Một số bài toán định tính của phương trình vi phân ngẫu nhiên không ôtônôm với nhiễu Brown phân thứ

Mục tiêu của luận án: Mục đích của luận án là chứng minh sự tồn tại duy nhất nghiệm của hệ phương trình vi phân ngẫu nhiên nhiễu bởi chuyển động Brown phân thứ, nghiên cứu tính chất của nghiệm, nghiên cứu số mũ Lyapunov của hệ tuyến tính và nghiên cứu tiêu chuẩn cho sự tồn tại tập hút pullback không autonome của dòng hai tham số ngẫu nhiên sinh bởi hệ. Nội dung của luận án gồm 3 phần chính. Phần 1: Phương trình vi phân ngẫu nhiên nhiễu bởi chuyển động Brown phân thứ. Phần 2: Phổ Lyapunov của hệ phương trình tuyến tính không autonome. Phần 3: Tập hút ngẫu nhiên của hệ phương trình không autonome. Các kết quả chính của luận án: Luận án đã đạt được các kết quả chính sau đây: 1. Chứng minh định lý về sự tồn tại duy nhất nghiệm của hệ phương trình không autonome nhiễu bởi chuyển động Brown phân thứ và một số tính chất của nghiệm. Chứng minh sự sinh dòng hai tham số ngẫu nhiên từ hệ và đặc biệt là sự sinh hệ động lực ngẫu nhiên khi các hàm hệ số không phụ thuộc thời gian. 2. Một số kết quả về phổ Lyapunov của hệ tuyến tính: lược đồ rời rạc hoá để tính toán phổ, công thức hiển cho phổ của hệ tam giác chính quy, tính chính quy hầu chắc chắn (theo nghĩa một độ đo xác suất) của hệ. 3. Tiêu chuẩn cho sự tồn tại tập hút pullback không autonome của dòng hai tham số ngẫu nhiên sinh bởi hệ: trường hợp tổng quát và hai trường hợp đặc biệt với hệ số khuếch tán tuyến tính, bị chặn. Xây dựng dòng Bebutov từ hệ và chỉ ra sự tồn tại tập hút ngẫu nhiên một điểm của dòng theo nghĩa pullback và forward.

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

Phan Thanh Hong

SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS

STOCHASTIC DIFFERENTIAL EQUATIONS

DRIVEN BY FRACTIONAL BROWNIAN MOTIONS

DISSERTATION FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

HANOI - 2021

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

Phan Thanh Hong

SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS

STOCHASTIC DIFFERENTIAL EQUATIONS

DRIVEN BY FRACTIONAL BROWNIAN MOTIONS

Speciality: Probability and Statistics Theory Speciality code: 9 46 01 06

DISSERTATION FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

Supervisor: Dr Luu Hoang Duc

HANOI - 2021

This dissertation was written based on my research works at the Institute ofMathematics, Vietnam Academy of Science and Technology under the supervi-sion of Dr Luu Hoang Duc I declare hereby that all the presented results havenever been published by others

April, 2021The author

Phan Thanh Hong

First and foremost I am extremely grateful to my advisor Dr Luu Hoang Ducfor continuous support of my academic research, for his invaluable advice, pa-tience, motivation, and immense knowledge His guidance helped me in all thetime of research and writing of this thesis I thank him for his encouragementand recommendation to the IMU Breakout Graduate Fellowship

I would also like to express my special appreciation to Prof Dr.Sc NguyenDinh Cong for his enormous support I benefited a lot from his advices in thepast few years Despite numerous other interests and busy academic life, Prof.Cong has taken the time to read the draft and made precious suggestions forthe contents of my thesis

My sincere thanks also goes to all the members in the Probability and tics Department of the Institute of Mathematics I received many suggestionsand experience through the seminars of the Department

Statis-Furthermore, I thank my colleages at Thang Long University, for their port throughout my PhD study I specially thank Prof Dr.Sc Ha Huy Khoai forhis support and encouragement

sup-I gratefully acknowledge the sup-IMU Breakout Graduate Fellowship Programand the International Center for Research and Postgraduate Training in Math-ematics - Institute of Mathematics for their financial support It is my honor toreceive the grants

