Rough Path Theory to approximate Random Dynamical Systems
Hongjun Gao, María J. Garrido-Atienza, Anhui Gu, Kening Lu, Björn Schmalfuss
aa r X i v : . [ m a t h . P R ] F e b ROUGH PATH THEORY TO APPROXIMATE RANDOMDYNAMICAL SYSTEMS
H. GAO, M.J.GARRIDO–ATIENZA, A. GU, K. LU, AND B. SCHMALFUSS
Abstract.
We consider the rough differential equation dY “ f p Y q d ω where ω “ p ω, ω q is a rough path defined by a Brownian motion ω on R m . Underthe usual regularity assumption on f , namely f P C b p R d , R d ˆ m q , the roughdifferential equation has a unique solution that defines a random dynamicalsystem ϕ . On the other hand, we also consider an ordinary random differen-tial equation dY δ “ f p Y δ q dω δ , where ω δ is a random process with stationaryincrements and continuously differentiable paths that approximates ω . Thelatter differential equation generates a random dynamical system ϕ δ as well.We show the convergence of the random dynamical system ϕ δ to ϕ for δ Ñ Introduction
The theory of random dynamical systems (RDS) allows to study the qualitativelongtime behavior of differential equations containing random terms. For a com-prehensive overview of this theory we refer to Arnold [1]. There are at least twoclasses of these differential equations with random terms. One class is given byordinary differential equations satisfying (roughly speaking) existence and unique-ness conditions with noise dependent coefficients (the meaning of noise is given inDefinition 2 below). The second class are stochastic differential equations. Thesolutions of these equations in general contain a stochastic integral in the senseof Ito or Stratonovich, defined as a limit in probability of special Riemann sumsrelated to the integrand and the increments of the noise, see for a precise definitionKunita [17]. Remarkably, these integrals are defined only almost surely. This leadsto the fact that the solutions are defined almost surely where exceptional sets maydepend on the initial condition, causing that the solutions of these equations do notgenerate an RDS, see Definition 4. To derive from an Ito (Stratonovich) equationan RDS some effort is necessary. In particular one has to combine the fact thata stochastic differential equation generates a stochastic flow, see Kunita [17], tofurther apply a perfection technique that gives a version of the solution randomfield which is an RDS, see Kager and Scheutzow [16].A way to avoid exceptional sets is to consider a pathwise definition of the stochas-tic integral based on fractional derivatives. For H¨older continuous driven processeswith H¨older exponent bigger than 1 { et al. [4], Gao et al. [7], Garrido-Atienza etal. [9], [10] and [8]. For less regular processes, much less is known. Nevertheless, there are already some results establishing that the solution to evolution equationsdriven by H¨older continuous functions with H¨older exponent in p { , { s generatesan RDS, see Garrido-Atienza et al. [11]. The main tool used in that paper is acompensation of fractional derivatives. Note that the case of a Brownian motion isincluded in these considerations. An alternative method to obtain from a stochas-tic differential equation an RDS is to interpret such an equation in the rough pathsense. Indeed the stochastic integral related to this theory is pathwise as well, sothat one can avoid exceptional sets depending on the initial states. The cocycleproperty then follows easily from the additivity and a shift property of the roughpath integral. In this context we would like to mention the recent papers Bailleul et al. [2] and Hesse and Neamtu [15].The main goal of this article is to approximate the RDS generated by a roughequation by RDS generated by random ordinary differential equations. In order todo that, we shall have to calculate the difference of solutions of two rough differentialequations driven by different noises, for which it will be necessary to estimate thedistance of the two noises in a particular metric, that in fact measures the distance ofthe two noise paths via the H¨older norm but also the distance of the correspondingL´evy areas (that are second order processes) of the lifted noises in H¨older norm.Then in a first step we have to show that these L´evy areas are well-defined. Tobe more precise, for a path ω of the Brownian motion we define the stationaryGaussian noise ω δ such that ω δ p r q : “ δ θ r ω p δ q such that δ ą θ r is defined by(2) below, and consider the Gaussian process X δ : “ ω δ ´ ω . We then show thatthis random process has a particular increment covariance so that the stochasticintegration theory by Friz and Hairer [5] or by Friz and Victoir [6] can be applied.As a result, we obtain that the moments of the L´evy areas of X δ converge to zerowhen δ goes to zero. This then gives us the convergence of ω δ to ω and theircorresponding L´evy areas as well. Hence the solution of the differential equationdriven by ω δ converges to the solution of the rough equation driven by a path of theBrownian motion ω and the same is true for the RDS generated by these equations.This kind of approximation by ω δ has been used for stochastic ordinary and partialdifferential equations when the noise is very simple, see Gu et al. [13] and [14]. Inthese papers the authors prove the convergence of the attractors of the RDS drivenby ω δ to the attractor of the RDS of the original equation. Due to the fact that thenoise in these papers is trivial (additive noise or linear multiplicative noise), theauthors apply a transformation technique rewriting the original stochastic equationinto a random equation. In particular, with this strategy the convergence of theL´evy area is not necessary. With the techniques developed in the current paper,in forthcoming investigations we will try to obtain similar results regarding conver-gence of random attractors, but without transformations of the stochastic systeminto a random one and for more complicated nontrivial diffusion coefficients.The article is organized as follows: In Section 2 we introduce the definitions ofrough path, metric dynamical system and RDS. In Section 3 we define stochasticintegrals based on the increment covariance and formulate the Kolmogorov criterionfor rough paths. In Section 4 we apply these results to obtain that the momentsof the L´evy area of X δ converge to zero to further establish that the area relatedto ω δ converges to the area related to ω when δ Ñ
0. This interesting fact willbe exploited to prove the convergence of the RDS generated by the solution of theequation driven by ω δ to the RDS generated by the solution of the equation driven OUGH PATH THEORY TO APPROXIMATE RANDOM DYNAMICAL SYSTEMS 3 by ω , in the case that ω is the Brownian motion. In the Appendix we calculate thecovariance of X δ and ω δ when ω is a fractional Brownian motion with any Hurstparameter H P p , q .2. Preliminaries on Rough Path Theory and Random DynamicalSystems
For a given T ă T P R , denote by ∆ r T , T s the simplex ∆ r T , T s “ tp s, t q : T ď s ď t ď T u . Let X be a mapping from r T , T s to R m and X be a mapping from∆ r T , T s to R m ˆ m . These mappings are called H¨older continuous if the seminormsgiven by ||| X ||| β : “ sup p s,t qP ∆ r T ,T s ,s ‰ t | X p t q ´ X p s q|p t ´ s q β , ||| X ||| β : “ sup p s,t qP ∆ r T ,T s ,s ‰ t | X p s, t q|p t ´ s q β are finite. In that case, we write X P C β pr T , T s , R m q and X P C β p ∆ r T , T s , R m ˆ m q . Definition 1. A β –rough path on an interval r T , T s with values in R m consists ofa function X P C β pr T , T s , R m q , β P p { , { q , as well as a second order process X P C β p ∆ r T , T s , R m ˆ m q , subject to the Chen’s algebraic relation (1) X p s, t q ´ X p s, u q ´ X p u, t q “ p X p u q ´ X p s qq b p X p t q ´ X p u qq for T ď s ď u ď t ď T . X is usually referred as the path component whereas X is the (L´evy) area component .Note that when X is smooth, we can define X by X p s, t q “ ż ts p X p r q ´ X p s qq b X p r q dr. However, in the rough case we need to find sufficient conditions under which X canbe enhanced with a well-defined area component X , see Section 3. This property isalso known as X can be lifted to a rough path X “ p X, X q and X is the lift of theprocess X .For β P p { , { q we denote by C β pr T , T s , R m q the space of H¨older rough paths X : “ p X, X q .If, in addition, the symmetric part of X is given bySym p X p s, t qq “ p X p t q ´ X p s qq b p X p t q ´ X p s qq then we will say that X is a geometric H¨older rough path and denote the corres-ponding space by C βg pr T , T s , R m q .It is interesting to point out that the space of geometric rough paths can be definedas the closure of smooth paths, enhanced with their iterated Riemann integrals,with respect to the metric ρ β defined below, in C β pr T , T s , R m q , see [5], Exercise2.12.Moreover, given two rough paths X , Y P C β pr T , T s , R m q we define the H¨olderrough path metric by ρ β,T ,T p X , Y q : “ sup p s,t qP ∆ r T ,T s ,s ‰ t ˆ | X p t q ´ X p s q ´ Y p t q ` Y p s q|p t ´ s q β ` | X p s, t q ´ Y p s, t q|p t ´ s q β ˙ H. GAO, M.J.GARRIDO–ATIENZA, A. GU, K. LU, AND B. SCHMALFUSS and the β -H¨older homogeneous rough path norm of X by ||| X ||| β : “ ||| X ||| β ` b ||| X ||| β . In what follows we introduce the concept of an RDS. We start with the definitionof a general noise.
