Lyapunov Exponents of Linear Stochastic Functional-Differential Equations. II. Examples and Case Studies

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1 Southern Illinois University Carbondale OpenSIUC Articles and Preprints Department of Mathematics 997 Lyapunov Exponents of Linear Stochastic Functional-Differential Equations. II. Examples and Case Studies Salah-Eldin A. Mohammed Southern Illinois University Carbondale, Michael K. R. Scheutzow Technical University of Berlin Follow this and additional works at: Part of the Mathematics Commons, Ordinary Differential Equations and Applied Dynamics Commons, and the Probability Commons Published in Annals of Probability, 53, -4. Recommended Citation Mohammed, Salah-Eldin A. and Scheutzow, Michael K. "Lyapunov Exponents of Linear Stochastic Functional-Differential Equations. II. Examples and Case Studies." Jan 997. This Article is brought to you for free and open access by the Department of Mathematics at OpenSIUC. It has been accepted for inclusion in Articles and Preprints by an authorized administrator of OpenSIUC. For more information, please contact

2 The Annals of Probability 997, Vol. 5, No. 3, 4 LYAPUNOV EXPONENTS OF LINEAR STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS. PART II. EXAMPLES AND CASE STUDIES By Salah-Eldin A. Mohammed and Michael K. R. Scheutzow Southern Illinois University, Carbondale and Technical University of Berlin We give several examples and examine case studies of linear stochastic functional differential equations. The examples fall into two broad classes: regular and singular, according to whether an underlying stochastic semiflow exists or not. In the singular case, we obtain upper and lower bounds on the maximal exponential growth rate λ σ of the trajectories expressed in terms of the noise variance σ. Roughly speaking we show that for small σ, λ σ behaves like σ /, while for large σ, it grows like log σ. Inthe regular case, it is shown that a discrete Oseledec spectrum exists, and upper estimates on the top exponent λ are provided. These estimates are sharp in the sense that they reduce to known estimates in the deterministic or nondelay cases.. Introduction and some preliminaries. Lyapunov exponents for linear stochastic ordinary differential equations without memory have been studied by many authors; compare, for example, [], [], [3], [7], [8], [7] and the references therein. Issues of asymptotic stability of stochastic functional differential equations sfde s were treated by Kushner [9], Mizel and Trutzer [], Mohammed [] [5], Mohammed and Scheutzow [6], Scheutzow [] and Kolmanovskii and Nosov [8]. In [], Chapter 5, several results are given concerning the top exponential growth rate for a class of stochastic delay differential equations driven by C-valued semimartingales. The results in [] assume among other things that the second-order characteristics of the driving semimartingales are time-dependent and decay to zero exponentially fast in time, uniformly in the space variable. In an earlier paper, we established the existence of stochastic flows and an Oseledec Lyapunov spectrum for a large class of linear sfde s in Euclidean space [6]. The present article is a sequel to [6]. Our aim is to examine closely a variety of examples of linear sfde s. These examples fall into two distinct classes: regular and singular, depending on whether stochastic flows exist or not. We shall be concerned with whether our examples are regular Received November 995; revised July 996. Research supported in part by NSF Grants DMS , DMS , DMS and by NATO Collaborative Grant 88-. Research supported in part by NATO Collaborative Grant 88-. AMS 99 subject classifications. Primary 6H, 6H, 34K5; secondary 93E5, 6H5. Key words and phrases. Stochastic functional differential equations, Lyapunov exponents, Brownian motion, exponential growth rate, semimartingale, Oseledec spectrum, stochastic semiflow, stochastic delay equations, Poisson process, singular equations, regular equations.

3 LYAPUNOV EXPONENTS OF LINEAR SFDE S or singular. For singular equations we obtain estimates on the maximal exponential growth rate λ as defined at the end of this section. In the regular case we shall study questions of existence of the stochastic semiflow and its Lyapunov spectrum. Furthermore, we obtain sharp upper bounds on the top Lyapunov exponent λ. We begin by summarizing the main results from [6] that we will need in this article. In order to do so, we will adopt the following assumptions and notations. Let t t P be a complete filtered probability space satisfying the usual conditions. Consider the following class of linear one-dimensional sfde s: { } dx t = x t + s ν t ds dt r I + K t s x t + s ds dn t +x t dl t t > r x =v R x s =η s r <s< r driven by continuous one-dimensional semimartingales N and L on, t t P with stationary ergodic increments. The process ν is stationary, measure-valued and satisfies mild regularity hypotheses. The process K is stationary and C r R -valued. The reader may refer to Hypotheses C in [6] Section, pages for full details of the conditions needed. These hypotheses allow for a multidimensional version of I, together with jumps in the process N and the bounded-variation part of L. Let M = R L r R denote the Delfour Mitter Hilbert space with the norm [ / v η M = v + η s ds] v R η L r R r Define the trajectory of I by x t x t, t, where x t denotes the segment x t s =x t + s s r t Recall the following definition of regularity introduced in [5]. Definition. A linear sfde is said to be regular with respect to M if its trajectory random field x t x t : x x = v η M t admits a Borel R + Borel M Borel M -measurable version X: R + M M with a.a. sample functions continuous on R + M. The sfde is said to be singular otherwise. Theorem 4. in [6] identifies a large class of regular sfde s. Indeed, one gets a semiflow consisting of bounded linear operators X t ω, t, ω, on M. Furthermore, for t r, and a.a. ω, these operators are compact. Under fairly general moment and ergodicity conditions on the driving processes ν, K, N, L in I, one can establish the regularity of I and the existence