And last but not least, I could not have finished this work without the ditional support from my parents, my husband and my little children I wouldlike to express my sincere gratitude to all of them

1.1 Fractional Brownian motions 1

1.1.1 Nonsemimartingale properties 2

1.1.2 Canonical spaces 3

1.2 Pathwise stochastic integrals with respect to fractional Brownian motions 6

1.2.1 Young integrals 7

1.2.2 Fractional integrals and fractional derivatives 8

1.2.3 Stochastic integrals w.r.t fractional Brownian motions 10

1.2.4 Young integrals on infinite domains 10

1.3 Greedy sequences of times 11

1.4 Stochastic flows 12

Chapter 2 Stochastic differential equations driven by fractional Brow-nian motions 14 2.1 Assumptions 15

2.2 Existence and uniqueness theorem for deterministic equations 16

2.2.1 Existence and uniqueness of a global solution 17

2.2.2 Estimate of the solution growth 27

2.2.3 Special case: linear equations 29

2.3 Continuity and differentiability of the solution 31

2.3.1 The continuity of the solution 32

2.3.2 The differentiability of the solution 33

2.4 The stochastic differential equations driven by fBm 38

2.5 The generation of stochastic two parameter flows 40

2.6 Conclusions and discussions 43

Chapter 3 Lyapunov spectrum of nonautonomous linear fSDEs 44

3.1 The generation of stochastic flow of linear operators 44

3.2 Lyapunov exponent of Young integrals w.r.t BH 46

3.3 Lyapunov spectrum for nonautonomous linear fSDEs 50

3.3.1 Exponents and spectrum 51

3.3.2 Lyapunov spectrum of triangular systems 55

3.3.3 Lyapunov regularity 60

3.4 Almost sure Lyapunov regularity 62

3.5 Conclusions and discussions 66

Chapter 4 Random attractors for nonautonomous fSDEs 67 4.1 Nonautonomous attractors 69

4.2 Existence of random attractors 73

4.3 Special case: g linear 84

4.4 Special case: g bounded 86

4.5 Bebutov flow and its generation 94

4.6 Conclusions and discussions 102

Table of Notations

a∨b the maximum of a and b

∆n the closed interval[n, n+1], n ∈ Z

∆[a, b] the simplex {(s, t) ∈ [a, b]|s ≤ t}

kxk∞,[a,b] the supermum norm of function x on [a, b]

kxkp−var,[a,b] the p variation norm of function x on[a, b]

kxkβ−Hol,[a,b] the β−H ¨older norm of function x on[a, b]

kxkLp (a,b) Lp−norm of function x

C([a, b],Rd) the space of Rd-valued continuous function on[a, b]

C∞([a, b],Rd) the subspace of smooth functions in C([a, b],Rd)

Cp−var([a, b],Rd) the subspace of bounded p-variation

functions in C([a, b],Rd)

Cα-Hol([a, b],Rd) the subspace of α−H ¨older functions inC([a, b],Rd)

C0,p−var([a, b],Rd) the closure ofC∞([a, b],Rd) inCp−var([a, b],Rd)

C0,α−Hol([a, b],Rd) the closure ofC∞([a, b],Rd) inCα−Hol([a, b],Rd)

C00,p−var([a, b],Rd) the subspace of functions which vanish at 0

inC0,p−var([a, b],Rd)

C00,α−Hol([a, b],Rd) the subspace of functions which vanish at 0

inC0,α−Hol([a, b],Rd)

G(z) Gamma function

fBm fractional Brownian motions

SDE stochastic differential equation

fSDE SDE driven by fractional Brownian motions

RDS random dynamical system

w.r.t with respect to

dex β ∈ (0, H) a.s For H > 1/2, the increments are positive correlated andfor H < 1/2 they are negative correlated Moreover, it is a long memory pro-cess when H > 12 ( [71]) These significant properties make fractional Brownianmotions a natural candidate to model the noise in applications to mathematicalfinance ( [18], [50], [37]), in hydrology, communication networks and in otherfields (see for instance [48], [84]).