Definition 2.
Let p Ω , F , P q be a probability space. A metric dynamical system on p Ω , F , P q is given by a family of measurable transformations θ : p R ˆ Ω , B p R q b F q Ñ p Ω , F q such that (1) For t , t P R , θ t ` t “ θ t ˝ θ t “ θ t ˝ θ t . (2) The measure P is invariant with respect to θ . (3) P is ergodic with respect to θ t , that is, for any θ t -invariant set A P F , either P p A q “ or P p A q “ . Example 3.
A fractional Brownian motion B H in R m with Hurst parameter H Pp , q is a centered and continuous Gaussian process with covariance R B H p s, t q “ p| t | H ` | s | H ´ | t ´ s | H q id t, s P R , where id is the identical matrix in R m ˆ m , that is, we assume that B H has indepen-dent components. No matter the value of the Hurst parameter, we can consider acanonical version of this random process given by p C p R , R m q , B p C p R , R m qq , P H , θ q , where C p R , R m q is the topological space of continuous functions which are zero atzero equipped with the compact open topology. This topology is metrizable giving usa Polish space. B p C p R , R m qq denotes the Borel σ -algebra of C p R , R m q and theprobability P H is the Gaussian measure of B H generated by the covariance R B H .For θ we consider the so-called Wiener shift defined by (2) θ t ω p¨q “ ω p¨ ` t q ´ ω p t q , for t P R . Then p C p R , R m q , B p C p R , R m qq , P H , θ q is a metric dynamical system, see [12] . Inaddition there exists a θ -invariant set Ω of full P H –measure such that ω P Ω is β –H¨older continuous for every β ă H on any interval r´ T, T s . On Ω we choose F to be the trace-algebra of B p C p R , R m qq with respect to Ω , and for the restriction of P H to this new σ -algebra we use again the same symbol P H . In the following wewill work on p Ω , F , P H , θ q , that is also a metric dynamical system, see [3] .Finally, note that when H “ { the fractional Brownian motion reduces to theBrownian motion B { . Now we introduce the definition of a random dynamical system (RDS).
Definition 4.
Let p Ω , F , P , θ q be a metric dynamical system. An RDS over p Ω , F , P , θ q is given by a measurable mapping ϕ : R ` ˆ Ω ˆ R m ÞÑ R m such that (1) ϕ p , ω, ¨q “ id , for all ω P Ω . (2) The cocycle property holds true, that is, ϕ p t ` t , ω, ¨q “ ϕ p t , θ t ω, ϕ p t , ω, ¨qq for all ω P Ω and t , t P R ` . OUGH PATH THEORY TO APPROXIMATE RANDOM DYNAMICAL SYSTEMS 5
Later we will deal with rough differential equations, that is, differential equationsthat are driven by rough (non-differentiable) paths. We will use the rough paththeory to give a meaning to these equations. The stochastic integral will be givenin terms of compensated Riemann sums, thus the pathwise solution will enjoy theproperty that it generates an RDS, provided that the considered noise forms ametric dynamical system. This will be the case since in this paper we restrict ourconsiderations to Brownian motion, whose canonical interpretation forms a metricdynamical system as we have already stated in Example 3. The theory of RDS willbe further applied in forthcoming works to study longtime behavior properties ofsolutions of rough differential equations.We would like to refer to the papers [2] and [15], where it is also investigated thegeneration of RDS by the solutions of rough equations.3.
General results on the lift of Gaussian processes
In this section, we would like to remind several results (that the reader can findin the monograph [5]) which shall help in order to construct the lift of a Gaussianprocess.Let X be a one dimensional process on R and suppose that X p t q P L . Then R X p s, t q : “ E X p s q X p t q is the covariance of X . We define for s ă t , s ă t the incremental covariance by R ˆ s ts t ˙ : “ E p X p t q ´ X p s qqp X p t q ´ X p s qq and the ρ -variation over an rectangle r s, t s by } R X } ρ, r s,t s : “ ˆ sup P p s, t q , P p s, t q ÿ r σ, τ s P P , r σ , τ s P P ˇˇˇˇ R ˆ σ τσ τ ˙ ˇˇˇˇ ρ ˙ { ρ for 1 ď ρ ă P p s, t q denotes as usual a partition of the considered interval r s, t s and | P p s, t q| denotes the corresponding maximal mesh length.The finiteness of the ρ -variation over rectangles allows us to define a stochasticintegral for Gaussian processes, see Friz and Hairer [5], Proposition 10.3. Theorem 5.
Let
X, Y be two continuous centered independent Gaussian processessuch that for some ď ρ ă } R X } ρ, r s,t s , } R Y } ρ, r s,t s ă 8 . Then for every ď s ď t the stochastic integral ż ts p Y p r q ´ Y p s qq b dX p r q is defined as an L -limit of the Stieltjes integrals ş P p Y p r q ´ Y p s qq b dX p r q when | P p s, t q| Ñ . Moreover, E ˆ ż ts p Y p r q ´ Y p s qq b dX p r q ˙ ď C } R X } ρ, r s,t s } R Y } ρ, r s,t s holds, where the constant C depends on ρ . H. GAO, M.J.GARRIDO–ATIENZA, A. GU, K. LU, AND B. SCHMALFUSS
Theorem 6.
Let X “ p X , ¨ ¨ ¨ , X m q be a continuous centered Gaussian process in R m with independent components, for which there exist ρ P r , q and M ą suchthat for every i P t , ¨ ¨ ¨ , m u and all ď s ď t ď T } R X i } ρ, r s,t s ď M | t ´ s | { ρ . Define, for ď i ă j ď m and ď s ă t ď T , in the L -sense, the process X i,j p s, t q “ ż ts p X i p r q ´ X i p s qq dX j p r q and X i,i p s, t q “ p X i p t q ´ X i p s qq , X j,i p s, t q “ ´ X i,j p s, t q ` p X i p t q ´ X i p s qqp X j p t q ´ X j p s qq . Then there exists a version of X and X so that for any β P p , ρ q we have a roughpath X “ p X, X q P C βg pr , T s , R m q almost surely. For the proof of this result see Theorem 10.4 in [5].The following result, that is an adaptation of Theorem 3.3 in [5], will allow us tomeasure the distance between two lifts. This theorem is a version of the well-knownKolmogorov test.