4 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW of a discrete nonrandom Lyapunov spectrum lim t t log X t v ω η ω ω M a.a. ω v η L M See Theorem 5. of [6] and its hypotheses. In particular, the above limit takes up a countable fixed set of values λ i i=, called Lyapunov exponents. This Lyapunov spectrum is bounded above, and the top exponent λ is given by λ = lim t t log X t ω L M for almost all ω, where L M denotes the uniform operator norm on M. This result is proved in [6], using the compactness of X t ω : M M, t r, together with an infinite-dimensional version of Oseledec s multiplicative ergodic theorem due to Ruelle [9]. In the white noise case, the result was previously established by the first author in [4] Theorem 4. Note that the Lyapunov spectrum of I does not change if one uses the state space D r R of cadlag functions, with the supremum norm [6], Remark following Theorem 5.3. In Section 4 of this article, we derive upper bounds on λ for various special cases of I. These upper bounds are expressed in terms of the coefficients of the equation. See Theorems 4., 4., and 4.3. The upper bounds are sharp in the sense that they reduce to the corresponding well-known bounds in the deterministic and/or nondelay case. In the singular case, however, there is no stochastic flow Theorem. and we do not know whether a set of Lyapunov exponents λ v η = lim t t log x t v η x t v η M v η M exists. The existence of Lyapunov exponents for singular equations seems to be a hard problem, even in the linear case. Nevertheless, we can still define the maximal exponential growth rate, λ = sup lim sup v η M t t log x t v η x t v η M for the trajectory random field x t v η x t v η : t, v η M. In general λ may depend on ω. Note however that λ = λ in the regular case, as is evident from Proposition.. In Section, we look at possibly singular equations of the form dx t =σ x t + s dν s dw t t > x x M r where W is a Wiener process and ν is a fixed finite Borel measure on r. For the above equation we specify a growth condition on the Fourier coefficients of the measure ν under which the equation becomes singular Theorem.. Theorem. thus underscores the extremely erratic dependence on the

5 LYAPUNOV EXPONENTS OF LINEAR SFDE S 3 initial paths of solutions of one-dimensional delay equations driven by white noise. In contrast, it is shown in Theorem. that for small noise variance, uniform almost sure global asymptotic stability still persists in spite of the erratic solution field. Indeed for small σ, one has λ σ / + o σ uniformly in the initial path [Theorem. and Remark iv, p. 3]. For large σ and ν = δ r, we show that /r log σ +o log σ λ /r log σ [Theorem.3 and Remark ii, p. 8]. This result is also in sharp contrast with the nondelay case r =, where one has λ = σ / for all values of σ. The proofs of Theorems. and.3 involve very delicate constructions of new types of Lyapunov functionals on the underlying state space. In Section 3, we characterize the Lyapunov spectrum for delay equations driven by Poisson noise Theorem 3.. This class of equations is interesting because it is regular but does not satisfy the set-up in [6]. The characterization of the spectrum in this case study is effected without using the Oseledec theorem. We end this section by stating a proposition that we will use frequently in the sequel. Proposition.. Suppose that the sfde I satisfies the hypotheses of Theorem 5. in [6], and let λ be its top Lyapunov exponent. Suppose there exists a real number β such that P lim sup t t log x t v η x t v η M β = for all v η M, where β is independent of v η. Then λ β. In particular, λ = λ. Proof. Denote by X: R + M M the stochastic semiflow of I. Let E i ω i= be the underlying Oseledec spaces in M, with E ω =M and { } E ω = v η M lim sup t t log X t v η ω M <λ for almost all ω [6], Theorem 5.. Pick a countable dense subset H of M. Since E ω is a closed proper subspace of M and H is dense, it follows that for almost every ω there exist v ω η ω H E ω c. Therefore P lim t However, by our hypothesis, P lim t t log X t v η M <λ for all v η H t log X t v η M β for all v η H Therefore λ β. This proves the proposition. We now look at our examples in some detail. = =

6 4 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW. Delay equations driven by white noise. Consider the one-dimensional stochastic linear delay equation II dx t =σx t r dw t t > x x = v η M = R L r R driven by a standard Wiener process W, where σ R is fixed and r is a positive delay. It is known that II is singular with respect to M for all nonzero σ [], [3]. Here we will examine the regularity of the more general one-dimensional linear sfde: dx t = x t + s dν s dw t t > III r x x M = R L r R where W is a Wiener process and ν is a fixed finite real-valued signed Borel measure on r. In this section and throughout the paper, we shall denote by W t t the filtration generated by W, namely, W t is the σ-algebra generated by W u : u t for each t. It follows from Theorem 4. of [6] that III is regular if ν has a C density with respect to Lebesgue measure on r. The following theorem gives conditions on the measure ν under which III is singular. Theorem.. Let r>, and suppose that there exists ε r such that supp ν r ε. Suppose <t ε. For each k, set ν k = πiks t exp dν s Assume that r t. ν k x /ν k = k= for all x. Let Y: ε M R be any Borel-measurable version of the solution field x t : t ε, x x = v η M of III. Then for a.a. ω, the map Y t ω : M R is unbounded in every neighborhood of every point in M, and hence non-linear. Remark. i Condition. of the theorem is implied by lim ν k log k = k

7 LYAPUNOV EXPONENTS OF LINEAR SFDE S 5 ii For the delay equation II, ν = δ r, ε = r. In this case condition. is satisfied for every t r. Proof. This proof is joint work with Victor Mizel. The main idea is to track the solution random field of a complexified version of III along the classical Fourier basis: πiks. η k s =exp r s k t in L r C. On this basis, the solution field gives an infinite family of independent Gaussian random variables. This allows us to show that no Borel measurable version of the solution field can be bounded with positive probability on an arbitrarily small neighborhood of in M, and hence on any neighborhood of any point in M cf. [], [3]. In order to simplify the computations, we shall complexify the state space in III by allowing v η to belong to M C = C L r C ; that is, we consider the sfde dx t = x t + s dν s dw t t > III C r x x = v η M C where x t C, t r, and ν, W are real-valued. We prove the theorem using a contradiction argument. Let Y: ε M R be any Borel-measurable version of the solution field x t : t ε, x x = v η M of III. Suppose, if possible, that there exists a set of positive P-measure, v η M and a positive δ such that for all ω, Y t ω is bounded on the open ball B v η δ in M of center v η and radius δ. Define the complexification Z ω : M C C of Y t ω : M R by Z ξ + iξ ω =Y t ξ ω +iy t ξ ω i = for all ξ, ξ M, ω. Let v η C denote the complexification v η C = v η +i v η. Clearly Z ω is bounded on the complex ball B v η C δ in M C for all ω. Now define the sequence Z k k= of complex random variables by Z k ω =Z η k η k ω η k ω k Then Z k = t r ε η k u + s dν s dw u k By standard properties of the Itô integral, together with Fubini s theorem, one has t EZ k Z l = η k u + s η l u + s du dν s dν s = r ε r ε

