SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS

SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS.SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS. SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS. SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS. SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS. SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS. SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS. SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS. SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS. SOME QUALITATIVE PROBLEMS OF NONAUTONOMOUS STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS.

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: 46 01 06 DISSERTATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisor: Dr Luu Hoang Duc Confirmation This dissertation was written based on my research works at the Institute of Mathematics, Vietnam Academy of Science and Technology under the supervi- sion of Dr Luu Hoang Duc I declare hereby that all the presented results have never been published by others April, 2021 The author Phan Thanh Hong Acknowledgment First and foremost I am extremely grateful to my advisor Dr Luu Hoang Duc for continuous support of my academic research, for his invaluable advice, pa- tience, motivation, and immense knowledge His guidance helped me in all the time of research and writing of this thesis I thank him for his encouragement and recommendation to the IMU Breakout Graduate Fellowship I would also like to express my special appreciation to Prof Dr.Sc Nguyen Dinh Cong for his enormous support I benefited a lot from his advices in the past few years Despite numerous other interests and busy academic life, Prof Cong has taken the time to read the draft and made precious suggestions for the contents of my thesis My sincere thanks also goes to all the members in the Probability and Statis- tics Department of the Institute of Mathematics I received many suggestions and experience through the seminars of the Department Furthermore, I thank my colleages at Thang Long University, for their sup- port throughout my PhD study I specially thank Prof Dr.Sc Ha Huy Khoai for his support and encouragement I gratefully acknowledge the IMU Breakout Graduate Fellowship Program and the International Center for Research and Postgraduate Training in Math- ematics Institute of Mathematics for their financial support It is my honor to receive the grants And last but not least, I could not have finished this work without the unconditional support from my parents, my husband and my little children I would like to express my sincere gratitude to all of them Contents Table of Notation Introduction vi Chapter 1.2 Preliminaries 1.1 Fractional Brownian motions 1.1.1 Nonsemimartingale properties 1.1.2 Canonical spaces Pathwise stochastic integrals with respect to fractional Brownian motions 1.2.1 Young integrals 1.2.2 Fractional integrals and fractional derivatives 1.2.3 Stochastic integrals w.r.t fractional Brownian motions 1.2.4 Young integrals on infinite domains 1.3 Greedy sequences of times 1.4 Stochastic flows Chapter motions v 1 10 10 11 12 Stochastic differential equations driven by fractional Brow- nian 2.1 Assumptions 2.2 Existence and uniqueness theorem for deterministic equations 2.2.1 Existence and uniqueness of a global solution 2.2.2 Estimate of the solution growth 2.2.3 Special case: linear equations 2.3 Continuity and differentiability of the solution 2.3.1 The continuity of the solution 2.3.2 The differentiability of the solution 2.4 The stochastic differential equations driven by fBm 2.5 The generation of stochastic two parameter flows 2.6 Conclusions and discussions 14 15 16 17 27 29 31 32 33 38 40 43 Chapter 3.1 3.2 3.3 3.4 3.5 44 The generation of stochastic flow of linear operators 44 Lyapunov exponent of Young integrals w.r.t BH 46 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 Almost sure Lyapunov regularity 62 Conclusions and discussions .66 Chapter 4.1 4.2 4.3 4.4 4.5 4.6 Lyapunov spectrum of