Theorem 7.
Let q ě , { ρ ´ { q ą { and X “ p X, X q and Y “ p Y, Y q be tworough paths in R m such that for all ´ T ď s ă t ď T | X p t q ´ X p s q| L q ď C | t ´ s | {p ρ q , | X p s, t q| L q ď C | t ´ s | { ρ , | Y p t q ´ Y p s q| L q ď C | t ´ s | {p ρ q , | Y p s, t q| L q ď C | t ´ s | { ρ , for some constant C ą . Assume that for some ε ą and all ´ T ď s ă t ď T | X p t q ´ X p s q ´ p Y p t q ´ Y p s qq| L q ď εC | t ´ s | {p ρ q , | X p s, t q ´ Y p s, t q| L q ď εC | t ´ s | { ρ . Let β P p , ρ ´ q q . Then ||| X ||| β , ||| Y ||| β P L q and | ρ β, ,T p X , Y q| L q ď εM where M ą is independent on ε and depends increasingly on C . We would like to stress that Theorem 3.3 works for rough paths p X, X q defined on r´ T, T s ˆ ∆ r´ T, T s and not only for rough paths defined on r , T s ˆ ∆ r´ , T s asproved in [5].Further we will consider centered, continuous Gaussian processes with independentcomponents X “ p X , ¨ ¨ ¨ , X m q and stationary increments. As we have seen above,for the construction of a lift associated to X it suffices to estimate the ρ -variationof the rectangular incremental covariance R X i , for all i P t , ¨ ¨ ¨ , m u , see Theorem6. To this end, it is enough to focus on one component, hence, in the next result X denotes a one-dimensional path, whose law is determined by σ p τ q “ E p X p t ` τ q ´ X p t qq (3)for t P R , τ ě OUGH PATH THEORY TO APPROXIMATE RANDOM DYNAMICAL SYSTEMS 7
Theorem 8.
Let X be a real-valued continuous centered Gaussian process with sta-tionary increments. Assume that σ p¨q given by (3) is concave and non–decreasingon r , h s , for some h ą . Assume also (4) σ p t q ď Lt { ρ for L ą , ρ ě and t P r , h s . Then the incremental covariance R X of X satisfies } R X } ρ, r s,t s ď LM p t ´ s q { ρ for all intervals r s, t s with length smaller than h and M “ M p ρ, h q ą .If now X “ p X , ¨ ¨ ¨ , X m q is a centered continuous Gaussian process with indepen-dent components such that each X i satisfies the above assumptions with commonvalues of h, L and ρ P r , q , then X can be lifted to X P C βg pr , T s , R m q . For the proof of this result we refer to Theorem 10.9 and Corollary 10.10 in [5].Note that revisiting the proof of Theorem 10.9 in [5], on account of (4) it turns outthat for an interval r s, t s of length smaller than h , for Π “ t t i u and Π “ t t j u twopartitions of r s, t s , if we fixed i then ÿ t j P Π ˇˇˇ E p X p t i ` q ´ X p t i qqp X p t j ` q ´ X p t j qq ˇˇˇ ρ ď ´ ρ ` σ p t i ` ´ t i q ˘ ρ ` L ρ ´ ρ | t i ` ´ t i | ď ´ ` ρ ` ´ ρ ¯ L ρ | t i ` ´ t i | , hence summing over t i and taking the supremum we obtain } R X } ρ, r s,t s ď L ´ ` ρ ` ´ ρ ¯ { ρ p t ´ s q { ρ “ : LM p t ´ s q { ρ that is, the value of the constant M above is given by(5) M : “ ´ ` ρ ` ´ ρ ¯ { ρ . Approximation of the Brownian motion by a stationary process
Let us consider the metric dynamical system p Ω , F , P , θ q introduced in Example3 in Section 2 and denote by ω : “ B { p ω q P Ω the canonical Brownian motion on R with values in R m .First of all, in this section we want to enhance ω to obtain the lift ω : “ p ω, ω q .Note that any of the components ω i of ω is such that σ ω i p u q “ u, for u ě
0, that obviously is non–decreasing and concave on any time interval r , T s .Since ρ P p , s we can estimate σ ω i p u q “ u ď T ´ { ρ u { ρ , hence, applying Theorem 8 we obtain } R ω i } ρ, r s,t s ď T ´ { ρ M | t ´ s | { ρ (6)with M given by (5). For ω i,j p s, t q we choose the interpretation of the integral in[5], Chapter 10, namely ω i,j p s, t q : “ ż ts p ω i p r q ´ ω i p s qq dω j p r q , s ď t P R ` a.s.(7) H. GAO, M.J.GARRIDO–ATIENZA, A. GU, K. LU, AND B. SCHMALFUSS
Despite the fact that the results of Section 3 are formulated in R ` , for our purposeswe need all the lifts to be defined on R . Therefore, we extend the rough path ω to r´ T, T s so that Chen’s equality holds, that is, for all s ă ă t ω i,j p s, t q : “ ω i,j p s, q ` ω i,j p , t q ´ ω j p t q ω i p s q , (8)which means that we can restrict to check whether ω i,j p s, q is well-defined for s ă
0. Taking into account that ż P p s, q p ω p r q ´ ω p s qq b dω p r q “ ż P p , ´ s q θ s ω p r q b dθ s ω p r q (9)and that θ s ω is a Brownian motion with the same incremental covariance as ω , thenthe limit when | P p s, q| Ñ L sense, which in turn impliesthat ω p s, t q exists for every s ă t P R due to Theorem 5.Furthermore, by Theorem 6 we can find a rough path ω : “ p ω, ω q P C βg pr´ T, T s , R m q for any T ą
0. Defining on a set of measure zero ω p s, t q “ ω P C βg pr´ T, T s , R m q for all ω P Ω.Moreover, in virtue of Theorem 6 we can also define a version(10) θ τ ω : “ p θ τ ω, θ τ ω q P C βg pr´ T, T s , R m q for all T ą , a.s.We emphasize that θ τ ω represents the area of the path θ τ ω .Now for δ P p , s we define the approximations(11) ω δ p t q : “ δ ż t θ r ω p δ q dr “ ż t ω p δ ` r q ´ ω p r q δ dr, t P R . We can interpret p t, ω q ÞÑ ω δ p t q as a random process on p Ω , F , P , θ q . Also, it is readily seen that R ˆ Ω Q p t, ω q ÞÑ ω δ p t q “ δ θ t ω p δ q is well-defined, hence ω δ can be enhanced by defining the Riemann integral(12) ω δ p s, t q : “ ż ts p ω δ p r q ´ ω δ p s qq b ω δ p r q dr, s ď t P R . Then pp s, t q , ω q Ñ ω δ p s, t q for s ď t P R is a random field on p Ω , F , P , θ q . Moreover, it is straightforward toprove the corresponding Chen’s relation (1), henceforth the lift ω δ : “ p ω δ , ω δ q is arough path.We also define θ τ ω δ p s, t q : “ ż ts p θ τ ω δ p r q ´ θ τ ω δ p s qq b p θ τ ω δ q p r q dr, τ P R , s ď t P R , ω P Ω . Then it is easy to see θ τ ω δ p s, t q “ ω δ p s ` τ, t ` τ q . (13)Now we define(14) θ τ ω δ : “ p θ τ ω δ , θ τ ω δ q , τ P R . OUGH PATH THEORY TO APPROXIMATE RANDOM DYNAMICAL SYSTEMS 9
It is an immediate exercise to check that θ τ ω δ is a rough path on r´ T, T sˆ ∆ r´ T, T s ,for any T ą δ P p , s we define the random process R ˆ Ω Q p t, ω q ÞÑ X δ p t, ω q : “ ω δ p t q ´ ω p t q P R m . Lemma 9. X δ and ω δ are centered Gaussian processes on R with stationary in-crements.Proof. Note that the integrand defining ω δ is continuous on any r´ T, T s for anyfixed δ and any T ą
0. Also the second moments of the integrand are uniformlybounded in r´ T, T s . Then ω δ (and therefore X δ as well) is a Gaussian process, see[18], Pages 91 and 297.Next, we prove that X δ has stationary increments. A simply computation showsthat, for s, t P R , X δ p t ` s, ω q ´ X δ p t, ω q “ δ ż t ` st θ r ω p δ q dr ´ p ω p t ` s q ´ ω p t qq“ δ ż s θ r θ t ω p δ q dr ´ θ t ω p s q“ X δ p s, θ t ω q . This means that the increment X δ p t ` s, ω q ´ X δ p t, ω q has the same distribution as X δ p s, ω q due to the fact that θ t is measure-preserving. The same holds for ω δ . (cid:3) Our next goal is to check that X δ can be enhanced, in such a way that we obtaina rough path p X δ , X δ q on r´ T, T s ˆ ∆ r´ T, T s , for any T ą Theorem 10.