8 6 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW for k l, because t η k u + s η l u + s du = whenever k l, for all s, s r. Furthermore t πik s s η k u + s η k u + s du = t exp for all s, s r. Hence E Z k = r ε = t = ν k r r ε πik s s t exp dν s dν s t πiks exp dν s t Now Z ω : M C C is bounded on B v η C δ for all ω, and η k, η k = r + for all k. Hence by the linearity property Z v η C δ + r + η k η k = Z v η C δ + r + Z η k η k k a.s., it follows that.3 P sup Z k < k > It is easy to check that Re Z k,imz k : k are independent νk / distributed Gaussian random variables. We now follow the computation in [4], page 37, in order to reach a contradiction to.3. More specifically, for each integer N, we have.4 P sup Z k <N k P Re Z k <N k [ π = k { exp π N /ν k exp t k= exp N /ν k x x ] dx } dx It is easy to see that there exists N > independent of k such that x exp dx ν k N.5 N /ν k N exp νk for all N N and all k.

9 LYAPUNOV EXPONENTS OF LINEAR SFDE S 7 Now, combining.4 and.5 and using hypothesis. of the theorem immediately yields P sup Z k <N k = for all N N. Hence P sup Z k < = k This clearly contradicts 3. Since Y t ω is locally unbounded, it must be nonlinear because of Douady s theorem [], Part II, pages The proof of the theorem is now complete. Note that the pathological phenomenon in Theorem. is peculiar to the delay case r>. The proof of the theorem suggests that this pathology is due to the Gaussian nature of the Wiener process W coupled with the infinitedimensionality of the state space M. Because of this, one may expect similar difficulties in certain types of linear stochastic partial differential equations driven by multidimensional white noise [5]. Problem. is regular. Classify all finite signed measures ν on r for which III Note that II automatically satisfies the conditions of Theorem., and hence its trajectory field explodes on every small neighborhood of M. In view of the singular nature of II, one notes the following striking fact when the variance σ of the noise is small. It is shown in Theorem. that the maximal exponential growth rate λ of II is negative for small σ and is bounded away from zero independently of the choice of the initial path in M. Theorem.. Let ν be a probability measure on, and consider dx t =σ x t + s dν s dw t t III x = η where σ R\, η C = C R, W t, t, standard onedimensional Brownian motion, and x η is the solution of III through η. Then there exists σ > and a continuous, strictly negative function φ: σ R such that P lim sup t t log x t η φ σ = for all η C and all σ with < σ <σ, where denotes the sup norm on C. The constant σ and the function φ can be chosen independently of ν and of η C.

10 8 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW Remark. statement: P i The conclusion of the theorem is equivalent to the following lim sup t t log x t v η x t v η M φ σ = for all v η M and all σ with < σ <σ. See the remark following Theorem 5.3 in [6]. ii If III is regular, then Proposition. and Theorem. imply that the top Lyapunov exponent λ φ σ for < σ <σ and σ sufficiently small. Proof of Theorem.. Without loss of generality, we may and will assume throughout this proof that σ>. For η C define R η =η η, the diameter of the range of η, where η = sup s η s and η = inf s η s. Fixα and let β>bea parameter to be specified later. All future constants may depend on α. Define the continuous functional V: C R + by V η = R η η α + βr η α η C We will show that V is a Lyapunov functional for III in the sense that.6 EV x η δ σ V η for some continuous function δ: σ and all η C. Let us first show that if.6 holds for <σ<σ, then the assertion of the theorem follows. Suppress η C and write x = x η. By the Markov property of x n n,.6 implies that δ σ n V x n, n, is a nonnegative W n n -supermartingale. Therefore there exists a random variable Z taking values in such that δ σ n V x n Z as n, almost surely. Hence lim sup n /n log V x n log δ σ a.s. Furthermore, lim sup t log x t lim sup t n n log x n + R x n = α lim sup n This implies the assertion of the theorem. We now prove.6. Fix η C and define [ { η = η ν s + n log V x n log δ σ = φ σ < α s η s + u dν u } ds] / Let Y t =x t η, t. It follows from [], Theorem 4.3, pages 5, 5, that sup t E Y t n < for all positive integers n. By standard

11 LYAPUNOV EXPONENTS OF LINEAR SFDE S 9 properties of the Itô integral, we get var Y = E Y =σ E x t + s dν s dt = σ E η t + s dν s.7 t + η ν t + Y t + s dν s dt σ η t because E t Y t + s dν s =. Next we want to show that E Y 4 Dσ 4 η 4 where D> is a constant not depending on η, ν and σ. A simple application of Itô s formula gives.8 EY 4 t =6σ t [ ] E Y s x s + u dν u ds t σ EY s + σ t [ E Y s η s + u dν u +η ν s ds s ] Y s + u dν u s for t. Define the continuous functions g, h: R + by g s = η s + u dν u +η ν s s s h t = sup EY 4 s t s t Then, using Hölder s inequality,.8 implies t t h t σ h s ds + σ h s / g s ds t σ h s ds + σ h t / η for all t. Dividing through by h t / when nonzero and using the fact that h t is nondecreasing in t, we obtain the inequality t h t / σ h s / ds + σ η for all t. Hence applying Gronwall s lemma to the above inequality, we get.9 sup EY 4 s =h Dσ 4 η 4 s ds

12 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW for all σ, where D is a positive constant independent of the choice of η C ν and σ. If η =, then x t = a.s. for all t, and the assertion of the theorem follows trivially. Hence we will assume that η >. Without loss of generality, we also assume that η. Define q = η / η. We will devote the rest of this proof to establishing the inequality R Y E η q + Y α R Y α η + βe. η δ σ q α q α +β q α for all q, and suitably chosen β, σ, δ σ. It is not hard to see that R η η η Using this,. clearly implies.6. First, observe that P R Y u P Y u/ + P Y u/ for all u>. Then, by Chebychev s and Doob s inequalities, we have P Y u 6E Y4 D E Y 4 u 4 u 4 for all u>, where D = 4 6 /3 4. Similarly, P Y u/ Du 4 E Y 4, u>. Hence. P R Y u DDσ 4 η 4 u 4 for all u>. Next, observe that R Y α E = η. R Y P t /α dt η D Dσ 4 dt = D t 4/α σ α where D = D D α/4 α/4. Furthermore, since R Y Y, then R Y E η q + Y α η I + I where I = I = R Y / η q/ R Y / η < q/ q + R Y α dp η q + Y α dp η