When modelling real data which often include noises, stochastic differentialequations is a powerful tool If noises are assumed to be fractional Brownianmotions the problem of modelling becomes a stochastic differential equationdriven by fBms which is understood in the integral form This leads to the need

of definition of integral w.r.t fractional Brownian motions

However, BH is not a semimartingale if H 6= 12, one cannot apply the classicalIto theory to construct a stochastic integral w.r.t the fBm by taking the limit inthe sense of probability convergence of a sequence of Darboux sums A moderndevelopment in the field of Stochastic Analysis deals with stochastic integratorswhich are more general than semimartingales Among a numerate attempts todefine a (stochastic) integral with respect to fractional Brownian motion, thedeterministic approach consists of two directions of development: rough paththeory and fractional calculus, in which the integrals can be defined in the path-wise sense A comprehensive presentation of these theories can be found in Frizand Victoir [42] and in Samko et al [88] Both theory relies on properties of thesample paths For the case H > 1/2, the integral defined by rough path theory

is understood in the Young sense and coincides with that defined by fractionalderivative on the space of H ¨older continuous functions.

In the last decades, after the successful construction of integral w.r.t fBm,stochastic differential equations driven by fractional Brownian motions (in shortfSDE) have attracted a lot of research interest In this thesis we study the nonau-tonomous stochastic differential equations driven by m−dimensional fractionalBrownian motions with Husrt index H > 1/2 of the form

The first important question is on the existence and uniqueness of solution

to (1) The first study on the differential equations driven by rough signalsdates back to [61] which is then generalized to introduce rough path theory( [62], [63]) Using this approach, the existence of the solution of equations in

a certain space of continuous functions with bounded p-variation is proved in[61] and [30], [78] The results are then generalized for the case 2 < p < 3 by [63]and [42], see also recent work by [77], [40] According to their settings, f , g aretime - independent and/or g is often assumed to be differentiable and bounded

in itself and its derivatives All can be applied to the stochastic differentialequations driven by fBm (fSDE) Another approach follows Z¨ahle [86] by usingfractional derivatives where the non autonomous systems are treated (see [74]).Similar results are established for system in infinite dimensional case, see forinstance in [66], [7]

Since our target is the equations driven by fBm with H > 1/2 which can bestudied by these two approaches, we aim to close the gap between the twomethods and develop techniques to study more on the infinite dimensionalcases ( [34]) and on the dynamic of these systems ( [23], [22])

At first we prove that, under similar assumptions to those in [74], the tence and uniqueness theorem for system (1) still holds in the space of contin-uous functions with bounded p-variation norm When applying to stochasticdifferential equations driven by fractional Brownian motions, by considering

exis-an appropriate probability space, it is proved that the system generates a rexis-an-dom dynamical system (in short RDS, see [16], [44], and [5]) However in thenonautonomous situation, one only expects the system to generate a stochastictwo-parameter flow on the phase space.

ran-These results allow us to study some qualitative problems of the systemsunder the framework of RDS theory with typical topics: random attractor, sta-bility, invariant manifolds and so on (see for instane [72], [3], [4]) In the scope

of this thesis we focus on studying the Lyapunov spectrum of linear systemsand the random attractor of semilinear equations Note that these problems arestill open even for the case H > 1/2 (see recent results in [42], [32])

Random attractor is one of the most important notation of random cal system Its generalization, nonautonomous random attractor is introduced

dynami-to sdynami-tochastic flow where the state of the system depends on both the initial andpresent time ( [24]) We develop the semi-group technique to study the exis-tence of the random pullback attractor provided that the linear part has negativeeigenvalue and the nonlinear pertubations are small In the case g is linear, theattractor is singleton and also forward attractor For the nonlinear case, undersome additional conditions we point out that the attractor is one point in thesense of the Bebutov flow generated by the equation which is a RDS on the ap-propriate space of noise These techniques show the capability to deal with therough equation in the work by [39], or in the paper for infinite dimensional case

by [22], [43]