The variance of any of the components X iδ of X δ satisfies σ X iδ p u q ď δ ´ { ρ u { ρ , for u ě , δ P p , s , ρ P r , q . As a result, } R X iδ } ρ, r s,t s ď δ ´ { ρ M | t ´ s | { ρ with M defined by (5).Proof. We know from Corollary 24 in the Appendix, that the variance σ X iδ of eachcomponent X iδ is given by(15) σ X iδ p u q “ " u ´ u δ , ď u ă δ, δ, u ě δ, for 1 ď i ď m . Obviously, σ X iδ p u q is a continuous function and it is non-decreasing.Moreover, σ X iδ p¨q is concave and from (15) it is easily seen that σ X iδ p u q ď Lu { ρ , with L : “ L p δ, ρ q “ δ ´ { ρ for u ě
0. Furthermore, as a consequence of Theorem 8 and the estimate (5), wecan derive the following explicit estimate for the ρ -variation norm of R X iδ :(16) } R X iδ } ρ, r s,t s ď δ ´ { ρ ˆ ρ ` ` ´ ρ ˙ { ρ | t ´ s | { ρ . (cid:3) Corollary 11.
For δ P p , s the area X δ p s, t q : “ ż ts p X δ p r q ´ X δ p s qq b dX δ p r q is well-defined for all ď s ď t . In particular we have | X i,jδ p s, t q| L ď Cδ p ´ { ρ q M | t ´ s | { ρ where M has been defined in Theorem 10, and C only depends on ρ .Proof. For i ‰ j the above statement follows directly from Theorem 5 and (16).On the other hand, for i “ j note that, for all 0 ď s ď t , | X i,iδ p s, t q| L ď E | X iδ p t q ´ X iδ p s q| ď p E | X iδ p t q ´ X iδ p s q| q ď δ p ´ { ρ q M | t ´ s | { ρ . (cid:3) Remark 12.
In fact, we can consider X i,jδ p s, t q for all s ă t P R . For s ď ď t we define for ď i, j ď m (17) X i,jδ p s, t q : “ X i,jδ p s, q ` X i,jδ p , t q ´ X jδ p t q X iδ p s q , hence we only need to give meaning to X i,jδ p s, q . From Lemma 9 we know that X δ has stationary increments, therefore the incremental covariance of X δ and θ s X δ arethe same. We then could follow exactly the same steps as we did to define ω p s, q for s ă .In addition, in virtue of Corollary 11 we have that, for any ď i, j ď m and s ď ď t , | X i,jδ p s, t q| L ď ´ | X i,jδ p s, q| L ` | X i,jδ p , t q| L ` ` E | X iδ p s q| E | X jδ p t q| ˘ { ¯ ď Cδ p ´ { ρ q M pp´ s q { ρ ` t { ρ q ď Cδ p ´ { ρ q M p t ´ s q { ρ ă 8 , where in the penultimate inequality we have used the fact that the mapping p a, b q ÞÑ| b ´ a | θ for θ ě is a control function, see [6] , Page 22. Note that the last term ofthe right hand side above can be estimate in this way because ` E | X iδ p s q| E | X jδ p t q| ˘ { ď Cδ ´ { ρ p´ s q { ρ t { ρ ď Cδ ´ { ρ pp´ s q { ρ ` t { ρ q . Note that for s ă t ď , by shifting the time we have | X i,jδ p s, t q| L “ | X i,jδ p , t ´ s q| L which can be considered similarly to the case in Corollary 11. Theorem 13.
Let δ P p , s , ρ ą and q ą so that { ρ ´ { q ą { . Then forevery β P p , ρ ´ q q , we have ||| ω ||| β , ||| ω δ ||| β P L q . Furthermore, there exists apositive constant C q,ρ,T that may depend on T , q and ρ such that | ρ β,s,t p ω δ , ω q| L q ď C q,ρ,T δ { p ´ { ρ q . Therefore, lim δ Ñ E ` ρ β,s,t p ω δ , ω qq q “ , for any r s, t s Ă r´ T, T s . OUGH PATH THEORY TO APPROXIMATE RANDOM DYNAMICAL SYSTEMS 11
Proof.