13 LYAPUNOV EXPONENTS OF LINEAR SFDE S Using. we have [ I 3 R Y ] α dp 3 α D R Y / η q/ η σ α From. we also get R Y q I = P t /α q dt η [ q 4 DDσ 4 t /α q ] dt [ 3q α q 4 ] = DDσ 4 + u q 4 αu α du 3q ] DDσ [q 4 α 4 4 α 3 α + α u q 4+α du = ˆDq α 4 σ 4 where ˆD = 5 α DD 3 α + α/ 4 α. Next, we derive two estimates for I. The first one is the obvious inequality 3q α I The second estimate will be required in order to handle large values of q. To derive this estimate, we first observe that there exists c > depending on α such that 3q + y α + αy cy for all y / /. Assume that q>. Setting Z = Y / q η and using.9, we have + Z α dp + αz cz dp Z / Z / + αz dp = + αz dp + Z 4 dp Z / and, by.7, Z dp = EZ Z / for all q>. Therefore Z >/ + EZ 4 + Dσ 4 q 4 Z >/ I q α [ + D σ4 q 4 c Z >/ Z dp σ q 4EZ 4 σ q 4Dσ 4 q 4 σ q 4Dσ4 q 4 + ]

14 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW Next choose D > 3 α c, β = β σ =D σ and δ σ = 3 cσ. Using.,. and the above estimates for I and I,. will follow once we have proved that there exists σ such that.3 3 α D σ α ˆDq α 4 σ 4 + 3q α q α [ + D σ4 q 4 c σ q 4Dσ4 q 4 + ] + D D σ +α δ σ q α q α +β q α for all q and <σ<σ. We prove.3 by pure elementary analysis using a case-by-case argument for different ranges of values of q. In each case below, σ is sufficiently small and independent of the choice of q in the respective range. Case i. q /3. The left-hand side of.3 is dominated by 3q α 3 α D σ α + + D D σ +α 3 cσ q α = δ σ q α for all σ σ where σ > is sufficiently small. Case ii. /3 <q<3/. The left-hand side of.3 is dominated by ˆDq α 4 σ 4 + q α c σ + q α 4 Dσ c +D q D σ +α δ σ q α for all σ σ, and σ > sufficiently small. Case iii..4 3/ q<. Observe that δ σ + β σ = 3 3 α cσ + D σ 3 α = + σ D 3 c cd α 3 3 α+ σ4 Therefore, there is a positive constant D 3 = D 3 α and a sufficiently small σ > both independent of q, such that the left-hand side of.3 is dominated by ˆDq α 4 σ 4 + q α + D σ4 q 4 + D D σ +α D 3 σ 4 + q α + D D σ +α q α δ σ + β σ 3 α δ σ q α + β σ q α for all σ σ and sufficiently small σ >. Note that the second inequality follows from.4 and the fact that D > 3 α c. The last inequality holds because q 3. This completes the proof of the theorem.

15 LYAPUNOV EXPONENTS OF LINEAR SFDE S 3 Remark. i For any positive delay r>, we can rescale the equation dx t =σ x t + s dν s dw t t III r x = η by a linear time scale t t/r. This reduces the question of stability of III to the case r =. Therefore III also satisfies the conclusion of Theorem.. Now consider the delay case ν = δ r, where r> is fixed. Then II also satisfies Theorem.. The choice of σ depends on r. The above scaling argument has the effect of replacing r by and σ by σ r. In fact, if λ r σ is the maximal exponential growth rate of II, then it is easy to see that λ r σ = /r λ σ r. Hence σ decreases like / r as r increases. In particular, for a fixed σ, a small delay r tends to stabilize equation III or II, while on the other hand and in view of Theorem.3 a large delay in II has a destabilizing effect. ii We conjecture that, even in the singular case, the following almost sure lim sup, lim sup t t log x t v η x t v η M is actually a limit whose value is nonrandom and independent of all η C with η >. iii Using a Lyapunov functional argument, we will show in Theorem.3 that for sufficiently large σ, the singular delay equation II is unstable. This result is in sharp contrast with the corresponding result in the nondelay case r =, where lim t t log x t = σ / < for all σ R, even when σ is large. iv The growth rate function φ in Theorem. satisfies.5 φ σ = σ / + o σ as σ +. This agrees with the nondelay case r =. In order to see.5, one may modify the proof of Theorem. by making the following choices: V η = MR η η α + βr η α η C c = α/ α ε δ σ = cσ ε for sufficiently small ε, α> and sufficiently large M. II Theorem.3. Consider the equation dx t =σx t r dw t t > x x = v η M = R L r R

16 4 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW driven by a standard Wiener process W with a positive delay r and σ R. Then there exists a continuous function ψ: R which is increasing to infinity such that.6 P lim inf t t log x t v η x t v η M ψ σ = for all v η M \ and all σ. The function ψ is independent of the choice of v η M \. Remark. It is easy to see that M can be replaced by the sup norm on C. Proof of Theorem.3. We break the proof up into two main steps. In Step, we prove a discrete version of.6. In Step, we use an interpolation argument to obtain the continuous-time limit.6. Step. We shall first prove that.7 P lim inf n n log x n v η x n v η M ψ σ = for all v η M \ and all σ. In view of Remark i, following the proof of Theorem., we will assume that r =, without loss of generality. We will also assume that σ>. Define the continuous functional V: M \ by V v η = We will show that for v η /4 v + σ η s ds.8 EV x v η x v η Kσ /4 V v η for all σ >, where K is a positive constant suitably chosen independently of v η M \. We show first that.8 implies the assertion of the theorem. Set X t = x t v η x t v η, t. Then.8 and the Markov property imply that K n σ n/4 V X n, n, is a nonnegative W n n - supermartingale, which therefore converges almost surely to a nonnegative random variable. Therefore lim sup n /n log V X n log K log σ. 4 Hence lim inf n n log X n M log σ log K almost surely. This in turn implies.7 with ψ σ = log σ log K, σ>. Let us now show that.8 holds for v η. Define η = η s ds /.If η =, then.8 holds with K =. Without loss of

