GENERATION OF ONE-SIDED RANDOM DYNAMICAL SYSTEMS BY STOCHASTIC DIFFERENTIAL EQUATIONS

: Let Z be an RI m -valued semimartingale with stationary increments which is realized as a helix over a (cid:12)ltered metric dynamical system S . Consider a stochastic di(cid:11)erential equation with Lipschitz coe(cid:14)cients which is driven by Z . We show that its solution semiﬂow (cid:30) has a version for which ’ ( t; ! ) = (cid:30) (0 ; t; ! ) is a cocycle and therefore ( S; ’ ) is a random dynamical system. Our results generalize previous results which required Z to be continuous. We also address the case of local Lipschitz coe(cid:14)cients with possible explosions in (cid:12)nite time. Our abstract perfection theorems for semiﬂows are designed to cover also potential applications to in(cid:12)nite dimensional equations.


Introduction
We start by defining the concept of a random dynamical system which has been introduced by L. Arnold and his school and which we adapt slightly according to our needs (cf. Arnold (1997), Arnold and Scheutzow (1995)).
Definition 1 Let (Ω, F , P I ) be a probability space and θ a Borel R I -action on Ω i.e. θ : If in addition N ∈ F, P I (N) = 0 implies P I (θ −1 t N) = 0 for all t ∈ R I i.e. P I is quasiinvariant under θ, then (Ω, F , P I , θ) is called quasimetric dynamical system (QDS). If even more P I (θ −1 t A) = P I (A) for all A ∈ F, t ∈ R I i.e. P I is invariant under θ, then (Ω, F , P I , θ) is called metric dynamical system (MDS).
ϕ : I × Ω → H is called a crude cocycle if for every s ∈ I there exists a P I -null set N s such that ϕ(t + s, ω) = ϕ(t, θ s ω) • ϕ (s, ω) for all t, s ∈ I, ω ∈ N s .
We will always assume that θ is defined on R I × Ω even if we consider cocycles which are defined on a proper subset of R I . This will be convenient and does not seem to be too restrictive provided one is willing to change the (e.g. one-sided) QDS on which the cocycle ϕ is defined without changing the law of ϕ (cf. (Arnold, Scheutzow (1995), Theorem 13) for a closely related question).
An important class of RDS arises via solution flows of stochastic differential equations (SDEs) driven by semimartingales Z with stationary increments. More precisely we assume that the m-dimensional semimartingale Z has the helix property on the MDS (Ω, F , P I , θ) i.e.
Z t+s (ω) − Z s (ω) = Z t (θ s ω), s,t≥ 0, ω ∈ Ω i.e. Z is a cocycle with (H, •) = ( R I m , +) and I = [0, ∞). We can always extend Z to a helix with index set R I in a unique way (Arnold, Scheutzow (1995)) but there will be no need to do this. Let f : R I n → R I n×m satisfy a global Lipschitz condition and let us assume that Z has continuous paths. Consider the n-dimensional SDE By results of Kunita (Kunita (1984) we know that (1) and such that φ s,t (ω)(x) solves (1) for each fixed x ∈ R I n and s ≥ 0.
The helix property of Z can be shown to imply that for every s ≥ 0 there exists a set M s of measure zero such that for all t ≥ 0 and all ω ∈ M s (Arnold, Scheutzow (1995)). If we define ϕ(t, ω) := φ 0,t (ω), t ≥ 0, ω ∈ Ω, then ϕ is obviously a crude cocycle with N s = M s (assuming that M s and N s are chosen as small as possible). Since one is interested in getting a cocycle ϕ rather than a crude cocycle and since φ is only uniquely defined up to sets of measure zero the question arises if there exists a version of φ (or of ϕ) such that N s = M s = ∅ for all s ≥ 0 i.e. such that ϕ is a cocycle. This is usually referred to as the perfection problem of crude cocycles. Note that the union of all sets M s need not be a set of measure zero in general. A positive solution of the perfection problem thus guarantees that (1) generates the RDS (Ω, F , P I , θ, ϕ).
The perfection problem in the above set-up (with continuous Z) was solved in (Arnold, Scheutzow (1995)) -even for more general equations than (1). In fact (Arnold, Scheutzow (1995)) and (Mohammed, Scheutzow (1996)) contain abstract perfection results for group-valued crude cocycles which enjoy continuity properties in the "time" variable (with different proofs). A more general perfection result for group-valued cocycles without continuity properties is given in (Scheutzow (1996)).