We are also interested in studying Lyapunov spectrum of nonautonomous ear systems Notice that Lyapunov spectrums and its splitting are the main con-tent of the celebrated multiplicative ergodic theorem (MET) by Oseledets [75] Itwas also investigated by Millionshchikov in [67–70] for linear nonautonomousdifferential equations In the stochastic setting, the MET is also formulatedfor random dynamical systems in [4, Chapter 3] Further investigations can

lin-be found in [19, 20] for stochastic flows generated by nonautonomous linearstochastic differential equations driven by standard Brownian motions To ourknowledge there has not been any works on this topic for the stochastic systemdriven by fBm

We use the approach developed in [19] to study the Lyapunov spectrum ofthe system We show that Lyapunov exponents can be computed based on thediscretization scheme And moreover, the spectrum is bounded by a nonran-dom constant We are also interested in the question on the non-randomness

of Lyapunov exponents In case the system is driven by standard Brownian

noises, it is proved in [19] that Lyapunov exponents are nonrandom However,the techniques used in [19] can not be applied to case of fractional Browniannoises, hence we develop further techniques to deal with this case We showthat the answer is affirmative for some special cases For example, autonomoussystems can generate random dynamical systems satisfying the integrabilitycondition, thus the Lyapunov spectrum is non-random by MET [4] Our in-vestigation shows that the Lyapunov spectrum of triangular systems that areLyapunov regular are non-random Finally, we prove a Millionshchikov theo-rem stating that almost all (in a sense of an invariant measure) linear fSDEs areLyapunov regular.

The thesis consists of 4 chapters

Chapter 1 recalls the definition and properties of fBm as well as the struction of the canonical space for fBm, some basic results on Young integrals,greedy sequences of times, the concepts of stochastic flows and random dynam-ical systems

con-In Chapter 2 we present the existence and uniqueness theorem for system (1)

as well as its backward equation The main tools in use are greedy sequences

of times, Shauder-Tychonoff fixed point theorem and a Gronwall-type lemma

We establish the estimate for the norm of the solution then prove finiteness ofits moments The continuity and differentiability of the solution are proved.The final part of this chapter presents the generation of stochastic flows whichbecome random dynamical systems for autonomous systems

In Chapter 3 we discuss Lyapunov spectrum of linear systems We showthat Lyapunov exponents can be computed based on the discretization scheme

Moreover, the spectrum can be computed independently of ω for triangular

systems (i.e both A, C are upper triangular matrices) which are Lyapunov ular Finally, under some further conditions on coefficient matrices we construct

reg-a probreg-ability mereg-asure together with reg-a Bebutov flow generreg-ated by the system.The regularity almost all of the spectrum is proved as a consequence of the MET

In Chapter 4 we extend the results in [35] and [36] for nonautonomous tions We establish criteria for the existence of a nonautonomous pullback at-tractor of the system Moreover we show that the attractor is a singleton and isalso a forward attractor in case g is linear or bounded At the end of this chapter,

equa-we show the generation of the random dynamical system w.r.t the metric namical system generated by Bebutov flows in the product space The flow thenpossesses a singleton random attractor in both pullback and forward directions

dy-as a consequence of the results presented at the beginning of this chapter.

The dissertation is written based on refereed papers [21], [34], [23], [35], [22]and also a preprint [36]

To conclude the introduction we would like to note that although we workwith system driven by fBm, many results still hold for equations driven by pro-cesses which are of stationary increments and have local p−variation boundedsample paths

Chapter 1

Preliminaries

In this chapter we recall some notations and summarize some standard sults which we will need in the sequel

re-1.1 Fractional Brownian motions

The fractional Brownian motion ( f Bm in short) was first introduced by mogorov in [57] and then further developed by Mandelbrot and Van Ness in[65] It is a family of Gaussian process defined as follows (see more in [73])

Kol-Definition 1.1 ( [73, Kol-Definition 2.1], [71, Kol-Definition 1.2.1]) A two-sided one-dimensional

fractional Brownian motion of Hurst parameter H ∈ (0, 1) is a centered continuousGaussian process BH = (BtH), t ∈ R with covariance function

|s| − |t−s|) Therefore B12 is a classical Brownian motion

(ii) It can be seen from the covariance function that E(BtH−BsH)2 = |t−s|2H Since

BH is a Gaussian process we obtain

2 (Symmetry) (−BtH)t∈R = (d BtH)t∈R,

3 (Stationary increments)(Bt+hH −BhH)t∈R = (d BtH)t∈R.