Assume first that 0 ď s ď t . To prove this result we are going to applythe version of Kolmogorov’s theorem, see Theorem 7, hence we have to check thecorresponding moment conditions for ω δ , ω and its difference (with a uniformconstant that does not depend on δ ).Throughout the proof C is a constant that may depend on ρ and/or T but not on δ , and may change from line to line. We will emphasize the dependence of thesevalues when we think it is suitable to do it.Notice that for i ‰ j we can consider the following splitting X i,jδ p s, t q “ ż ts rp ω iδ ´ ω i qp r q ´ p ω iδ ´ ω i qp s qs d p ω jδ ´ ω j qp r q“ ż ts r ω iδ p r q ´ ω iδ p s qs dω jδ p r q ´ ż ts r ω i p r q ´ ω i p s qs dω jδ p r q´ ż ts rp ω iδ ´ ω i qp r q ´ p ω iδ ´ ω i qp s qs dω j p r q“ : ω i,jδ p s, t q ´ I ,i,jδ p s, t q ´ I ,i,jδ p s, t q . The integral I ,i,jδ can be rewritten as I ,i,jδ “ ż ts r ω i p r q ´ ω i p s qs dω j p r q ` ż ts r ω i p r q ´ ω i p s qs d p ω jδ ´ ω j qp r q“ : ω i,j p s, t q ` I ,i,jδ p s, t q . Hence, from the previous considerations we obtain that ω i,jδ p s, t q ´ ω i,j p s, t q “ X i,jδ p s, t q ` I ,i,jδ p s, t q ` I ,i,jδ p s, t q . (18)According to Theorem 5 we can estimate the second moments of I ,i,jδ and I ,i,jδ bythe respective variation seminorms of the incremental covariances. In fact, E | I ,i,jδ p s, t q| “ E ˆˇˇˇˇ ż ts r X iδ p r q ´ X iδ p s qs dω j p r q ˇˇˇˇ ˙ ď C } R X iδ } ρ, r s,t s } R ω j } ρ, r s,t s (19)and E | I ,i,jδ p s, t q| “ E ˆˇˇˇˇ ż ts r ω i p r q ´ ω i p s qs dX jδ p r q ˇˇˇˇ ˙ ď C } R ω i } ρ, r s,t s } R X jδ } ρ, r s,t s . (20)Note that (16) and (6) together with (19) and (20) imply that E | I k,i,jδ p s, t q| ď Cδ ´ { ρ T ´ { ρ M | t ´ s | { ρ , for k “ ,
3. Going back to (18), taking into account Corollary 11, we obtain that E | ω i,jδ p s, t q ´ ω i,j p s, t q| ď E | X i,jδ p s, t q| ` E | I ,i,jδ p s, t q| ` E | I ,i,jδ p s, t q| ď CM δ ´ { ρ p δ ´ { ρ ` T ´ { ρ q| t ´ s | { ρ ď CM δ ´ { ρ p ` T ´ { ρ q| t ´ s | { ρ . (21)In fact (21) can be rewritten as | ω i,jδ p s, t q ´ ω i,j p s, t q| L ď Cε | t ´ s | { ρ , (22) where ε : “ δ { p {´ { ρ q and C above is a constant that depends on T and ρ , but noton δ .Moreover, for i ‰ j , on account of Theorem 5 and (6), we obtain E | ω i,j p s, t q| ď CT p ´ { ρ q M | t ´ s | { ρ . As a result of the previous inequality and (21) we also obtain the correspondingestimate for ω δ , namely E | ω i,jδ p s, t q| ď E | ω i,jδ p s, t q ´ ω i,j p s, t q| ` E | ω i,j p s, t q| ď CM ´ δ p ´ { ρ q ` δ ´ { ρ T ´ { ρ ` T p ´ { ρ q ¯ | t ´ s | { ρ , and therefore | ω i,j p s, t q| L ď C | t ´ s | { ρ , | ω i,jδ p s, t q| L ď C | t ´ s | { ρ . (23)On the other hand, since ω i p t q ´ ω i p s q and ω iδ p t q ´ ω iδ p s q are Gaussian variables,see Lemma 9, taking into account that ω iδ p t q “ X iδ p t q ` ω i p t q we can estimate E | ω iδ p t q ´ ω iδ p s q ` ω i p t q ´ ω i p s q| ď E | ω iδ p t q ´ ω iδ p s q| ` E | ω i p t q ´ ω i p s q| ď p E | ω iδ p t q ´ ω iδ p s q| q ` p E | ω i p t q ´ ω i p s q| q ď p E | X iδ p t q ´ X iδ p s q| ` E | ω i p t q ´ ω i p s q| q ` p E | ω i p t q ´ ω i p s q| q ď C | t ´ s | { ρ , where C is uniform with respect to δ P p , s , and depends on T and ρ . Note nowthat, by a direct computation, when i “ j we obtain E p ω i,iδ p s, t q ´ ω i,i p s, t qq “ E ˆ p ω iδ p t q ´ ω iδ p s qq ´ p ω i p t q ´ ω i p s qq ˙ “ E ˆ p ω iδ p t q ´ ω iδ p s q ` ω i p t q ´ ω i p s qqp X iδ p t q ´ X iδ p s qq ˙ ď ˆ E |p ω iδ p t q ´ ω iδ p s q ` ω i p t q ´ ω i p s q| E |p X iδ p t q ´ X iδ p s q| ˙ { and therefore | ω i,iδ p s, t q ´ ω i,i p s, t q| L ď Cε | t ´ s | { ρ , (24)where we remind that ε “ δ { p {´ { ρ q . According to (6) we obtain | ω i,i p s, t q| L ď C | t ´ s | { ρ . (25)Again, by using triangle inequality | ω i,iδ p s, t q| L ď C | t ´ s | { ρ (26)and therefore, taking into account (22)-(26), all the moment conditions for the areasare satisfied in the particular case that q “ OUGH PATH THEORY TO APPROXIMATE RANDOM DYNAMICAL SYSTEMS 13
Now by the equivalence of L q - and L -norms on Wiener–Ito chaos, see [6], AppendixD, for every q ą | ω p s, t q| L q ď C q,ρ,T | t ´ s | { ρ , | ω δ p s, t q| L q ď C q,ρ,T | t ´ s | { ρ , | ω δ p s, t q ´ ω p s, t q| L q ď εC q,ρ,T | t ´ s | { ρ . On the other hand, by Theorem 10 | ω δ p t q ´ ω δ p s q ´ p ω p t q ´ ω p s qq| L q “ | X δ p t q ´ X δ p s q| L q ď εC q,ρ,T | t ´ s | {p ρ q , and by (6) | ω p t q ´ ω p s q| L q ď C q,ρ,T | t ´ s | {p ρ q . These last two inequalities trivially imply that | ω δ p t q ´ ω δ p s q| L q ď C q,ρ,T | t ´ s | {p ρ q , see also Lemma 22 in the Appendix.Now for every β P p { , { q we can find ρ ą q ą { ρ ´ { q ą { ||| ω ||| β , ||| ω δ ||| β P L q and | ρ β,s,t p ω δ , ω q| L q ď C q,ρ,T δ { p {´ { ρ q . (27)Hence, lim δ Ñ | ρ β,s,t p ω δ , ω q| L q “ . So far, we have considered only the case 0 ď s ď t , but the other cases are qual-itatively the same. In fact, if s ď ď t by (8) and (17) we should only estimatethe second moments of I ,i,jδ and I ,i,jδ defined above. For instance, for I ,i,jδ weobserve I ,i,jδ p s, t q “ ż s r X iδ p r q ´ X iδ p s qs dω j p r q ` ż t X iδ p r q dω j p r q ´ X iδ p s q ω j p t q , that is, I ,i,jδ p s, t q “ I ,i,jδ p s, q ` I ,i,jδ p , t q ´ X iδ p s q ω j p t q , and the second moments of this splitting can be estimated using Theorem 5 andthe fact that the increments of X δ and ω are stationary.The case s ă t ď | X i,jδ p s, t q| L “ | X i,jδ p , t ´ s q| L , therefore we omit it here. (cid:3) Theorem 14.