17 LYAPUNOV EXPONENTS OF LINEAR SFDE S 5 generality, we shall assume that v and η >. Define µ = v/ σ η. Then.8 is equivalent to.9 E σ µ + t η K + σµ /4 η u dw u dt + µ + B /4 for all µ, where B = / η η u dw u. Note that B is an - Gaussian random variable. Case i. σµ. The left-hand side of.9 is dominated by E µ + B / = π µ + x / exp x / dx /4 K K + σµ /4 where K = /4 sup y + x / exp x / dx < π y R + Case ii. σµ >. It suffices to show that there exists K < such that [ E σ µ + t /4. η u dw u dt] K σµ /4 η That is, [ sup E + t µ η η u dw u dt] /4 < where the supremum is taken over all η L R \, and µ>. Set Y t = + t η u dw u µ η t Then Y t is Gaussian with mean. It is easy to see that there exists a positive constant β independent of t [and the variance of Y t ] such that P Y t u βu for all <u /. Hence P Y t u β u for u /4. Now /4 E Y t dt = P Y t dt u 4 du

18 6 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW and P Y t dt z P l s Y s z / l P s ω Y s ω z = P Y s z ds β z for all z /8. The symbol l denotes Lebesgue measure on. Therefore /4 E Y t dt 8 /4 + β u 4 du 8 /4 This proves.7. = 8 /4 + 8 /4 βu du = K < Step.. where We now show.6. In view of.7, it suffices to show that a.s. lim sup n n Y n Y n = log For v η M \ define Fix β>. We will show that X t M inf n t n+ X n M Z v η = inf t log X t M v η M. sup E Z v η β < v η M \ Then by. and the Markov property, we get E Y β n P Y n αn αn sup E Z v η β αn β < n= n= β v η M \ n= for every α>. Therefore if we prove., then. will follow from the above relation and the Borel Cantelli lemma. We now prove.. Consider E Z v η β v + σ t E log inf η s dw s + t η s ds β t v + η s ds

19 LYAPUNOV EXPONENTS OF LINEAR SFDE S 7 When η =, the right-hand side of the above inequality is equal to. So we will assume that η. Defining µ as before, the right-hand side of the above inequality is equal to µ + / η t E log inf η s dw s + / σ η t η s ds β.3 t µ + /σ For different σ and σ the ratio of the corresponding terms in the above expression is bounded above and below uniformly in µ η. Therefore for the rest of this proof we will assume without loss of generality that σ =. We will also assume that µ. By a time-change argument,.3 equals [ µ + B t ] + t β E log inf t µ + where B t, t, is an W t t -standard Brownian motion. Now M t = µ + B t + t, t, is a continuous nonnegative martingale. Let M = inf t M t. Using a stopping-time argument and the martingale property, it follows that for all x>. Therefore.4 EZ v η β = P M x P M x [ ] β M P log x dx µ + M µ + exp x /β dx [ µ + µ + / exp x /β / µ µ + / exp x /β / ] dx where x = / π x exp y / dy. Now fix x > such that 4 exp x /β. Then the integrand in.4 is dominated by K exp x /β / for some K>, uniformly for all µ and all x x. Therefore.4 implies that sup E Z v η β < v η M \ This completes the proof of the theorem. Remarks. i Observe that the functional V in the above proof does not satisfy an inequality of the type lim t + t EV x t x t V v η KV v η for some K> and all v η M \. To see this, simply pick v = η on /3 /3 and η on /3 /3.

20 8 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW ii For the singular equation II, one can show that.5 λ inf δ + δ + δ R σ exp δr + where λ is its maximal exponential growth rate. To see.5, we modify a technique due to Kushner [9]. Consider the equation II dx t =µx t dt + σx t r dw t x x = v η M = R L r R Define the functional V M R + by V v η = v + σ η s ds r t > v η M Set x t =x t v η, t r. A simple application of Itô s formula gives EV x t x t = V v η + t EQ x u du t > where Q z = µ + σ z, z R. The above relation, together with the Markov property, implies that the process exp µ+σ t V x t x t t is a nonnegative W t t -supermartingale. Therefore the maximal exponential growth rate for II is less than or equal to µ + σ / +. Using an exponential shift cf. Section 4 and setting µ = immediately gives.5. Now put δ = /r log σ in.5. This gives λ /r log σ for large σ. On the other hand, one can show that a.s. lim inf σ lim inf t /t log x σ t x σ t log σ / r where x σ is the solution of II for a given σ R. This follows from the proof of Theorem.3. The above discussion shows that /r log σ +o log σ λ /r log σ for large σ. It is interesting to observe that the estimate.5 does not yield the stability of II for small σ that was demonstrated by the conclusion of Theorem.. 3. Delay equations with Poisson noise. This class of equations was studied in []. Consider the one-dimensional linear delay equation IV dx t =x t dn t t > x = η D = D R The process N t Ris a Poisson process with i.i.d. interarrival times T i i= which are exponentially distributed with the same parameter µ. The jumps Y i i= of N are i.i.d. and independent of all the T i s. Let { } j j t =sup j T i t i=

21 Then LYAPUNOV EXPONENTS OF LINEAR SFDE S 9 j t N t = Y i Equation IV can be solved a.s. in forward steps of lengths, using the relation j t x η i t =η + Y i x T j a.s. i= j= It is easy to see that the trajectory x t t of IV is a Markov process in the state space D with the supremum norm. It is interesting to observe that equation IV does not satisfy the general set-up in [6]. In spite of this, the above relation implies that IV is regular in D; that is, it admits a measurable flow X R + D D with X t ω continuous linear for all t and a.a. ω. It is interesting to compare this with the singular equation II of Section. Furthermore, we will show that IV has an a.s. Lyapunov spectrum which can be characterized directly without appealing to the Oseledec theorem. This can be done by interpolating between the sequence of random times: τ ω = { } k τ ω =inf n T j / n n for all k j= { τ i+ ω =inf n>τ i ω k j= i= } T j / n n for all k i It is easy to see that τ, τ τ, τ 3 τ are i.i.d. Furthermore Eτ = e µ. Theorem 3.. Let ξ D stand for the constant path ξ s = for all s. Suppose E log X τ ξ exists possibly =+ or. Then the a.s. Lyapunov spectrum λ η =lim t t log X t η ω η D ω of IV is λ where λ = e µ E log X τ ξ In fact, { λ lim t t log X t η ω η / Ker X τ ω ω = η Ker X τ ω ω Proof. For simplicity of notation, we shall denote the sup norm by throughout this proof. If P X τ ξ >, then it is easy to see that lim /t log X t η = t

22 3 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW for all η D, a.s. So we will assume that P X τ ξ = Define the i.i.d. sequence of random variables S i i= by S i = X τ i ξ X τ i ξ i = with τ =. The law of large numbers implies that lim n τ n log X τ n ξ ω = e µ E log S for a.a. ω. For t and ω, let n = n ω t satisfy τ n t<τ n+. Then t log X t ξ ω = t log X τ n ξ ω + X t ξ ω log t X τ n ξ ω Since τ n /t ast, the term /t log X τ n ξ ω converges to e µ E log S for a.a. ω by what we showed above. Let us now assume that E log S <. We will show that, for a.a. ω, 3. Let lim t X t ξ ω t log X τ n ξ ω = M ω = sup s τ X s ξ ω m = inf s τ X s ξ ω ω Then there is a positive number q such that P S >α M>α q> for all α. This holds because after a jump to a value greater than α in absolute value, the probability of staying in that state for another two units of time is positive independently of α. Using this and the fact that E log S <, implies that E log M<. Similarly it follows that E log m>. Now by the first Borel Cantelli lemma, we get lim n n sup τ n t<τ n+ for a.a. ω. Hence 3. follows. X t ξ ω log X τ n ξ ω =

23 LYAPUNOV EXPONENTS OF LINEAR SFDE S 3 The cases E log S = or, respectively, can be treated similarly. The remaining assertions of the theorem can now easily be proved. Remarks. i Let p D be a measurable functional, satisfying a p λη = λ p η for all η D homogeneity. b There exist γ γ > such that γ inf η s p η γ s sup s η s for all η D. For example, p can be any L p -norm p<, but also, for example, the functional p η =inf s η s. Using the same arguments as in the proof of the theorem, it can be shown that the Lyapunov spectrum lim log p X t η ω t t with respect to the functional p satisfies the conclusion of Theorem 3.. In particular, the above spectrum is nonrandom and is independent of the choice of the functional p satisfying a and b. ii The same methods work with a few obvious changes for dx t = x t + s dµ s dn t r for any finite signed measure µ whose support is bounded away from zero. iii By the same methods we can also treat the equation dx t =αx t dt + x t dn t for a fixed α R. Just define z t =e αt x t and apply Itô s formula to get dz t =z t e α dn t The above equation can be analyzed using the method of proof of Theorem 3.. Let p: D R be as in Theorem 3., Remark i. Suppose η D and define η D by η s =e αs η s, s. Let Z t η ω =z t η ω. Then lim log p X t η ω = α + lim log p Z t η ω t t t t Hence the Lyapunov spectrum of X is that of Z shifted by α. 4. A class of regular equations. In this section we shall outline a general scheme to obtain estimates on the top Lyapunov exponent for a class of one-dimensional regular linear stochastic functional differential equations. After outlining our scheme, we will apply it to several specific examples within the above class. Our scheme can be applied to multidimensional linear equations with multiple delays.

24 3 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW The reader may note that the approach in [9] yields strictly weaker estimates than ours in all cases. Let t t P be a complete filtered probability space satisfying the usual conditions. Consider the following class of one-dimensional linear sfde s: { } dx t = ν x t +µ x t r + x t + s σ s ds dt r V { } + ν x t + x t + s σ s ds dm t r where r >, σ, σ C r R, and M is a continuous t t -local martingale with stationary ergodic increments. By the ergodic theorem, we have the a.s. deterministic limit β = lim t M t /t. Assume that β< and M L R. It is easy to see that equation V is regular with respect to M because of Theorem 4. of [6]. Hence V has a sample-continuous stochastic semiflow X R + M M. Furthermore, equation V satisfies all the hypotheses of Theorem 5. of [6]. Therefore the stochastic semiflow X has a fixed nonrandom Lyapunov spectrum. Let λ be its top exponent. We wish to develop an upper bound for λ. In view of the proof of Theorem 5. in [6], there is a shift-invariant set of full P-measure and a measurable random field λ M R, 4. λ v η ω =lim t t log X t v η ω M v η M ω giving the Lyapunov spectrum of V. We introduce the following family of equivalent norms: 4. v η α = on M. Clearly { / αv + η s ds} v η M α> r 4.3 λ v η ω =lim t t log X t v η ω α v η M ω for each α>; that is, the Lyapunov spectrum of V with respect to α is independent of α>. Let x be the solution of V starting at v η M, and define the family of processes ρ α, α>, by 4.4 t ρ α t = X t α = αx t + x u du t r t > α> For each fixed v η M, define the set by = ω ρ α t ω for all t>. IfP =, then by uniqueness there is a random time τ such that a.s. X t v η = for all t τ. This implies

25 LYAPUNOV EXPONENTS OF LINEAR SFDE S 33 that λ =. Hence we may suppose that P >. Using Itô s formula and a simple stopping-time argument gives 4.5 t log ρ α t =log ρ α + Q α a u b u I u du t + Q α a u I u d M u + t R α a u I u dm u t > a.s. on, where Q α z z z 3 =ν z + αµ z z + αz z 3 + z α z 4.6 Q α z z 3 =α z ν α z + z 3 R α z z 3 =ν z + αz z 3 and 4.7 for i =,, t>, a.s. on. In view of the estimate I i t ρ α t { / σ i = σ i s ds} i = r αx t a t = ρ α t r b t =x t ρ α t r I i t = x t + s σ i s ds ρ α t / x t + s ds σ i = r a t σ i i = a.s. on the variables z z z 3 z 3 appearing in 4.6 must satisfy z z R z 3 z σ z 3 z σ Let τ = inf t > ρ α t =. Then the local martingale t τ R α a u I u dm u t > is a time-changed possibly stopped Brownian motion. In view of this and the fact that a.s. R α a u I u ν + α σ for all u τ, it follows that 4.8 lim t t t τ R α a u I u dm u = a.s.