We will show that if Z is a semimartingale helix which is just cadlag (i.e. right continuous with left limits for all ω ∈ Ω) then there still exists a solution map φ with properties (2) and (3). It does not take values in the group of homeomorphisms in general but just in the semigroup (C n , •) of continuous functions from R I n to R I n . If we define ϕ(t, ω) := φ 0,t (ω) as before then ϕ is still a crude cocycle but the perfection theorems above can not be applied. In fact all proofs of the perfection theorems mentioned above heavily use the existence of inverse elements in the group H.
Let us point out the difference between the perfection problem in the group-valued and the semigroup-valued case. We call φ a crude semiflow if it satisfies (2) and (3) and semiflow or perfect semiflow if it satisfies (2) and (3) without exceptional sets.
There is a one-to-one correspondence between H-valued semiflows φ and H-valued cocycles ϕ given by ϕ(t, ω) := φ 0,t (ω) and φ s,t (ω) := ϕ(t − s, θ s ω) which holds also if H is just a semigroup. For the corresponding crude objects φ and ϕ and the perfection problems there is still a close correspondence if H is a group but no longer if H is just a semigroup. Indeed if we start with a group-valued crude semiflow φ, define ϕ(t, ω) := φ 0,t (ω), find a perfection ϕ of ϕ and define φ s,t (ω) := ϕ(t − s, θ s ω), then φ is a semiflow which is indistinguishable from φ i.e. φ and φ agree identically up to a set of measure zero. We call φ a perfection of φ. Conversely if we start with a group-valued crude cocycle ϕ and define φ s,t (ω) := ϕ(t, ω) • ϕ −1 (s, ω), then φ is a crude semiflow (observe that ϕ(0, ω) equals the identity e H of H almost surely). Note that φ s,t (ω) := ϕ(t − s, θ s ω) will not be a crude semiflow in general! If φ is a perfection of φ then ϕ(t, ω) := φ 0,t (ω) is a perfection of ϕ. So the perfection problems for ϕ and φ are equivalent in the group-valued case. This is no longer true in case H is just a semigroup.
If we start with a semigroup-valued crude semiflow φ, then φ as defined above is still a semiflow but it need not be indistinguishable from φ (see Example 10). In general the best we can say is that for every fixed s ≥ 0 there exists a P I -null set A s such that φ s,t (ω) = φ s,t (ω) for all t ≥ s and all ω ∈ A s . If we start with a semigroup-valued crude cocycle then we can not even define φ as above due to the possible nonexistence of inverses. So the perfection problems for crude cocycles and crude semiflows are different problems in the general semigroup-valued case.
Here we will be interested in obtaining perfect versions for solution semiflows of SDEs. Therefore we will concentrate on perfection results for crude semiflows (Theorems 3 and 4) which we will apply to solutions of SDEs in Chapter 3. In Theorem 5 we provide the existence of a nice solution semiflow of (1) without requiring Z to have stationary increments. Even though Theorem 5 and its counterpart Proposition 8 for the case of locally Lipschitz coefficients look somewhat "classical" we could not find them in the literature. Corollaries 6 and 9 then show (in particular) that if Z is a helix then (1) generates a RDS in case f is Lipschitz resp. locally Lipschitz.
The perfection problem for crude semigroup-valued cocycles has been solved in certain cases in (Kager (1996)) -even without continuity assumptions. We mention that a number of authors have treated the related perfection problem for multiplicative functionals of Markov processes which are by definition cocycles which take values in the semigroup [0, ∞) with respect to multiplication (Meyer (1972), Walsh (1972), Sharpe (1988), Getoor (1990)). The restriction to the semigroup [0, ∞) simplifies the problem. On the other hand it is complicated by the fact that in the Markov process literature θ is usually only one-sided.