Consider a sequence p δ i q i P N that converges sufficiently fast to zerowhen i Ñ 8 . Then we have the following convergences lim i Ñ8 ω δ i “ ω in C β pr´ T, T s , R m q , lim i Ñ8 ω δ i “ ω in C β p ∆ r´ T, T s , R m ˆ m q (28) for every T ą , for all ω P Ω and β ă { .Proof. The assumption that the sequence p δ i q i P N converges sufficiently fast to zerohas the meaning that we can apply the Borel-Cantelli lemma, therefore we willrestrict here to consider δ i “ ´ i . In addition, it is sufficient to fix an arbitrary T “ n P N . Taking into account (27), for every β P p { , { q we can find ρ ą q ą { ρ ´ { q ą { C n such that E ` ρ β, ´ n,n p ω δ i , ω q ˘ q ď C q,ρ,n δ q p ´ { ρ q i “ C q,ρ,n iq p { ρ ´ q Ñ , (29)as i Ñ 8 . Now, by the Chebyshev inequality and (29), we have that P ` ω : ρ β, ´ n,n p ω δ i , ω q ą i { p { ρ ´ q ˘ ď iq { p { ρ ´ q E ` ρ β, ´ n,n p ω δ i ´ ω q ˘ q ď C q,ρ,n iq { p { ρ ´ q . By Borel-Cantelli lemma, we conclude that there exist a set Ω p n q Ă Ω of full P –measure and i “ i p ω, n q ě ω P Ω p n q , ρ β, ´ n,n p ω δ i , ω q ď i { p { ρ ´ q for i ě i . Now, taking ˆΩ “ Ş n ě Ω p n q , we have P p ˆΩ q “
1. Replacing ω by θ τ ω we can introduce the full set ˆΩ τ , τ P R .In order to find an invariant set Ω where the convergences (28) take place, letΩ τ be the measurable set with respect to the sigma-algebra F of all ω such that p θ τ ω δ i q i P N converges in C β pr´ n, n s , R m q for any n P N , i.e. the sequence forms aCauchy sequence for any n . The measurability of Ω τ follows becausesup ´ n ď s ă t ď n | θ τ ω δ i p s, t q|| t ´ s | β “ sup t´ n ď s ă t ď n uX Q | θ τ ω δ i p s, t q|| t ´ s | β , and C p R , R m q Q ω ÞÑ θ τ ω δ i p s, t q is measurable. For the path component we can argue in a similar way, hence weonly consider the area component. In addition the set of ω for which a sequenceof random variables is a Cauchy sequence is measurable. The sets Ω τ containˆΩ τ , hence Ω τ are sets of full measure. Furthermore, if ω P Ω τ , the limit is in C β pr´ n, n s , R m q for n P N and an indistinguible version of θ τ ω is given by (10),which follows by the Borel-Cantelli argument as above. On the zero-measure set p Ω τ q c we set ω ” ω is the zero path. We denotethis new version by the same symbols: θ τ ω “ p θ τ ω, θ τ ω q . In virtue of (13), for any q, τ P R , θ τ ` q ω δ i p s, t q “ θ τ ω δ i p s ` q, t ` q q , and thus, taking limits we obtain θ τ ` q ω p s, t q “ θ τ ω p s ` q, t ` q q for all s ă t P R , such that Ω τ Ă Ω τ ` q . In a similar way, θ τ ω δ i p s, t q “ θ τ ` q ´ q ω δ i p s, t q “ θ τ ` q ω δ i p s ´ q, t ´ q q which implies that Ω τ Ą Ω τ ` q , and therefore taking τ “ q “ Ω . Then for q P R θ ´ q Ω “ θ ´ q Ω “ t ω P Ω : θ q ω δ i converges for n P N with metric ρ β, ´ n,n u “ Ω q “ Ω , OUGH PATH THEORY TO APPROXIMATE RANDOM DYNAMICAL SYSTEMS 15 hence Ω : “ Ω is θ invariant. We have defined ω ” ω to be the zero path,hence the convergences of the statement holds true for all ω P Ω. (cid:3) Rough path stability of the random dynamical systems
In this section we present the main result of this paper, concerning the convergencewhen δ Ñ ω δ to thesolution process of the rough equation driven by ω . We will present this conver-gence result under the cocycle formulation.As we already mentioned, we use the rough path theory to define the integral withintegrator ω . We recall therefore some of the basic facts regarding this theory,namely the definition of a controlled rough path and the sewing lemma. Definition 15.
For β P p { , { q , given ω P C β pr , T s , R m q we say that Y P C β pr , T s , R d q is controlled by ω if there exist Y P C β pr , T s , R d ˆ m q and R Y P C β p ∆ r , T s , R d q such that Y p t q ´ Y p s q “ Y p s qp ω p t q ´ ω p s qq ` R Y p s, t q , for ď s ď t ď T . Y is known as the Gubinelli derivative of Y and R Y is theremainder term. We introduce the space of controlled rough paths p Y, Y q P D βω pr , T s , R d q , with the norm } Y, Y } ω,β : “ ˇˇˇˇˇˇ Y ˇˇˇˇˇˇ β ` ˇˇˇˇˇˇ R Y ˇˇˇˇˇˇ β ` | Y p q| ` | Y p q| . Then p D βω pr , T s , R d q , } ¨ } ω,β q is a Banach space, see [5], Page 56.Taking into account the definition of the Wiener shift (2), it is straightforward tosee that that p Y, Y q P D βω pr τ, τ ` T s , R d q if and only if p Y p¨ ` τ q , Y p¨ ` τ qq P D βθ τ ω pr , T s , R d q .Furthermore, the composition of a smooth function with a controlled rough path isyet a controlled rough path. Lemma 16.
Assume that p Y, Y q P D βω pr , T s , R d q and f P C b p R d , R d ˆ m q . Then f p Y q is also controlled by ω with p f p Y p t qqq “ f p Y p t qq Y p t q ,R f p Y q p s, t q “ f p Y p t qq ´ f p Y p s qq ´ f p Y p s qq Y p s qp ω p t q ´ ω p s qq . For the proof we refer to [5], Lemma 7.3.In what follows we would like to define the integral of a controlled rough path. Theidea is to define it by using compensations of Riemann sums, which makes sensedue to the following result, known as sewing lemma.Define the space C β,γ p ∆ r , T s , R m q of functions Ξ from ∆ r , T s into R m with Ξ p t, t q “ } Ξ } β,γ “ ||| Ξ ||| β ` ||| δ Ξ ||| γ ă 8 , where δ Ξ p s, u, t q : “ Ξ p s, t q ´ Ξ p s, u q ´ Ξ p u, t q with ||| δ Ξ ||| γ : “ sup s ă u ă t | δ Ξ p s,u,t q|| t ´ s | γ . Lemma 17. (Sewing lemma, [5, Lemma 4.2] ). Assume β P p { , { q and γ ą .Then there exists a unique continuous map I : C β,γ p ∆ r , T s , R m q Ñ C β pr , T s , R m q such that p I Ξ qp q “ and (30) p I Ξ qp t q ´ p I Ξ qp s q “ lim | P p s,t q|Ñ ÿ r u,v sP P p s,t q Ξ p u, v q . As a consequence, it is easy to derive that under the assumptions of Lemma 16,considering Ξ p u, v q “ f p Y p u qqp ω p v q ´ ω p u qq ` f p Y p u qq Y p u q ω p u, v q we can define the integral as follows(31) ż ts f p Y p r qq d ω p r q “ lim | P p s,t q|Ñ ÿ r u,v sP P p s,t q Ξ p u, v q . Note that ÿ r u,v sP P p s ` τ,t ` τ q ` f p Y p u qqp ω p v q ´ ω p u qq ` f p Y p u qq Y p u q ω p u, v q ˘ “ ÿ r u,v sP P p s,t q ` f p Y p u ` τ qqp ω p v ` τ q ´ ω p u ` τ qq ` f p Y p u ` τ qq Y p u ` τ q ω p u ` τ, v ` τ q ˘ “ ÿ r u,v sP P p s,t q ` f p Y p u ` τ qqp θ τ ω p v q ´ θ τ ω p u qq ` f p Y p u ` τ qq Y p u ` τ q θ τ ω p u, v q ˘ “ ÿ r u,v sP P p s,t q ` f p ˜ Y p u qqp θ τ ω p v q ´ θ τ ω p u qq ` f p ˜ Y p u qq ˜ Y p u q θ τ ω p u, v q ˘ , where in the last equality ˜ Y denotes the function defined as ˜ Y p u q : “ Y p u ` τ q , for u P r s, t s . Now, taking the limit as | P p s, t q| Ñ | P p s ` τ, t ` τ q| Ñ θ τ ω “ p θ τ ω, θ τ ω q , we have ż t ` τs ` τ f p Y p r qq d ω p r q “ ż ts f p Y p r ` τ qq dθ τ ω p r q “ ż ts f p ˜ Y p r qq dθ τ ω p r q . (32)The expression (32) represents the behavior of the integral under a change of varia-ble, and this will be the key property, together with the pathwise character of theintegral, to further establish the cocycle property for the RDS generated by roughdifferential equations, see Lemma 19 below.Now we would like to solve rough differential equations driven by the rough path ω . For T ą
0, consider the equation " dY p t q “ f p Y p t qq d ω p t q , t P r , T s Y p q “ ξ P R d . (33)The integral against ω has to be understood as in (31) above. The following resultregarding the existence and uniqueness of a solution to (33) can be found in [5],Theorem 8.4. Theorem 18.
