, ChinaE-mail address: [email protected]

Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF244AG, UK

E-mail address: [email protected]

Received 15.09.2017

ÄÐÎÁÎÂÅ ÐIÂÍßÍÍß ÑÒÎÊÑÀ�ÁÓCÑIÍÅÑÊÀ�ËÀÍÆÅÂÅÍÀÒÀ ÌIÒÒÀÃ-ËÅÔÔËÅÐIÂÑÜÊÅ ÑÏÀÄÀÍÍß ÊÎÐÅËßÖI

Â. Â. ÀÍ, Ì. Ì. ËÅÎÍÅÍÊÎ

Àíîòàöiÿ. Ðîçãëÿäàþòüñÿ ñòàöiîíàðíi ïðîöåñè, ÿêi ¹ ðîçâ'ÿçêàìè äðîáîâîãî ðiâíÿííÿ Ñòîêñà �Áóñciíåñêà �Ëàíæåâåíà. Òàêi ïðîöåñè ìàþòü âëàñòèâîñòi ðåôëåêòîðíî¨ ïîçèòèâíîñòi òà ¨õ êîðå-ëÿöiéíi ôóíêöi¨ óçãîäæåíi ç åôåêòîì Àäëåðà �Âàéíðàéòà, àáî ìàþòü äîâãîòåðìiíîâó çàëåæíiñòü,ÿêà çîáðàæà¹òüñÿ ó òåðìiíàõ ôóíêöi¨ Ìiòòàã-Ëåôôëåðà. Òàêi âëàñòèâîñòi äîñÿãàþòüñÿ âèêîðè-ñòàííÿì òåîði¨ ðiâíÿííÿ Êóáî �Ìîði �Îêàáå(KMO) �Ëàíæåâåíà ó êîìáiíàöi¨ ç ôóíêöiÿìè Ìiòòàã-Ëåôôëåðà òà äðîáîâèìè ïîõiäíèìè. Äîñëiäæåíî òàêîæ çâ'ÿçîê iç äðîáîâèì ðóõîì Ðiññà �Áåññåëÿ.Öåé çâ'ÿçîê äîçâîëÿ¹ âèâ÷àòè îäíî÷àñíî åôåêò äîâãîòåðìiíîâî¨ çàëåæíîñòi òà ïåðåìiæíîñòi.

ÄÐÎÁÍÎÅ ÓÐÀÂÍÅÍÈÅ ÑÒÎÊÑÀ�ÁÓÑÑÈÍÅÑÊÀ�ËÀÍÆÅÂÅÍÀÈ ÌÈÒÒÀÃ-ËÅÔÔËÅÐÎÂÑÊÎÅ ÓÁÛÂÀÍÈÅ ÊÎÐÐÅËßÖÈÈ

Â. Â. ÀÍ, Í. Í. ËÅÎÍÅÍÊÎ

Àííîòàöèÿ. Îáñóæäàþòñÿ ñòàöèîíàðíûå ïðîöåññû, ÿâëÿþùèåñÿ ðåøåíèåì äðîáíîãî óðàâíåíèÿÑòîêñà �Áóññèíåñêà �Ëàíæåâåíà. Ýòè ïðîöåññû èìåþò ñâîéñòâî îòðàæåííîé ïîçèòèâíîñòè è èõêîððåëÿöèîííûå ôóíêöèè ìîãóò áûòü ñîãëàñîâàíû ñ ýôôåêòîì Àäëåðà �Âàéíðàéòà, èëè èìåþòäîëãîñðî÷íóþ çàâèñèìîñòü, âûðàæàåìóþ â òåðìèíàõ ôóíêöèè Ìèòòàã-Ëåôôëåðà. Ýòè ñâîéñòâàóñòàíîâëåíû ñ èñïîëüçîâàíèåì òåîðèè óðàâíåíèÿ Êóáî �Ìîðè �Îêàáå(ÊÌÎ) �Ëàíæåâåíà, êîìáè-íèðîâàííîé ñ ôóíêöèÿìè Ìèòòàã-Ëåôôëåðà è äðîáíûìè ïðîèçâîäíûìè. Èññëåäîâàíà òàêæå ñâÿçüñ äðîáíûì äâèæåíèåì Ðèññà �Áåññåëÿ. Ýòà ñâÿçü ïîçâîëÿåò èçó÷àòü îäíîâðåìåííî ýôôåêò äîëãî-ñðî÷íîé çàâèñèìîñòè è ïåðåìåæàåìîñòè.

Òåîðiÿ éìîâiðíîñòåé Teoriya �Imovirnoste��òà ìàòåìàòè÷íà ñòàòèñòèêà ta Matematychna StatystykaÂèï. 1(98)/2018, ñ. 29�50 No. 1(98)/2018, pp. 29�50

UDC 519.21

WAVELET ANALYSIS OF A MULTIFRACTIONAL PROCESS

IN AN ARBITRARY WIENER CHAOS

A. AYACHE, Y. ESMILI

Abstract. The well-known multifractional Brownian motion (mBm) is the paradigmatic example of

a continuous Gaussian process with non-stationary increments whose local regularity changes from

point to point. In this article, using a wavelet approach, we construct a natural extension of mBmwhich belongs to a homogeneous Wiener chaos of an arbitrary order. Then, we study its global and

local behavior.