Perfection of a crude semiflow
We will use the following notation: = Lebesgue measure on R I (or its restriction to a subset of R I ) P I * = inner measure associated with P I Q I + = set of nonnegative rationals D I n = space of R I n -valued cadlag functions on [0, ∞) equipped with the topology of uniform convergence on compacts.

Remark:
It is no restriction to assume in (ii) that the exceptional null sets do not depend on t because if they do then by assumption (v) and the Hausdorff property of H has also measure zero for every s ≥ 0.
We claim that φ has all the properties stated in the theorem.
(vii) follows from (5) and the definition of φ.
(viii) follows from (5) and the definition of φ.

Remark:
Contrary to the group-valued case for which a perfection theorem without continuity conditions in the "time" variable was proved in (Scheutzow (1996)) Theorem 3 is wrong if we drop (v) in both the assumption and conclusion. We will give an example in Chapter 4. Note that we made use of (v) twice in the proof: the first time to show M ∈ B([0, ∞)) ⊗ F and then in (6). The crucial step in which we can not avoid (v) completely is (6) -even if we define φ(s, s, ω) differently (see Example 10).
Let us show that (v) can be avoided to prove (4) and (5)  onto the first two components. By the projection theorem (Cohn (1980), Prop. 8.4.4) M is measurable w.r.t. the completion of B([0, ∞)) ⊗ F w.r.t. λ ⊗ P I . So we can find M 1 ⊂ M ⊂ M 2 such that M 1 , M 2 ∈ B([0, ∞)) ⊗ F and λ ⊗ P I (M 2 \ M 1 ) = 0. Clearly M 2 has full measure and therefore M 1 as well. If we define Ω as before but with M 1 instead of M then (4) and (5) follow as before.
In a number of cases some of the assumptions in Theorem 3 will not be satisfied. In particular the continuity assumptions are rather strong in infinite dimensional cases if one uses the usual (strong) topologies on H (even in the deterministic case). We will therefore formulate a version of Theorem 3 which assumes only pathwise right continuity and which may well be applicable for certain infinite dimensional equations.
Theorem 4 Let E be a Hausdorff second countable topological space with E = B(E) and (H, •) a semigroup of maps from E to E such that either E is Polish (i.e. second countable and complete metric) or there exists a countable set D ⊆ E such that h,h ∈ H, h| D =h| D implies h =h. Let (Ω, F , P I , θ) be a QDS and assume that φ : ∆ × Ω → H satisfies (i), (ii) of Theorem 3 and (v') t → φ(s, t, ω, x) is right continuous for every s ≥ 0, ω ∈ Ω, x ∈ E on [s, ∞).

Proof:
Define M as in the proof of Theorem 1. If a set D as in Theorem 4 exists, then So M ∈ B([0, ∞)) ⊗ F as before. If E is Polish then define M 1 as in the previous remark with t replaced by the pair (t, x) ∈ [0, ∞)×E. The rest of the proof is completely analogous to that of Theorem 3 except that we always add an additional argument x ∈ E of φ and limits and the integral are taken for fixed x.

Application to stochastic differential equations
Let m, n ∈ N I and let (Z t ) t≥0 be an m-dimensional semimartingale w.r.t. the stochastic basis (Ω, F , (F t ) t≥0 , P I ) which we assume to satisfy the usual conditions (Protter (1992) )measurable and such that F is the P I -completion of F . We could take F = F here but as soon as we will introduce θ and require that θ be (B([0, ∞)) ⊗ F, F )-measurable then it is a severe restriction of generality to assume that F is complete.
Let f : R I n → R I n×m satisfy a global Lipschitz condition. We will first show that there exists a version of the solution map φ : ∆ × Ω → H of the SDE which satisfies (i), (iii), (iv) and (v) of Theorem 3 if we take H = C n = C( R I n , R I n ) equipped with the compact-open topology (which is the same as the one generated by uniform convergence on compact sets). It is well-known that H is a second countable and metrizable topological semigroup (Dugundji (1966), Chap. XII, Th. 2.2, Th. 5.2, 8.5 ). Then we will assume in addition that Z is a helix and we will show that φ automatically satisfies (ii). By Theorem 3 we get a version φ of φ which is in particular a perfect semiflow. If we define ϕ(t, ω) := φ(0, t, ω), then (Ω, F , P I , θ, ϕ) is an RDS which is generated by (7).
Theorem 5 Let Z be an R I m -valued semimartingale and let f : R I n → R I n×m satisfy a global Lipschitz condition. Then there exists a map φ : ∆ × Ω → H(= C n ) which satisfies (i) and (iii) of Theorem 3 and also (ix) (s, t) → φ(s, t, ω) is cadlag for every ω ∈ Ω, (xiii) If Z has continuous paths then (s, t) → φ(s, t, ω) is continuous for every ω ∈ Ω and φ(s, t, ω) is a homeomorphism from R I n to R I n for every (s, t) ∈ ∆ and ω ∈ Ω.

Remark:
We assumed that the stochastic basis satisfies the usual conditions, so φ will automatically be progressively measurable for fixed s and x.

Proof:
It is known that for a given Lipschitz map f there exists > 0 such that for every semimartingale Z satisfying Z H ∞ < there exists a mapφ : [0, ∞) × Ω → H such that a) φ(t, ω)(x) solves (7) for s = 0 and every x ∈ R I n .
A straightforward contradiction argument shows that b) and c) imply d) and e): Let us further show Fix t ≥ 0, a compact set K ⊆ R I n and an open set G ⊆ R I n . Then a) and c) imply i.e. ω → φ(t, ω) is (F , H)-measurable for every t ≥ 0. Using d) and the fact that H is metrizable we get f).
For the given semimartingale Z (which need not satisfy Z H ∞ < ) we can find a sequence of finite stopping times 0 = T 0 < T 1 < T 2 < · · · such that P I (T k ↑ ∞) = 1 and Z T k − −Z T k−1 H ∞ < for all k ∈ N I , where is chosen as before (depending on f). Here Z T and Z T − stand for the semimartingales Z T t := Z T ∧t and Z T − t := Z t 1I {T >t} +Z T − 1I {T ≤t} resp. (Protter (1992), p. 192/193). We can now apply the previous results to the semimartingales . Denote the corresponding maps byφ k . Defineφ k : R I n × Ω → R I n by where T k−1 ≤ s < T k and T l−1 ≤ t < T l . On the null set on which T k does not converge to ∞ we letφ(s, t, ω) = e H for all (s, t) ∈ ∆. Obviouslyφ takes values in H.
(i) and (xi) follow from the definition of φ.
(ix) follows from d), e) and the definition of φ.
(x) From the definition ofφ, it is clear thatφ (and hence φ) solves (7) for every s ≥ 0, x ∈ R I n .
(xii) is a consequence of (ix). In fact for a function ψ : [0, ∞) → C( R I n , R I n ) the following properties are equivalent: (xiii) By (Kunita (1990), Theorem 4.5.1) we know that if Z has continuous paths there exists a map φ : ∆ × Ω → H with values in the group of homeomorphisms which satisfies (i), (x), (xi) and such that (s, t, x) → φ(s, t, ω)(x) is continuous for every ω ∈ Ω. The last property implies that (s, t) → φ(s, t, ω) is continuous -even in the stronger topology of the group of homeomorphisms (Kunita (1990), p. 115), so (xii) and (xiii) follow. To show (iii) we may have to change φ on a P I -null set as above (without destroying any of the other properties).
Let us now consider the particular case when Z has stationary increments. More precisely, let (Ω, F , (F t ) t≥0 , P I , θ) be a filtered MDS (FMDS) in the sense that (Ω, F , P I , θ) is an MDS, (Ω, F , (F t ) t≥0 , P I ) is a stochastic basis (satisfying the usual conditions) and θ s is (F t+s , F t )-measurable for every s, t ≥ 0. As before F denotes the P I -completion of F .
Corollary 6 Let Z be an R I m -valued semimartingale helix over the FMDS (Ω, F , (F t ) t≥0 , P I , θ) and assume that f : R I n → R I n×m satisfies a global Lipschitz condition. Then the map φ in Theorem 5 satisfies all assumptions of Theorem 3. In particular there exists a solution semiflowφ of (7) i.e. (7) generates an RDS.

Proof:
We only need to check assumption (ii) of Theorem 3. Fix s ≥ 0 and x ∈ R I n . We have a.s.
We want so show that almost surely because this implies that for almost all ω we have φ(0, t − s, θ s ω, x) = φ(s, t, ω, x) for all t ≥ s due to the fact that (7) has a unique solution.
where g(u, ω) := f(φ(0, u − , ω, x)). Observe that the integral on the left side of (8) is welldefined because the integrand is caglad and adapted due to the measurability assumptions on θ. The fact that (8) holds follows by first checking (8) in case g is simple predictable (which follows immediately from the helix property of Z) and then approximating the given g by simple predictable processes uniformly on [0, t − s] in probability. Alternatively we could derive (8) from (Protter (1986), Theorem 3.1 (vi)) which is more general (but has the disadvantage that his set-up is slightly different).
Remark (about the linear case): Let us briefly specialize to the linear case i.e. f ij (x) = a ij x, where a T ij ∈ R I n , i = 1, · · · , n, j = 1, · · · , m. Then the solution of (7) for s = 0 is given by X t = A t x, where A t , t ≥ 0 is an R I n×n -valued solution of where Z is an R I n×n -valued semimartingale whose components are linear combinations of the components of Z and where I is the n × n-identity matrix. Any solution of (9) is called stochastic exponential of Z and is usually denoted by E( Z) t , t ≥ 0. A glance at the proof of Theorem 5 shows that we can choose φ in such a way that φ(s, t, ω) is linear for all (s, t) ∈ ∆, ω ∈ Ω. If Z is a helix then Theorem 5 and (vii) of Theorem 3 show that there exists a perfectionφ of φ which takes values in the subsemigroup of linear maps.
We will now consider the case in which f : R I n → R I n×m is only locally Lipschitz continuous. Until further notice Z is an R I m -valued semimartingale which need not be a helix. Let ∂ be an element not contained in R I n . We follow the idea of the proof of Theorem V.38 in (Protter (1992)) and choose global Lipschitz functions f k , k ∈ N I such that f k (x) = f(x) for all |x| ≤ k. Let φ (k) : ∆ × Ω → H(= C n ) be the corresponding maps of Theorem 5. Then there exists a P I -null set N such that for all k, l ∈ N I , x ∈ R I n , (s, t) ∈ ∆, ω ∈ N as long as sup s≤u<t | φ (k) (s, u, ω)(x) |≤ k ∧ l. Changing the φ (k) on a global set of measure zero if necessary, we can and will assume that N is empty. Now define φ(s, t, ω)(x) := lim k→∞ φ (k) (s, t, ω)(x), if sup k∈IN sup s≤u≤t | φ (k) (s, u, ω)(x) |< ∞ ∂, otherwise (10) Note that the limit exists in R I n and that we could extend the sup over s ≤ u < t instead of s ≤ u ≤ t without changing φ. φ is called strictly conservative if there exists a P I -null set N such that φ(s, t, ω)(x) = ∂ for all (s, t) ∈ ∆, x ∈ R I n , ω ∈ N . In this case we again can and will assume that N is empty.
Clearly φ is well-defined (i.e. the limit exists). Obviously φ is indistinguishable from φ, so b) follows. All other statements either follow from the corresponding properties for the φ (k) or we have proved them already.
Remark on a generalization of Theorems 3 and 4: The group-valued perfection results in (Arnold, Scheutzow (1995)) and (Scheutzow (1996)) are formulated for more general groups than R I as the "time" variable of the underlying MDS, namely for locally compact second countable Hausdorff (LCCB) groups G. Therefore one might ask whether Theorems 3 and 4 admit a similar generalization. This is indeed possible provided that the index set I ⊆ G of the crude semiflow is a measurable subsemigroup of the LCCB group G and that I has strictly positive Haar measure. Of course we need to say what we mean by "right continuous" in this case. If we assume for simplicity that φ is jointly continuous in the first two variables, then the proof of Theorem 3 goes through with a few obvious modifications (replace λ by Haar measure on G and t ≥ 0 by t ∈ I).