In the following we recall some properties of f Bm, we refer to [65], [73], [71]

or other references there in for the details

H ¨older continuity of paths

The property of H ¨older continuity paths of fBm is important one to deal withthe integral w.r.t fBm This is deduced by Kolmogorov Theorem

Theorem 1.1 ( [55, Theorem 2.8, p 53]) Let Xt, t ∈ [0, T] be a stochastic process.Suppose that there exist p > 0, c > 0, ε > 0 so that for every s, t ∈ [0, T]

E(|Xt−Xs|p) ≤ c|t−s|1+ε.Then there exists a continuous modification ˜X of X which is locally α−H¨older contin-

uous for every α ∈ (0, ε/p)

Proposition 1.2 ( [73, Proposition 1.6]) For each 0 < ν < H fixed, there exists amodification of BH with paths of ν−H¨older continuity on every compact sets ofR Proof. Since

for n ∈ N∗ Choose n such that nH−1n > ν Due to, Kolmogorov Theorem

BH has a continuous modification and ν−H ¨older modification on any compact

Recall that a real-valued continuous process is a semimartingale if it can be composed as the sum of a local martingale and a continuous adapted process of

de-locally bounded variation ( [55, Definition 3.1, p 149]) It is known that a martigale is of locally bounded quadratic variation If X is a semimartingale onthe interval [0, 1],

semi-• VΠ,[0,1](2) (X) converges in probability as n → ∞ and

• if lim

| Π|→0V

(2) Π,[0,1](X) = 0, limΠ→0V

(1) Π,[0,1](X) < ∞ almost sure

For the case fBm, due to [17, Proposition 3],

lim

| Π|→0EhVΠ,[0,1](1/H)(BH) −E(|G|1/H)i2 = 0And similar to the case of B12 ( [76, Corollary 2.5, p 29]), the fractional Brownianpaths a.s have infinite variation on any interval Namely, since H1 > 1 and

0 < lim

| Π|→0V

(1/H) Π,[0,1](BH) < ∞, we have lim

| Π|→0V

(1) Π,[0,1](BH) = ∞, a.s

If H > 12, by choosing p = 2 > H1 one has lim

| Π|→0V

(2) Π,[0,1](BH) = 0 If BH is a

semimartingale, a.s lim

Π→0V

(1) Π,[0,1](BH) < ∞ which is a contradiction

If H < 12, since 0 < lim

| Π|→0V

(1/H) Π,[0,1](BH) < ∞ and 1

H > 2, the continuity of BHpaths leads to

lim

| Π|→0V

(2) Π,[0,1](BH) = ∞Therefore BH with H < 12 can not be a semimartingale either

1.1.2 Canonical spaces

It is known that a stochastic process(Xt)t∈Rdefined on the probability(Ω,F,P)valued in Rm with continuous paths can be viewed as a measurable map X :(Ω,F,P) → (C(R, Rm),B) where C(R, Rm) is the spaces of continuous func-tions from R to Rm and B is the σ−algebra generated by cylinder sets, the

smallest σ−algebra which ensures the measurability of all the projections πt :C(R, Rm) → Rm, πt(x) := xt To use the theory of random dynamical system,

it is better to work with the canonical version (C(R, Rm),B,PX), where PX isthe image measure XP and with the coordinate canonical process

Yt : (C(R, Rm),B,PX) → Rm, Yt(x) =πt(x) = xt

Definition 1.2 ( [4]) Let (Ω,F,P) be a probability space A family (θt)t∈R of

map-pings of (Ω,F,P) into itself is called a metric dynamical system if it satisfies the lowing conditions

fol-(i) (ω, t) 7→ θt(ω) is measurable,

(ii) θ0 = IdΩ,

(iii) θt+s = θt ◦θs for all s, t ∈R,

(iv)P is(θt)ưinvariant, i.e P = θtP, for all t ∈R.