26 34 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW Now divide 4.5 by t, let t, and use the above relation to obtain 4.9 λ v η ω lim sup t t + lim sup t t Q α a u b u I u du t t Q α a u I u d M u a.s. on, for all α>. Scheme. If ε d M t /dt ε a.s. for all t, then maximize Q α z, z, z 3 +γ Q α z, z 3 over all z, z R z 3 z σ, z 3 z σ and ε γ ε for fixed α>. Hence for a.a. ω, 4.9 gives an upper bound Q α ν, ν, µ, σ, σ for λ v η ω. It is easy to see from the cocycle property that λ v η ω = for all ω c. Therefore one has λ v η ω Q α ν, ν, µ, σ, σ for a.a. ω. Since the upper bound Q α ν, ν, µ, σ, σ isindependent of v η M and ω, it follows from Proposition. that Q α ν, ν, µ, σ, σ is also an upper bound for the top exponent λ of V for each α>. Hence λ inf α> Q α ν, ν, µ, σ, σ. This upper bound can be refined further by applying an exponential shift of the Lyapunov spectrum by an arbitrary amount δ and then minimizing the resulting upper bound over δ R [cf. proofs of Theorems ]. The special case ε = ε = corresponds to M being a standard Brownian motion. Scheme. Compute the maximal values of Q α and Q α for fixed α> separately over z, z R, z 3 z 3 as above. Let the maximal values of Q α Q α thus obtained be denoted by q α and q α, respectively. Then, as in Scheme, we have λ inf α> q α + q α β Again use the exponential shift by δ and then minimize over δ R, in order to further refine the above estimate cf. proof of Theorem 4.3. The above two schemes will now be used to develop upper bounds on λ in the following special cases. Consider the one-dimensional linear sfde: { } VI dx t = ν x t +µ x t r dt + x t + s σ s ds dw t t > r with real constants ν, µ and σ C r R. It is a special case of V. Hence VI is regular with respect to M. Observe that the process r x t + s σ s ds has C paths in t, and so the stochastic differential dw with respect to the one-dimensional Brownian motion W in VI may be interpreted in the Itô or Stratonovich sense without changing the solution x.

27 Theorem 4.. the function LYAPUNOV EXPONENTS OF LINEAR SFDE S 35 θ δ α = δ + Suppose λ is the top a.s. Lyapunov exponent of VI. Define ν + δ + αµ eδr + α α σ r eδ+ for all α R +, δ R, where δ + = max δ. Then 4. λ inf θ δ α δ R α R + Proof. We follow Scheme. Putting ν = σ = in V, it is easy to see from 4.6 that Q α z z + Q α z z 3 ν + z α + αµ z z z α z z σ for all z, z R z 3 z σ. For fixed z, the right-hand side of 4. attains its absolute maximum when z = αµz. Hence 4. sup Q α z z + Q α z z z z 3 A 3 z ν + αµ α α z z σ where A = z z 3 : z R, z 3 z σ. We further maximize the right-hand side of 4. over z. This gives sup Q α z z + Q α z z z z z 3 B 3 ν + αµ + α 4.3 α σ where the supremum is taken over the set B = z z z 3 z z R z 3 z σ The inequality 4.3 implies that λ ν + αµ + α 4.4 α σ for every α ν µ σ. Finally we apply an exponential shift to VI by an arbitrary amount δ R. Shift the Lyapunov spectrum = λ i i= of VI by setting y t =e δt x t t r, for a fixed δ R. Then y solves the linear sfde VI dy t = ν + δ y t +µ e δr y t r dt + where ˆσ C r R is given by ˆσ s =e δs σ s r s r y t + s ˆσ s ds dw t

28 36 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW Let = λ i i= denote the Lyapunov spectrum of VI. Then it is easy to see that λ i = λ i + δ for all i. Furthermore, applying 4.4 to VI, and noting that ˆσ σ eδ+r, we immediately get 4.. This completes the proof of the theorem. The following corollary shows that the estimate in Theorem 4. reduces to a well-known estimate in the deterministic case σ [6], pages 7, 8. Corollary 4.. In VI, suppose µ and let δ be the unique real solution of the transcendental equation 4.5 ν + δ + µ e δr = Then λ δ + σ 4.6 e δ r µ If µ = and ν, then λ ν + ν + σ.ifµ = and ν <, then λ ν + σ e νr. Proof. Suppose µ. Denote by f δ, δ R, the left-hand side of 4.5. Then f δ is an increasing function of δ. A simple application of Rolle s theorem shows that f has a unique real zero δ. Using 4., we may put δ = δ and α = µ e δ r in the expression for θ δ α. This immediately yields 4.6. Now suppose µ =. Put δ = ν + in θ δ α and minimize the resulting expression over all α>. This proves the last two assertions of the corollary. Remarks. i Although the upper bounds for λ in Theorem 4. and its corollary agree with the corresponding bounds in the deterministic case for µ, they still do not yield the optimal bound when µ = and σ is strictly positive and sufficiently small; compare Theorem., Remark ii. ii It would be interesting to study the asymptotics of λ for small delays r. Our second example is the stochastic delay equation VII dx t = ν x t +µ x t r dt + x t dm t t > where M is the local martingale appearing in V and satisfying the conditions therein. In particular, VII is regular with respect to M. Furthermore, we have the following estimate on its top exponent. Theorem 4.. In VII define δ as in Corollary 4. Then the top a.s. Lyapunov exponent λ of VII satisfies 4.7 λ δ + β 6

29 LYAPUNOV EXPONENTS OF LINEAR SFDE S 37 Proof. We follow Scheme. Here, put σ = σ, ν = in V. Therefore λ inf α> q α + q α β where q α = sup Q α z z z z R q α = sup z Q α z and Q α z z =ν z + αµ z z + z α z Q α z = z z z z R An elementary computation shows that q α = ν + αµ + + and q α α = 6 Using an exponential shift, as in the proof of Theorem 4., gives λ β [ 6 + inf δ + ν + δ + α R + αµ eδr + + ] α for all δ R. For fixed δ R, the infimum on the right-hand side of the above inequality is attained when α = e δr / µ. Therefore λ β 6 + inf [ δ + ν + δ + µ e δr +] δ R The above minimum is attained if δ solves the transcendental equation 4.5. Hence the conclusion of the theorem. Remark. The above estimate for λ is sharp in the deterministic case β = and µ, but is not sharp when β ; for example, M = W, onedimensional standard Brownian motion in the nondelay case µ =. On the other hand, when M = ν W for a fixed real ν, the above bound may be considerably sharpened as in the following theorem. Theorem 4.3. VIII For the equation dx t = ν x t +µ x t r dt + ν x t dw t set 4.8 φ δ = δ + [ µ 4ν e δr + ν + δ + + ] ν

30 38 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW for ν. Then 4.9 λ inf δ R φ δ In particular, if δ is the unique solution of the equation 4. ν + δ + µ e δr + ν = then λ δ. Proof. We use Scheme. Take σ = σ =, M = W in V. Then 4.6 implies that 4. Q α z z + Q α z = ν + α + ν z + αµ z z z ν z4 for z, z R. For fixed z, the right-hand side of 4. attains its maximum when z = αµ z. Therefore 4. λ sup Q α z αµ z + Q α z z [ = sup z ν + α + µ α + ν { ν 6ν + + } α + µ α + ν ] z ν z4 for all α R +. Now set h α =ν + α + µ α + ν α > Note that inf α> h α =h µ =ν + µ +ν Hence 4.3 λ [ ν + µ 6ν +ν +] The first assertion of the theorem follows by applying an exponential shift to VIII and using 4.3. The last assertion of the theorem is obvious. This completes the proof of the theorem.

31 LYAPUNOV EXPONENTS OF LINEAR SFDE S 39 Remark. The reader may check that the estimate in Theorem 4.3 agrees with the nondelay case µ = whereby λ = ν ν = inf δ Rφ δ. See also [], [3] and []. Acknowledgments. The authors are grateful for fruitful discussions with S. T. Ariaratnam, L. Arnold, P. Baxendale, W. Kliemann, V. J. Mizel, M. Pinsky and V. Wihstutz. The authors thank the anonymous referee for a careful reading of the manuscript and very helpful suggestions. REFERENCES [] Arnold, L., Kliemann, W. and Oeljeklaus, E Lyapunov exponents of linear stochastic systems. Lyapunov Exponents. Lecture Notes in Math Springer, Berlin. [] Arnold, L., Oeljeklaus, E. and Pardoux, E Almost sure and moment stability for linear Itô equations. Lyapunov Exponents. Lecture Notes in Math Springer, Berlin. [3] Baxendale, P. H Moment stability and large deviations for linear stochastic differential equations. In Proceedings of the Taniguchi Symposium on Probabilistic Methods in Mathematical Physics N. Ikeda, ed Kinokuniya, Tokyo. [4] Dudley, R. M The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal [5] Flandoli, F. and Schaumlöffel, K.-U. 99. Stochastic parabolic equations in bounded domains: random evolution operator and Lyapunov exponents. Stochastics Stochastic Rep [6] Hale, J. K Theory of Functional Differential Equations. Springer, New York. [7] Has minskiǐ, R. Z. 98. Stochastic Stability of Differential Equations. Sijthoff and Noordhoff. [8] Kolmanovskii, V. B. and Nosov, V. R Stability of Functional Differential Equations. Academic Press, London. [9] Kushner, H. J On the stability of processes defined by stochastic differential difference equations. J. Differential Equations [] Mao, X. R Exponential Stability of Stochastic Differential Equations. Dekker, New York. [] Mizel, V. J. and Trutzer, V Stochastic hereditary equations: existence and asymptotic stability. J. Integral Equations 7 7. [] Mohammed, S.-E. A Stochastic Functional Differential Equations. Pitman, Boston. [3] Mohammed, S.-E. A Non-linear flows for linear stochastic delay equations. Stochastics 7 7. [4] Mohammed, S.-E. A. 99. The Lyapunov spectrum and stable manifolds for stochastic linear delay equations. Stochastics Stochastic Rep [5] Mohammed, S.-E. A. 99. Lyapunov exponents and stochastic flows of linear and affine hereditary systems Survey article. In Diffusion Processes and Related Problems in Analysis M. Pinsky and V. Wihstutz, eds Birkhäuser, Boston. [6] Mohammed, S.-E. A. and Scheutzow, M. K. R Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. Part I. The multiplicative ergodic theory. Ann. Inst. H. Poincaré Probab. Statist [7] Pardoux, E. and Wihstutz, V Lyapunov exponent and rotation number of twodimensional stochastic systems with small diffusion. SIAM J. Appl. Math [8] Pinsky, M. and Wihstutz, V Lyapunov exponents of nilpotent Itô systems. Stochastics [9] Ruelle, D. 98. Characteristic exponents and invariant manifolds in Hilbert space. Ann. of Math

32 4 S.-E. A. MOHAMMED AND M. K. R. SCHEUTZOW [] Scheutzow, M. K. R Stationary and Periodic Stochastic Differential Systems: A Study of Qualitative Changes with Respect to the Noise Level and Asymptotics. Habiltationsschrift, Univ. Kaiserslautern, Germany. [] Schwartz, L Radon Measures on Arbitrary Topological Vector Spaces and Cylindrical Measures. Academic Press, London. Department of Mathematics Southern Illinois University Carbondale, Illinois 69 salah@math.siu.edu Fachbereich 3, MA 7-5 TU Berlin, Strasse des 7 Juni 36 D-63 Berlin Germany ms@math.tu-berlin.de