Key words and phrases. Wiener chaos, self-similar processes, modulus of continuity, wavelet bases,

fractional processes.

2010 Mathematics Subject Classification. 60G17, 60G22.

1. Introduction

A fractional Brownian motion (fBm) of an arbitrary Hurst parameter H ∈ (0, 1),denoted by {BH(t) : t ∈ R}, is defined, up to a multiplicative constant, as the unique (indistribution) Gaussian process with stationary increments which is globally self-similarof order H. Recall that a stochastic process {F (t) : t ∈ R} is said to have stationaryincrements if, for any fixed point t0 ∈ R, one has:

{F (t0 + u)− F (t0) : u ∈ R

} law={F (u)− F (0) : u ∈ R

}; (1.1)

and it is said to be globally self-similar of order H if, for each fixed positive real numberν, one has:

{ν−HF (νt) : t ∈ R

} law={F (t) : t ∈ R

}. (1.2)

The representation of an fBm as a well-balanced moving average is given, for every t ∈ R,by the Wiener integral over R:

BH(t) =

∫

R

[|t− s|H−1/2 − |s|H−1/2

]dB(s), (1.3)

with the convention that |t − s|0 − |s|0 = log |t − s| − log |s|. FBm was first introducedby Kolmogorov in 1940 as a way for generating Gaussian spirals in Hilbert spaces [14].Later, in 1968, the well-known article [16] by Mandelbrot and Van Ness emphasisedits importance as a model in several areas of application: hydrology, geology, finance,and so on. Since then many applied and theoretical aspects of this stochastic processhave been extensively explored in the literature and, among many other things, its pathbehavior has been well understood. Despite its importance in modeling, fBm does notalways succeed in giving a sufficiently reliable description of real-life signals. Indeed, fBmsuffers from two main limitations:

(a) its Gaussian character,

The authors are very grateful to the anonymous referee for his valuable comments which have led to

improvements of the article. This work has been partially supported by the Labex CEMPI (ANR-11-LABX-0007-01) and the GDR 3475 (Analyse Multifractale).

30 A. AYACHE, Y. ESMILI

(b) local roughness of its paths remains everywhere the same; more precisely, theirlocal and pointwise Hölder exponents are everywhere equal to the Hurst param-eter H.

In order to overcome the limitation (b) of fBm, the so-called multifractional Brownianmotion (mBm) was introduced, about twenty years ago, independently by Benassi, Jaf-fard and Roux [5] and by Lévy Véhel and Peltier [20]. The latter continuous Gaussianprocess with non-stationary increments can be obtained by substituting the constantHurst parameter H in (1.3) by a deterministic continuous function H(·) depending on tand with values in the open interval (0, 1). Nowadays mBm has become a quite usefulmodel in the fields of financial modeling and signal processing (see for instance [6–8]).

In order to overcome the limitations (a) and (b) together, the extensions of mBmwhose Hurst parameter is a stochastic process or more generally a sequence of stochasticprocesses were introduced in [2, 3]. Other extensions of mBm to frames of heavy-tailedstable distributions were proposed in [10, 22, 23]. More recently, [21] constructed a mul-tifractional generalized Rosenblatt process belonging to the second order homogeneousWiener chaos. In our present article, we construct a multifractional process, denoted by{Z(t) : t ∈ R}, which belongs to a homogeneous Wiener chaos of an arbitrary integerorder d ≥ 2. The latter multifractional process is not a generalization of the Rosenblattprocess but of a process {YH(t) : t ∈ R} consisting in a very natural chaotic extensionof the fBm in (1.3). Namely, it is defined, for all t ∈ R, through the multiple Wienerintegral on Rd:

YH(t) =

∫

Rd

[‖t∗ − x‖H−

d2

2 − ‖x‖H− d22

]dBx1 . . . dBxd , (1.4)

where t∗ = (t, . . . , t) ∈ Rd and ‖ · ‖2 denotes the Euclidian norm over Rd. A classof chaotic self-similar processes with stationary increments, which implicitly includes{YH(t) : t ∈ R}, had been first introduced and investigated in [18]. Long time later,{YH(t) : t ∈ R} was explicitly introduced and studied in its own right in [1] throughwavelet methods inspired by the ones in [2, 3].

Recall that a centred non-Gaussian square integrable real-valued random variable, onthe underlying probability space (Ω,F ,P), belongs to the homogeneous Wiener chaosof an arbitrary integer order d ≥ 2 when it can be represented by a multiple Wienerintegral over Rd. We always denote by Id(·) this stochastic integral, and use the classicalconvention that, for every f ∈ L2(Rd), one has Id(f) = Id(f̃); the function f̃ being thesymmetrization of f , defined, for all (t1, . . . , td) ∈ Rd, as

f�

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