Definition 1.3 A metric dynamical system (Ω,F,P,(θt)t∈R)is called ergodic if the

set A ∈ F which are(θt)ưinvariant, i.e θt(A) = A for all t ∈ R has measure one or

zero

In this section, we recall the construction of the canonical sample space for

BH following [45] which facilitates the random dynamical system approach tothe stochastic differential equations driven by fBm The canonical space forordinary Brownian motion is (C0(R, R),B,P1

2, θ) (see [4]) in which C0(R, R)

is the space of continuous functions on R with values in R vanishing at zero,

equipped with the compact open topology given by the uniform convergence

on compact intervals in R, B is the Borel σưalgebra,P1

2 is the Wiener measure

and θ is the Wiener shift operator, given by

θt(ω)(·) := ω(t+ ·) ưω(t)

for ω ∈ C0(R, R) It is known that the space(C0(R, R),B,P1

2, θ)is of ergodicity(see [4, p 546-547])

For each square integrable kernel k(t, s), the Ito integralRRk(t, s)dWs, where

W = {Wt, t ∈ R} is the two-sided Brownian motion, defines a Gaussian cess With the convenient kernel, fractional Brownian motion has the represen-tation in law RRk(t, s)dWs (see for example [1]) We introduce here the repre-sentation of fBm due to Mandelbrot and Van Ness [65] Put

pro-kH(t, u) := [(tưu) ∨0]Hư12 ư [(ưu) ∨0]Hư12, t, u ∈ R

and

cH =

r12H +

Z ∞ 0

de-BH : (C0(R, R),B) → (C0(R, R),B)

Then there exists (C0(R, R),B,PH, θ) the canonical space for fBm, in which

PH := BHP1

2 Moreover the space is an ergodic metric dynamical system

Theorem 1.2 ( [45, Theorem 1])(C0(R, R),B,PH,(θt))is an ergodic metric ical system

dynam-Multidimensional fractional Brownian motions

Definition 1.4 Let m be a positive integer An m−dimensional fractional Brownianmotion BH of Hurst index H ∈ (0, 1)is(B1H, B2H,· · · , BmH)where BiH are independentone-dimensional fractional Brownian motions of Hurst index H

Similar to the one-dimensional case, one can construct the canonical space(Ω,F,P, θ) for m−dimensional BH Here Ω is C0(R, Rm) equipped with the

Borel σ−algebra, P is the distribution of BHand θ is the Wiener shift operator.

Moreover, the space is ergodic

For our purpose, we need to work with the space of bounded p - variationpaths on any compact intervals For each p ≥ 1 denote by Cp−var(R, Rm),

C0,p−var(R, Rm) the spaces of all continuous functions ω : R → Rm such that

the restriction of ω on [−T, T] is in Cp−var([−T, T],Rm), C0,p−var([−T, T],Rm)respectively for any T > 0 EquipCp−var(R, Rm)with the metric

Assign

C00,p−var(R, Rm) := {x ∈ C0,p−var(R, Rm)| x0 = 0},Note that for x ∈ C00,p−var(R, Rm), |||x|||p−var,I and kxkp−var,I are equivalentnorms for every compact interval I containing 0 Due to [5, Proposition 1,Theorem 5] the space C00,p−var(R, Rm) is a completely separable metrizabletopological space Moreover, C00,p−var(R, Rm) is θ−invariant and for each t,

θt : (C00,p−var(R, Rm), d) → (C00,p−var(R, Rm), d) is continuous (see [5, rem 5])

Theo-Since BH has the 1p− H ¨older continuous version hence p−var one for 1 <

p < H1 one can take the trace of B on C00,p−var(R, Rm) and restrict PH on this

new σ−algebra to build a new metric dynamical system which we keep theold notation (Ω,F,P, θ) Moreover, this metric dynamical system is ergodic(see [12, Lemma 1, Remark 2])

Remark 1.2. (i)Identical results apply to ν− H¨older space, for 0 < ν < H

(ii)When dealing with the differential equation driven by H¨older continuous paths, the

noise is often denoted by x Throughout this thesis we would prefer using the notation

ω to emphasize the randomness of the objects.

In the following chapters, we fix p ∈ (1, H1), the fBm is understood to bedefined on the canonical space (Ω,F,P, θ) = (C00,p−var(R, Rm),B,P, θ) whenmentioned We identify BH(·, ω)with ω(·)

1.2 Pathwise stochastic integrals with respect to fractional

Brownian motions

In this section we recall two approaches to extend Stieljes integrals for infinitevariation integrators Both can be applied to define stochastic integral withrespect to fractional Brownian motions in the pathwise sense Throughout thisthesis we will only work with the first approach

We first review on Riemann - Stieltjes and Lebesgue - Stliejes integrals as amotivation for the extensions introduced later in this section The details can beseen in chapter 2 of [42]

Let f , g be functions on[a, b]value in Rd×m,Rm respectively LetΠ = {ti}be

a partition of[a, b]and ξi ∈ [ti, ti+1] If the sum

SΠ := ∑

ti∈ Π

f(ξi)[g(ti+1) −g(ti)] (1.3)

converges to a finite limit I as |Π| → 0 independent of the choice of Π and ξi

then I is called the Riemann - Stieltjes integral of f w.r.t g We then write

I =

Z b

a f(t)dg(t).Due to [42, Proposition 2.2] , if f is continuous and g is in C1−var([a, b],Rm)thenthe Riemann–Stieltjes integralRab f(t)dg(t)exists and

Z b

a f(t)dg(t)

≤ kfk∞,[a,b]|||g|||1−var,[a,b].For Lebesgue - Stliejes integral, let g be a function which is of finite variation

on [a, b] Then g can be expressed by the difference of two monotone functions

g1, g2 Denote by µ1, µ2the Borel measures associated with g1, g2 If f is a Borelfunction then the Lebesgue - Stliejes integral of f w.r.t g on [a, b]is defined by

which can be rewriten by

Z b

a f(t)dg(t) = −

Z b

a fa0+(t)gb−(t)dt+ f(a+)[g(b−) −g(a+)] (1.4)where fa+(t) = 1(a,b)(t)[f(t) − f(a+)], gb−(t) =1(a,b)(t)[g(t) −g(b−)]

It is known that if f is continuous, g is right continuous and of boundedvariation, then these two integrals exist and coincide

1.2.1 Young integrals

In this subsection we recall some facts about Young integral which is an tension of Riemann–Stieltjes integral to a more general function spaces, moredetails can be seen in [85], [42]

ex-Definition 1.5 Given f ∈ Cq−var([a, b],Rd×m) and g ∈ Cp−var([a, b],Rm) we saythat z ∈ C([a, b],Rd) is an indefinite Young integral of f against g if there exist asequence (fn, gn)inC1−var([a, b],Rd×m) × C1−var([a, b],Rm)satisfying

z = Ra· f(t)dg(t)and setRst f(u)dg(u) := Rat f(u)dg(u) −Ras f(u)dg(u)

Theorem 1.3 (Young-Loeve, [42])

Given f ∈ Cq−var([a, b],Rd×m) and g ∈ Cp−var([a, b],Rm) with 1p + 1q > 1, thereexists unique indefinite Young integral of f against g, Ra· f(t)dg(t) Moreover, theYoung-Loeve estimate

1−21−1p −1q |||g|||p−var,[s,t]|||f|||q−var,[s,t]

(1.5)holds for all [s, t] ⊂ [a, b]

From the Young-Loeve estimate (1.5), one can see that the Riemann - Stieltjessum in (1.3) converges to Rst f(u)dg(u)when the mesh|Π|tends to 0, i.e

Lemma 1.1 For p ≥ 1, q ≥ 1 such that 1p + 1q > 1 and f ∈ Cq−var([a, b],Rd×m),

g ∈ Cp−var([a, b],Rm), the following estimate holds

... [86]).

1.2.3 Stochastic integrals w.r.t fractional Brownian motions

Given a stochastic process Xt and a fractional Brownian motion BtH withHurst... e−iπα For a function f(t)with t ∈ [a, b], the left-sided and right-sided

Riemann-Liouville fractional derivative of f of order α at t ∈ (a, b)are defined