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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 HANOI - 2021 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 i 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, patience, 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 support 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 Mathematics - 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 ii Contents Table of Notation Introduction Chapter 1.1 Fractional Brownian motions 1.1.1 1.1.2 Pathwise stochastic integrals with r motions 1.2.1 1.2.2 1.2.3 1.2.4 Greedy sequences of times Stochastic flows 1.2 1.3 1.4 Chapter nian motions 2.1 2.2 Assumptions Existence and uniqueness theorem 2.2.1 2.2.2 2.2.3 Continuity and differentiability of th 2.3.1 2.3.2 The stochastic differential equation The generation of stochastic two pa Conclusions and discussions 2.3 2.4 2.5 2.6 iii Chapter Lyapunov spectrum of nonautonomous linear fSDEs 3.1 The generation of stochastic flow o 3.2 Lyapunov exponent of Young integ 3.3 Lyapunov spectrum for nonautonom 3.3.1 3.3.2 3.3.3 3.4 Almost sure Lyapunov regularity 3.5 Conclusions and discussions Chapter Random attractors for nonautonomous fSDEs 4.1 Nonautonomous attractors 4.2 Existence of random attractors 4.3 Special case: g linear 4.4 Special case: g bounded 4.5 Bebutov flow and its generation 4.6 Conclusions and discussions General Conclusions List of Author’s Related Papers References iv Table of Notations a_b Dn D[a, b] jj kxk ¥,[a,b] kxk p kxk var,[a,b] b Hol,[a,b] kxk p L (a,b) d C([a, b], R ) ¥ d C ([a, b], R ) p C -Hol var d Ca ([a, b], R ) 0,p var d C ([a, b], R ) 0, Hol d C a ([a, b], R ) C0 0,p d ([a, b], R ) var Hol C0 0,a d ([a, b], R ) d ([a, b], R ) G(z) a.s fBm SDE fSDE RDS w.r.t v Introduction A fractional Brownian motion (in short fBm) is a family of centered Gaussian H H processes B = Bt t R or R+, indexed by the Hurst parameter H (0, 1) with continuous sample paths and the covariance function RH (s, t) = 2H t 2H +s jt 2H sj It was originally defined and studied by Kolmogorov ( [57]) and then was developed by Mandelbrot and Van Ness in [65] It is a self-similar process with stationary increments and has Holderă continuous sample paths with in-dex b (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 pro-cess when H > ( [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 H However, B is not a semimartingale if H 6= , 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 vi is understood in the Young sense and coincides with that defined by fractional derivative on the space of Holderă 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 H dxt = f (t, xt)dt + g(t, xt)dBt , to take advantage of the simplicity of Young integral System (1), like Ito differential equations, is understood as an integral equation of form xt = x0 + d Z d d d m where f : [0, T] R ! R , g : [0, T] R ! R 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 Zahleă [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 existence and uniqueness theorem for system (1) still holds in the space of continuous functions with bounded p-variation norm When applying to stochastic differential equations driven by fractional Brownian motions, by considering vii an appropriate probability space, it is proved that the system generates a random dynamical system (in short RDS, see [16], [44], and [5]) However in the nonautonomous situation, one only expects the system to generate a stochastic two-parameter 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, stability, 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 dynamical 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 viii Due to [42, Corollary 5.33, p 98], for C Similarly, for all b > a p < p we have p Cb d n Other spaces of functions Define C(R R , R ) is the space of continuous d n functions on R R , valued in R Equip this space with compact open topology, i.e topo generated by metric d1 d where Kn = [ n, n] B(0, n) R R 1,0 d d m d d m Denote by C (R R , R ) the subspace of C(R R , R ) contains all