nonautonomous linear fSDEs Random attractors for nonautonomous fSDEs 67 Nonautonomous attractors 69 Existence of random attractors 73 Special case: g linear 84 Special case: g bounded .86 Bebutov flow and its generation 94 Conclusions and discussions .102 General Conclusions 103 List of Author’s Related Papers 104 References 114 Table of Notations a∨b ∆n ∆[a, b] |·| ǁxǁ∞,[a,b] ǁxǁp−var,[a,b] ǁ x ǁ β−Hol,[a,b] ǁ xǁ L p (a,b) C([ a, b], Rd) C ∞ ([ a, b], Rd) C p−var([a, b], Rd) C α-Hol ([ a, b], R d ) C0,p−var([a, b], Rd) C0,α−Hol([a, b], Rd) 0,p−va C0r ([a, b], Rd) 0C 0,α−Hol G(z) a.s fBm SDE fSDE RDS w.r.t ([a, b], Rd) the maximum of a and b the closed interval [n, n + 1], n ∈ Z the simplex {(s, t) ∈ [a, b]|s ≤ t} norm on the Euclide space Rd the supermum norm of function x on [a, b] the p variation norm of function x on [a, b] the Hoă lder norm of function x on [ a, b] Lp−norm of function x the space of Rd-valued continuous function on [a, b] the subspace of smooth functions in C([ a, b], R d) the subspace of bounded p-variation functions in C([ a, b], R d) the subspace of Hoă lder functions in C([ a, b], R d ) the closure of C ∞ ([ a, b], R d) in C p−var([a, b], R d) the closure of C ∞ ([ a, b], R d) in Cα−Hol([a, b], R d) the subspace of functions which vanish at in C0,p−var([a, b], R d) the subspace of functions which vanish at in C0,α−Hol([a, b], R d) Gamma function almost sure fractional Brownian motions stochastic differential equation SDE driven by fractional Brownian motions random dynamical system with respect to Introduction A fractional Brownian motion (in short fBm) is a family of centered Gaussian processes BH = BH tt ∈ R or R+, indexed by the Hurst parameter H ∈ (0, 1) with continuous sampleΣpaths and the covariance function R (s, t) = t2H + s2H t s 2H − | − | H It was originally defined and studied by Kolmogorov ( [57]) and then was developed Σ by Mandelbrot and Van Ness in [65] It is a self-similar process with stationary increments and has Hoă lder continuous sample paths with in- dex β ∈ (0, H) a.s For H > 1/2, the increments are positive correlated and for H < 1/2 they are negative correlated Moreover, it is a long memory process when H > 21 ( [71]) These significant properties make fractional Brownian motions a natural candidate to model the noise in applications to mathematical finance ( [18], [50], [37]), in hydrology, communication networks and in other fields (see for instance [48], [84]) When modelling real data which often include noises, stochastic differential equations is a powerful tool If noises are assumed to be fractional Brownian motions the problem of modelling becomes a stochastic differential equation driven 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 ƒ= , one cannot apply the classical Ito theory to construct a stochastic integral w.r.t the fBm by taking the limit in the sense of probability convergence of a sequence of Darboux sums A modern development in the field of Stochastic Analysis deals with stochastic integrators which are more general than semimartingales Among a numerate attempts to define a (stochastic) integral with respect to fractional Brownian motion, the deterministic approach consists of two directions of development: rough path theory 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 Friz and Victoir [42] and in Samko et al [88] Both theory relies on properties of the sample 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 fractional derivative on the space of Hoă lder 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 short fSDE) have attracted a lot of research interest In this thesis we study the nonau- tonomous stochastic differential equations driven by m−dimensional fractional Brownian motions with Husrt index H > 1/2 of the form dxt = f (t, xt)dt + g(t, x t ) dB H , t (1) to take advantage of the simplicity of Young integral System (1), like Ito differ- ential equations, is understood as an integral equation of form xt = x0 + t f (s, ∫ x )ds + s t g(s, ) dBH , t ∈ [0, T] ∫ x s (2) s where f : [0, T] × Rd → R d, g : [0, T] × Rd → Rd×m are time dependent coefficient functions, the first integral is of Riemann type and the second one is understood pathwise in the Young sense The first important question is on the existence and uniqueness of solution to (1) The first study on the differential equations driven by rough signals dates 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 < p < by [63] and [42], see also recent work by [77], [40] According to their settings, f , g are time independent and/or g is often assumed to be differentiable and bounded in itself and its derivatives All can be applied to the stochastic differential equations driven by fBm (fSDE) Another approach follows Zaăhle [86] by using fractional derivatives where the non autonomous systems are treated (see [74]) Similar results are established for system in infinite dimensional case, see for instance in [66], [7] Since our target is the equations driven by fBm with H > 1/2 which can be studied by these two approaches, we aim to close the gap between the two methods and develop techniques to study more on the infinite dimensional cases ( [34]) and on the dynamic of these systems ( [23], [22]) At first we prove that, under similar assumptions to those in [74], the exis- tence and uniqueness theorem for system (1) still holds in the space of contin- uous functions with bounded p-variation norm When applying to stochastic differential equations driven by fractional Brownian motions, by considering an appropriate probability space, it is proved that the system generates a ran- dom dynamical system (in short RDS, see [16], [44], and [5]) However in the nonautonomous situation, one only expects the system to generate a stochastic twoparameter flow on the phase space These results allow us to study some qualitative problems of the systems under 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 systems and the random attractor of semilinear equations Note that these problems are still 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 dynami- cal system Its generalization, nonautonomous random attractor is introduced to stochastic flow where the state of the system depends on both the initial and present time ( [24]) We develop the semi-group technique to study the exis- tence of the random pullback attractor provided that the linear part has negative eigenvalue and the nonlinear pertubations are small In the case g is linear, the attractor is singleton and also forward attractor For the nonlinear case, under some additional conditions we point out that the attractor is one point in the sense 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 the rough 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 lin- ear systems Notice that Lyapunov spectrums and its splitting are the main con- tent of the celebrated multiplicative ergodic theorem (MET) by Oseledets [75] It was also investigated by Millionshchikov in [67–70] for linear nonautonomous differential equations In the stochastic setting, the MET is also formulated for random dynamical systems in [4, Chapter 3] Further investigations can be found in [19, 20] for stochastic flows generated by nonautonomous linear stochastic differential equations driven by standard Brownian motions To our knowledge there has not been any works on this topic for the stochastic system driven by fBm We use the approach developed in [19] to study the Lyapunov spectrum of the system We show that Lyapunov exponents can be computed based on the discretization 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 For < α ≤ denote by C α-Hol ([ a, b], R d ) the space of all Hoă lder continuous paths x : [a, b] Rd with exponential α, equipped with the norm ǁxǁα-Hol,[a,b] α-Hol,[a,b] := |xa| + |x | = |xa| + sup a≤s p, xn converges along a subsequence to some x ∈ C p−var([a, b], Rd) (ii) If (x n) is bounded and supn |x |α−Hol,[a,b] < ∞ then for each αJ < α, xn converges along a subsequence to some x ∈ Cα−Hol([a, b], Rd) Closure of smooth paths in variation norm, Hoă lder norm Define C0,p−var([a, b], Rd), C0,α−Hol([a, b], Rd) as the closure of C∞([a, b], Rd)-the space of smooth functions on [a, b], in C p−var([a, b], Rd) and Cα−Hol([a, b], Rd) respectively Thus they are Banach space and so are the spaces 0C 0,p−var ([a, b], Rd) := {x ∈ C0,p−var([a, b], Rd)| x0 = 0}, and C00,α−Hol([a, b], Rd) := {x ∈ C0,α−Hol([a, b], Rd)| x0 = 0} Moreover, these are separable spaces which can be defined as C 0,p−var ([ a, b], R d ) = ,x ∈ C p−var ([ a, b], R d ) | lim sup and ∑ |xti δ→0 Π(a,b), |Π|≤δ ti ∈Π C 0,α−Hol ([ a, b], Rd ) = ,x ∈ C α−Hol ([ a, b], R d ) | lim sup δ→0 a≤s and put Kδ : = { c ∈ A¯ α,[−n,n] ≤ ε | m[a,b](c, δ) ≥ ε} The Kδ are closed for all δ Due to the fact that lim m[a,b](c, δ) = for all c ∈ C˜ δ →0 we have δ\> proves (3.33) Kδ = ∅ Then there exists δ = δ(ε) > such that Kδ = ∅, which For the ”only if” part, assume (3.32) and (3.33) and prove the compactness of H¯ Since C˜ is a complete metric space, it suffices to prove that every sequence {cn}∞n= ⊂ H has a convergent subsequence Now following the arguments Theorem 4.9, p 63] line by line, we can construct a convergent subseof [55, quence {c˜n }n∞= by the ”diagonal sequence” such that c˜n (r ) → c(r ) as n → ∞ number r ∈ Q With (3.32) and (3.33), H satisfies the condition for any rational in [55, Theorem 4.9, p 63], hence c˜n converge uniformly to a continuous func- tion c in every [a, b] ⊂ R Fix [a, b], by (3.33) for each ε > there exist δ0 > such that if δ ≤ δ0, |c˜n (t)−c˜n sup ≤ ε for all n Hence s,t∈[a,b] |s−t|≤δ ( s) | |t−s|α sup s,t∈[a, b] |s−t|≤δ |c(t) − c(s) ≤ε |t −| s|α and then c ∈ C˜ Finally, we prove that c˜n converge to c in the Hoă lder seminorm on every compact interval [a, b] Namely, with ε, δ0 given, there exist n0 such that for all n ≥ n0, ǁc˜n − cǁ∞,[a,b] ≤ δαε We then have for n ≥ n0 ≤ sup |( c˜ |(c˜n − c)(t) − (c˜n − c) n − c)(t) − (c˜n − c ) s,t∈[a,b sup ] (s)| (s)| s,t∈[a,b |t−s|≤δ0 α ] |t − s| |t − s|α + sup s,t∈[a,b] |t−s|≥δ0 ≤ m[a,b] (c˜n |(c˜n − c)(t) − (c˜n − c) (s)| |t − s|α [a,b] (c, ) + , )+m δ0 δ0 2ǁc˜n − cǁ∞,[a,b] δ0α ≤ 4ε This implies |c˜n − c |α−Hol,[a,b] converge to as n → ∞ This complete the proof Q Tempered variables Let (Ω, F , P) be a probability space equipped with an ergodic metric dynam- ical system θ, which is a P measurable mapping θ : T × Ω → Ω, T is either R or Z, and θt+s = θt ◦ θs for all t, s ∈ T Recall that a random variable ρ : Ω → [0, ∞) is called tempered if lim t→±∞ t + log ρ(θtω) = 0, a.s which, as shown in [52, p 220], [44], is equivalent to the sub-exponential growth lim t→±∞ e−c|t|ρ(θtω) = a.s ∀c > Note that our definition of temperedness corresponds to the notion of tempered- ness from above given in [4, Definition 4.1.1(ii)] Lemma A5 (i) If h1, h2 ≥ are tempered random variables then h1 + h2 and h1h2 are tempered random variables (ii) If h1 ≥ is a tempered random variable, h2 ≥ is a measurable random variable and h2 ≤ h1 almost surely, then h2 is a tempered random variable + (iii) Let h1 be a nonnegative measurable function If log h1 ∈ L1 then h1 is tempered Proof (i) See [4, Lemma 4.1.2, p 164] (ii) Immediate from the definition of tempered random variable (iii) See [4, Proposition 4.1.3, p 165] Lemma A6 Let c : Ω → [0, ∞) be a tempered random variable, and δ > be an arbitrary fixed positive number Put d ( ω ) := ∞ ∑e −δk c(θ −kω) k=1 Then d(·) is a nonnegative almost everywhere finite and tempered random variable Proof Put dn(ω) := ∑n k= e −δkc(θ−kω) Then dn(·), n ∈ N, is an increasing sequence of nonnegative random variable, hence converges to the nonnegative random variable d(·) By 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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:... subspace of functions which vanish at in C0,α−Hol([a, b], R d) Gamma function almost sure fractional Brownian motions stochastic differential equation SDE driven by fractional Brownian motions. .. for stochastic flows generated by nonautonomous linear stochastic differential equations driven by standard Brownian motions To our knowledge there has not been any works on this topic for the stochastic