Let us consider any T ą , β P p { , { q , f P C b p R d , R d ˆ m q and ξ P R d . Then there is a unique solution p Y, Y q P D βω pr , T s , R d q to (33) , that is, (34) Y p t q “ ξ ` ż t f p Y p s qq d ω p s q , @ t P r , T s OUGH PATH THEORY TO APPROXIMATE RANDOM DYNAMICAL SYSTEMS 17 with Y “ f p Y q . In the next result we establish that the solution operator given by (34) is a cocycle.
Lemma 19.
Over the metric dynamical system p Ω , F , P , θ q introduced in Example3, the solution of equation (33) generates a random dynamical system ϕ : R ` ˆ Ω ˆ R m ÞÑ R m given by ϕ p t, ω, ξ q “ Y p t q , for all t P r , T s and ω P Ω .Proof. First of all, the measurability of ϕ directly follows from the Picard iterationargument. On the other hand, it is trivial to see that ϕ p , ω, ξ q “ ξ. Therefore,it remains to prove the cocycle property. Taking into account that the integraldefined by (31) is additive, and its behavior under a change of variable is given by(32), we have the following chain of equalities ϕ p t ` τ, ω, ξ q “ ξ ` ż t ` τ f p Y p s qq d ω p s q“ ξ ` ż τ f p Y p s qq d ω p s q ` ż t ` ττ f p Y p s qq d ω p s q“ Y p τ q ` ż t f p Y p s ` τ qq dθ τ ω p s q . Considering again the auxiliary function ˜ Y p t q : “ Y p t ` τ q for t ě
0, the previousexpression reads ϕ p t ` τ, ω, ξ q “ ˜ Y p q ` ż t f p ˜ Y p s qq dθ τ ω p s q“ ϕ p t, θ τ ω, ϕ p τ, ω, ξ qq , which completes the proof. (cid:3) In what follows, we would like to compare the solution of (33) with that of thecorresponding system driven by the approximated lift ω δ , that is to say, we considerthe following rough differential equation " dY δ p t q “ f p Y δ p t qq d ω δ p t q , t P r , T s ,Y δ p q “ ξ δ P R d , (35)with the aim of establishing the relationship with system (33).We should remark that ω δ is in C pr , T s , R m q and therefore we can use the ordinarydifferential equations theory to interpret this equation. This actually means that(35) is equivalent to the following random ordinary differential equation " dY δ p t q “ f p Y δ p t qq dω δ p t q , t P r , T s ,Y δ p q “ ξ δ P R d , (36)Then under the regularity assumption f P C b p R d , R d ˆ m q (which indeed is too muchregularity for solving the ordinary differential equation (36)), there exists a unique Y δ P C β pr , T s , R d q solving (35), that furthermore generates an RDS ϕ δ given by ϕ δ p t, ω, ξ δ q “ Y δ p t q for all t P r , T s and ω P Ω.Now we are in position to establish the relationship between the two RDS ϕ δ and ϕ . Theorem 20.
Let f P C b p R d , R d ˆ m q . For any given T ą and β P p { , { q ,there exists a sequence of positive numbers p δ i q i P N (say dyadic numbers) convergingto zero such that if the sequence of initial conditions p ξ δ i q i P N converges to ξ , thenthe RDS ϕ δ i generated by the solution of system (35), converges to ϕ , the RDSgenerated by the solution of (33), in the space C β pr , T s , R d q when i Ñ 8 .Proof.
Since ω and ω δ i are rough paths, there exists a positive constant K “ K p ω q such that ||| ω ||| β, ,T ď K, ||| ω δ i ||| β, ,T ď K. Now, by Theorem 8.5 and Remark 8.6 in [5], we know that there exists C “ C p K, β, f q ą ||| Y δ i ´ Y ||| β, ,T ď C ´ | ξ δ i ´ ξ | ` ρ β, ,T p ω δ i , ω q ¯ hence Theorem 14 implies ||| ϕ δ i ´ ϕ ||| β, ,T ď C ´ | ξ δ i ´ ξ | ` ρ β, ,T p ω δ i , ω q ¯ Ñ , when i Ñ 8 , which completes the proof. (cid:3)
Appendix: Covariances of ω δ and X δ for the fBm with any H P p , q In this section, we would like to compute the explicit expression of the covariance of ω δ as well as that of X δ . In order to make the result as much general as possible, inthis section we assume that ω is a fractional Brownian motion with Hurst parameter H P p , q . In that way, in a forthcoming research, we will able to use the resultswhen considering a fractional Brownian motion as driven noise. Theorem 21.
The covariance of X δ is σ X δ p u q “ K p u q id with K p u q given by K p u q “ H ` δ H ` δ p H ` qp H ` q ¯ K p u q , where for u ě δ , ¯ K p u q “ p u ` δ q H ` ` p u ´ δ q H ` ´ u H ` ´ δ p H ` qpp u ` δ q H ` ´ p u ´ δ q H ` q ` δ p H ` qp H ` q u H , and for ď u ă δ , ¯ K p u q “ p u ` δ q H ` ` p δ ´ u q H ` ´ u H ` ´ δ p H ` qpp u ` δ q H ` ` p δ ´ u q H ` q ` δ p H ` qp H ` q u H . The proof of this result is based on the following lemma.
Lemma 22.
Assume that ω is a fractional Brownian motion with values in R m and Hurst parameter H P p , q . Then for u ě the cox of ω δ p u q is given by I p u q id ,where I p u q : “ δ p H ` qp H ` qˆ " pp u ` δ q H ` ´ δ H ` ´ u H ` ` p u ´ δ q H ` q , u ě δ pp u ` δ q H ` ´ δ H ` ´ u H ` ` p δ ´ u q H ` q , u ă δ Furthermore, for u ě , E ω δ p u q ω p u q is given by J p u q id , being J p u q : “ δ p H ` q ˆ " p u ` δ q H ` ´ δ H ` ´ p u ´ δ q H ` , u ě δ p u ` δ q H ` ´ δ H ` ` p δ ´ u q H ` , u ă δ. OUGH PATH THEORY TO APPROXIMATE RANDOM DYNAMICAL SYSTEMS 19
Proof.