functions h which is continuously differential w.r.t x and of which ¶xh continuous w.r.t (t, x) with seminorms h k k1,0;K := h k kƠ,K + h , kảx kƠ,K d where K is a compact subset in R R Then a complete metric is given by (see [4, Appendix B.2, p 552-553]) r( f , g) := ;1,0 d dm 1,0 d dm For < a < 1, consider the subspace Ca (R R , R ) C (R R , R containing functions h which is of local a Holder w.r.t t for each d d x R and moreover for each compact set K in R ) d sup jjjh( , x)jjja Hol,[a,b] < ¥, 8[a, b] R x2K a;1,0 We consider the following metric on C d2 (R d R ,R where kf jjj f d with K , K are compact sets in R, R respectively 109 d m ) which is denoted by Proposition A.7 (C a;1,0 d (R R , R d m), d2) is a complete metric space a;1,0 d d Proof That d2 is a metric on C (R R , R m) is evident due to the seminorm properties of the Holderă norm We only need to prove the n ;1,0 d d m complete-ness Let f be a Cauchy sequence in Ca (R R , R ) Since 1,0 d d (C (R R , R m), r) is complete, there exists a subsequence, which we still n 1,0 d dm use the notation f , converges to f in C (R R , R ), i.e n lim r( f , f ) = n!¥ d n We will prove that for each K , K compact sets in R, R , jjj f f jjja,K1 K2 ! d as n ! ¥ Fix K R compact, we have for each [a, b] R there exist a n constant M such that supn supx2K jjj f ( , x)jjja jf this implies that supx2K jjj f ( , x)jjja Now to complete the proof we show that f d n converges to f , in a Holderă norm on each K compact in R For each s < t [a, b], x K = which implies jjj f The proof is completed Proof.[Lemma 3.6] The if part is obvious since it can be proved that which shows the continuity of m on C continuous on a compact set, which shows (3.32) and (3.33) To be more precise, denote by C the space C ˜ compact in C , we prove (3.32) and (3.33) are fulfilled For each n N , put n 110 Then Gn is open in C H Since n0 such that n S H To prove (3.33), first note that for each c C and (see [42, Theorem 5.31,p 96]) Secondly Indeed, fix [a, b], d due to the definition of m[a,b](c, d) s0, t0 [a, b], < js0 On the other hand, m [a,b] (c , d) [a,b] m (c, d) m Exchange the role of c and c we since # is arbitrary This implies the continuity of the map In fact, fix [ If d(c, c0) < h we have kc Now, fix # > and put The are closed for all d [a,b] (c , d) Kd \ d0 K we have > proves (3.33) 111 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 c ¥ f ngn =1 H has a convergent subseque of [55, Theorem 4.9, p 63] line by line, we can construct a convergent subse¥ quence fc˜ng n=1 by the ”diagonal sequence” such that c˜n(r) ! c(r) as n ! ¥ for any rational number r Q With (3.32) and (3.33), H satisfies the condition in [55, Theorem 4.9, p 63], hence c˜n converge uniformly to a continuous function c in every [a, b] R Fix [a, b], by (3.33) for each # > there exist d0 > such that if d d0, s t j j js tj d ˜ Finally, we prove that ˜n converge to in the Holderă seminorm c c C c on every compact interval [a, b] that for all n n0, kc˜n s,t c)(t) j(c˜n sup jt [a,b] 4# This implies jjjc˜n cjjja Hol,[a,b] converge to as n ! ¥ This complete the proof Tempered variables Let (W, F, P) be a probability space equipped with an ergodic metric dynam-ical system q, which is a P measurable mapping q : T W ! W, T is either R or Z, and qt+s = qt qs for all t, s T Recall that a random variable r : W ! [0, ¥) is called tempered if lim t! ¥ t + log r(qtw) = 0, a.s 112 which, as shown in [52, p 220], [44], is equivalent to the sub-exponential growth cjtj r(qtw) = lim e t! a.s 8c > ¥ 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 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 (ii) + (iii) Let h1 be a nonnegative measurable function If log h1 L 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 : W ! 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sharp by giving an example on autonomous g deterministic equation with g C , < g < p having more than... 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, patience, motivation, and immense knowledge
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