We start calculating the covariance of ω δ . First, note that for 1 ď i, j ď m , ω iδ p u q ω jδ p u q “ δ ż u ż u θ r ω i p δ q θ q ω j p δ q dqdr. For i ‰ j , E p ω iδ p u q ω jδ p u qq “ ω . If i “ j , I p u q : “ E p ω iδ p u qq “ δ ż u ż u | r ´ δ ´ q | H ` | r ` δ ´ q | H ´ | r ´ q | H dqdr “ : I p u q ` I p u q ` I p u q . Notice that we have I p u q “ ´ δ ż u ż r p r ´ q q H dqdr “ ´ δ p H ` q ż u r H ` dr “ ´ δ p H ` qp H ` q u H ` . Now we study I . First of all, consider the case δ ă u and suppose r ´ q ą δ . Thenwe have ż uq ` δ p r ´ q ´ δ q H dr “ H ` p u ´ q ´ δ q H ` , that it is well-defined since δ ` q ă u . Considering now r ´ q ă δ we obtain ż q ` δ p q ` δ ´ r q H dr “ H ` p q ` δ q H ` for δ ` q ă u and ż u p q ` δ ´ r q H dr “ H ` ˆ p q ` δ q H ` ´ p q ` δ ´ u q H ` ˙ for δ ` q ą u . Collecting all the previous cases, when δ ă u we obtain that I equalsto I p u q “ δ p H ` q ż u ´ δ pp u ´ q ´ δ q H ` ` p q ` δ q H ` q dq ` δ p H ` q ż uu ´ δ pp q ` δ q H ` ´ p q ` δ ´ u q H ` q dq “ δ p H ` qp H ` q pp u ` δ q H ` ´ δ H ` ` p u ´ δ q H ` q . On the other hand, when δ ą u , I p u q “ δ ż u ż u p q ` δ ´ r q H drdq “ δ ż u H ` pp q ` δ q H ` ´ p q ` δ ´ u q H ` q dq “ δ p H ` qp H ` q pp u ` δ q H ` ´ δ H ` ` p δ ´ u q H ` q , and therefore for δ ą u we conclude that I p u q “ δ p H ` qp H ` q pp u ` δ q H ` ´ δ H ` ` p δ ´ u q H ` q . Finally we deal with I p u q . As before, assume in a first step that δ ă u . In addition,suppose that r ´ q ą ´ δ . Then for q ă δ , we have ż u p r ´ q ` δ q H dr “ H ` ˆ p u ´ q ` δ q H ` ´ p δ ´ q q H ` ˙ while for q ą δ , ż uq ´ δ p r ´ q ` δ q H dr “ H ` p u ´ q ` δ q H ` . If we suppose that r ´ q ă ´ δ , then we obtain ż q ´ δ p q ´ δ ´ r q H dr “ H ` p q ´ δ q H ` that it is well-defined since in that case q ą δ . Collecting all the previous cases,when δ ă u we obtain that I p u q “ δ p H ` q ż δ pp u ´ q ` δ q H ` ´ p δ ´ q q H ` q dq ` δ p H ` q ż uδ pp u ´ q ` δ q H ` ` p q ´ δ q H ` q dq “ δ p H ` qp H ` q pp u ` δ q H ` ´ δ H ` ` p u ´ δ q H ` q . On the other hand, when δ ą u , I p u q “ δ ż u ż u p r ` δ ´ q q H drdq “ δ p H ` q ż u pp u ` δ ´ q q H ` ´ p δ ´ q q H ` q dq “ δ p H ` qp H ` q pp u ` δ q H ` ´ δ H ` ` p δ ´ u q H ` q . As a result, it turns out that the covariance of ω δ is the m ˆ m matrix with diagonalelements I p u q and off-diagonal elements equal to zero.Now we consider E p ω δ p u q ω p u qq . As before, when i ‰ j , E p ω iδ p u q ω j p u qq “
0, hencewe only consider the case i “ j . We have that J p u q : “ E p ω iδ p u q ω i p u qq “ E ´ δ ż u θ r ω i p δ q ω i p u q dr ¯ “ δ ż u ` p r ` δ q H ´ | r ` δ ´ u | H ´ | r | H ` | r ´ u | H ˘ dr “ δ p H ` q pp u ` δ q H ` ´ δ H ` q ´ δ J p u q , where, for δ ă u , J p u q “ ż uu ´ δ p r ` δ ´ u q H dr ` ż u ´ δ p u ´ δ ´ r q H dr “ H ` p δ H ` ` p u ´ δ q H ` q OUGH PATH THEORY TO APPROXIMATE RANDOM DYNAMICAL SYSTEMS 21 and for δ ą u , J p u q “ H ` p δ H ` ´ p δ ´ u q H ` q . As a consequence, J p u q “ δ p H ` q ˆ " p u ` δ q H ` ´ δ H ` ´ p u ´ δ q H ` , u ą δ p u ` δ q H ` ´ δ H ` ` p δ ´ u q H ` , u ă δ. Therefore, E p ω δ p u q ω p u qq is an m ˆ m matrix with diagonal elements J p u q andoff-diagonal elements equal to zero. (cid:3) Now we can prove the main result of this section.
Proof. (of Theorem 21). Let us consider u ě
0. Since X δ “ ω δ ´ ω , then we knowthat for any 1 ď i ď j ď m , E p X iδ p u q X jδ p u qq “ E p ω iδ p u q ω jδ p u qq` E p ω i p u q ω j p u qq´ E p ω iδ p u q ω j p u qq´ E p ω jδ p u q ω i p u qq . Then the result follows simply taking into account Lemma 22 and that σ ω p u q “| u | H id. (cid:3) Lemma 23.
When u ă the formulas in Theorem 21 and Lemma 22 hold replacing u by ´ u .Proof. Let u ă
0. Then it is easy to check that E ˆ ż u θ r ω p δ q dr ż u θ r ω p δ q dr ˙ “ E ˆ ż ´ u θ r ` u ω p δ q dr ż ´ u θ r ` u ω p δ q dr ˙ “ E ˆ ż ´ u θ r ω p δ q dr ż ´ u θ r ω p δ q dr ˙ “ E ˆ ż ´ u θ r ω p δ q dr ż ´ u θ r ω p δ q dr ˙ . Similarly E ˆ ω p u q ż u θ r ω p δ q dr ˙ “ E ˆ p´ θ u ω p´ u qq ż ´ u θ r ` u ω p δ q dr ˙ “ ´ E ˆ ω p´ u q ż ´ u θ r ω p δ q dr ˙ “ E ˆ ω p´ u q ż ´ u θ r ω p δ q dr ˙ . (cid:3) In the particular case of dealing with the Brownian motion, that is, when H “ ,then Theorem 21 reads as follows. Corollary 24. If ω denotes the Brownian motion, then the corresponding cova-riance σ X iδ of each component X iδ is given by σ X iδ p u q “ " u ´ u δ , ď u ă δ, δ, u ě δ, for ď i ď m . Acknowledgements
M.J. Garrido-Atienza and B. Schmalfuß would like to thank VIASM in Hanoi (Viet-nam) giving them the possibility for many fruitful discussions about the objectivesof this article. The authors also would like to thank Robert Hesse for a criticalproofreading of the paper.H. Gao was supported by NSFC Grant No. 11531006.M.J. Garrido-Atienza was supported by grants PGC2018-096540-I00 and US-1254251.A. Gu likes to thank for a DAAD-K.C. Wong grant allowing him to work onesemester at the FSU Jena (Germany).
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Institute of Mathematics, School of Mathematical Sciences, NanjingNormal University, Nanjing 210046, China,
E-mail address , Hongjun Gao: (Mar´ıa J. Garrido–Atienza)
Facultad de Matem´aticas, Avenida Reina Mercedes, s/n, 41012,Sevilla, Spain,
E-mail address , Mar´ıa J. Garrido–Atienza: [email protected] (Anhui Gu)
School of Mathematics and Statistics, Southwest University, Chongqing400715, China
E-mail address , Anhui Gu: [email protected] (Kening Lu)
346 TMCB, Brigham Young University, Provo, UT 84602, USA
E-mail address , Kening Lu: [email protected] (Bj¨orn Schmalfuß)
Institut f¨ur Stochastik, Friedrich Schiller Universit¨at Jena, ErnstAbbe Platz 2, D-77043, Jena, Germany,
E-mail address , Bj¨orn Schmalfuß:, Bj¨orn Schmalfuß: