ALEA, Lat. Am. J. Probab. Math. Stat. 15, 1183–1213 (2018) DOI: 10.30757/ALEA.v15-44
Local Skorokhod topology on the space of cadlag processes
Mihai Gradinaru and Tristan Haugomat Universit´ede Rennes, CNRS, IRMAR - UMR 6625 F-35000 Rennes, France. E-mail address: { mihai.gradinaru , tristan.haugomat } @univ-rennes1.fr
Abstract. We modify the global Skorokhod topology, on the space of cadlag paths, by localising with respect to space variable, in order to include the eventual explo- sions. The tightness of families of probability measures on the paths space endowed with this local Skorokhod topology is studied and a characterization of Aldous type is obtained. The local and global Skorokhod topologies are compared by using a time change transformation.
1. Introduction
The study of cadlag L´evy-type processes has been an important challenge during the last twenty years. This was due to the fact that phenomena like jumps and unbounded coefficients of characteristic exponent (or symbol) should be taken in consideration in order to get more realistic models. To perform a systematic study of this kind of trajectories one needs, on one hand, to consider the space of cadlag paths with some appropriate topologies, e.g. Skorokhod’s topologies. On the other hand, it was a very useful observation that a unified manner to tackle a lot of questions about large classes of processes is the martingale problem approach. Identifying tightness is an important step when studying sequences of distributions of processes solving associated martingale prob- lems and the Aldous criterion is one of the most employed. The martingale problem approach was used for several situations: diffusion pro- cesses, stochastic differential equations driven by Brownian motion, L´evy processes, Poisson random measures (see, for instance, Stroock, 1975, Stroock and Varadhan, 2006, Kurtz, 2011...). Several technical hypotheses (for instance, entire knowledge of the generator, bounded coefficients hypothesis, assumptions concerning explo- sions...) provide some limitation on the conclusions of certain results, in particular, on convergence results.
Received by the editors February 2nd, 2018; accepted September 10th, 2018. 2010 Mathematics Subject Classification. 60B10, 60J75, 54E70, 54A10. Key words and phrases. cadlag processes, explosion time, local Skorokhod topology, Aldous tightness criterion, time change transformation. 1183 1184 M. Gradinaru and T. Haugomat
The present paper constitutes our first step in studying Markov processes with explosion and, in particular in the martingale problem setting. It contains the study of the so-called local Skorokhod topology and of a time change transformation of cadlag paths. The detailed study of the martingale problem, of L´evy-type processes and several applications will be presented elsewhere (see Gradinaru and Haugomat, 2017b and Gradinaru and Haugomat, 2017a). One of our motivations is that we wonder whether the solution of a well-posed martingale problem is continuous with respect to the initial distribution. The classical approach when one needs to take into consideration the explosion of the solution is to compactify the state space by one point, say ∆, and to endow the cadlag paths space by the Skorokhod topology (see, for instance, Ethier and Kurtz, 1986, Kallenberg, 2002). Unfortunately, this usual topology is not appropriate when we relax hypotheses on the martingale problem setting. The simplest example is provided by the differential equation x˙ = b(t, x ), t> 0, starting from x Rd, t t 0 ∈ where b : R Rd Rd is a locally Lipschitz function. The unique maximal + × → solution exists by setting xt = ∆, after the explosion time. In general, for some t > 0, the mapping x0 xt is not continuous, and in particular x0 x• is not continuous for the usual7→ (global) Skorokhod topology. As an illustration,7→ let us consider x˙ = (1 t)x2, t> 0, x R. t − t 0 ∈ For any initial condition x0, the unique maximal solution is given by
−1 , if x [0, 2), t2 1 ∞ 0 ∈ x = t + before t = 1 1 2/x0, if x0 2, t 2 − x max − − ≥ 0 1+ 1 2/x , if x < 0, p − 0 0 and xt := ∆, after tmax. Indeed, this trajectory isp not continuous with respect to the initial condition in the neighbourhood of x0 = 2. To achieve the continuity of the mapping x0 x• our idea will be to localise the topology on the paths space, not only with7→ respect to the time variable but also with respect to the space variable. More precisely, we need to consider uniform convergence until the exit d time from some compact subset of R+ R . We adapt this idea to cadlag paths by× following a similar approach as in Billings- ley (1999) and we get the local Skorokhod topology which is weaker than the usual (global) Skorokhod topology. Then, we describe the compactness and the tightness in connection with this topology. Furthermore, we state and prove a version of the Aldous criterion, which is an equivalence, as in Rebolledo (1979). As in Ethier and Kurtz (1986), pp. 306-311, we introduce a time change trans- formation of the cadlag path x by the positive continuous function g. Roughly speaking, it is defined by (g x) := x with τ the unique solution starting from 0 · t τt t of the equationτ ˙t = g(xτt ). Another novelty of our paper is the employ of the time change transformation to compare the local Skorokhod topology with the usual (global) Skorokhod topology. Our paper is organised as follows: the following section is mainly devoted to the study of the local Skorokhod topology on spaces of cadlag paths: the main result is a tightness criterion. Properties of the time change mapping, in particular the continuity, and the connection between the local and global Skorokhod topologies Local Skorokhod topology on the space of cadlag processes 1185 are described in Section 3. The last section contains technical proofs, based on local Skorokhod metrics, of results stated in Section 2.
2. Paths spaces
2.1. Local spaces of cadlag paths. Let S be a locally compact Hausdorff space with countable base. This topological feature is equivalent with the fact that S is sep- arable and can be endowed by a metric for which the unit balls are compact, so, S is a Polish space. Take ∆ S, and we will denote by S∆ S the one-point compactification of S, if S is6∈ not compact, or the topological sum⊃ S ∆ , if S is compact (so, ∆ is an isolated point). Clearly, S∆ is a compact Hausdorff⊔{ } space with countable base which could be also endowed with a metric. This latter metric will be used to construct various useful functions, compact and open subsets. For any topological space A and any subset B R, we will denote by C(A, B) ⊂ the set of continuous functions from A to B, and by Cb(A, B) its subset of bounded continuous functions. We will abbreviate C(A) := C(A, R) and Cb(A) := Cb(A, R). All along the paper we will denote C ⋐ A for a subset C which is compactly embedded in A. Similarly, C ⋐ A means either that C is not a subset of A or C is not compactly embedded in A6 . We start with the definition of our spaces of trajectories: Definition 2.1 (Spaces of cadlag paths). Define the space of exploding cadlag paths 0 ξ(x) , ≤ ≤ ∞ D t0 [0, ξ(x)) xt0 = lim xt, exp(S) := x : [0, ξ(x)) S ∀ ∈ t↓t0 . → t0 (0, ξ(x)) xt0− := lim xt exists in S ∀ ∈ t↑t0
When ξ(x) = 0, we assume that x is the empty subset of [0, 0) S. For a path × x from Dexp(S), ξ(x) is its lifetime or explosion time. We identify Dexp(S) with a subset of (S∆)R+ by using the mapping
∆ R+ ( Dexp(S) ֒ (S → with xt := ∆ if t ξ(x). x (xt)t≥0 ≥ 7→ We define the local cadlag space as the subspace
Dloc(S) := x Dexp(S) ξ(x) (0, ) and xs s<ξ(x) ⋐ S imply xξ(x)− exists . ∈ ∈ ∞ { } (2.1) We also introduce the global cadlag space as the subspace of Dloc(S) D(S) := x D (S) ξ(x)= SR+ . { ∈ loc | ∞} ⊂ We will always denote by X the canonical process on Dexp(S), Dloc(S) and D(S) without danger of confusion. We endow each of Dexp(S), Dloc(S) and D(S) with a σ-algebra := σ(Xs, 0 s< ) and a filtration t := σ(Xs, 0 s t). We will always skipF the argument≤X for∞ the explosion timeFξ(X) of the canonical≤ ≤ process. The following result provides a useful class of measurable mappings: Proposition 2.2. For t R , the mapping 0 ∈ + ∆ Dexp(S) [0,t0] S (x,× t) → x 7→ t 1186 M. Gradinaru and T. Haugomat is ([0,t ])-measurable. For t R∗ , the set Ft0 ⊗B 0 0 ∈ + ∆ A := (x, t) Dexp(S) (0,t ] x exists in S ∈ × 0 t− belongs to t0− ((0,t0]) and the mapping F ⊗B A S∆ (x, t) → x 7→ t− is ((0,t ])-measurable. For U an open subset of S and for t R , the set Ft0− ⊗B 0 0 ∈ + 2 B := (x,s,t) Dexp(S) [0,t ] x ⋐ U ∈ × 0 { u}s∧t≤u 2 Corollary 2.3. For any ( t)-stopping time τ0, an open subset of S , h R F D U ∈ C( , +) a continuous function and M : exp(S) [0, ] a τ0 -measurable map, theU mapping → ∞ F t ⋐ τ := inf t τ0 (Xτ0 ,Xs) τ0≤s≤t or h(Xτ0 ,Xs)ds M ≥ |{ } 6 U τ ≥ n Z 0 o is a ( t)-stopping time. In particular, ξ is a stopping time. Furthermore, if U S is anF open subset, ⊂ τ U := inf t 0 X U or X U ξ (2.2) { ≥ | t− 6∈ t 6∈ }≤ is a stopping time. Proof of Corollary 2.3: For each t 0, using Proposition 2.2 it is straightforward to obtain that ≥ 1 if τ0 > t, −t Y := h(Xτ ,Xs)ds if τ0 t and (Xτ ,Xs) τ ≤s≤t ⋐ , τ0 0 ≤ { 0 } 0 U otherwise. ∞R is -measurable. Hence Ft τ t = Y M = Y M τ t , { ≤ } { ≥ } { ≥ }∩{ 0 ≤ }∈Ft so, τ is a ( )-stopping time. Ft Proof of Proposition 2.2: Let d be a complete metric for the topology of S, note that A = q1, q2 [t δ, t) d(xq1 , xq2 ) ε ∗ { ∈ − ⇒ ≤ } ε∈Q δ∈Q∗ Q \+ [+ q1,q2∈\+∩[0,t0) so, A belongs to ((0,t ]). It is clear that for each n N Ft0− ⊗B 0 ∈ A S∆ → (x, t) x t0 nt 7→ n+1 j t0 k Local Skorokhod topology on the space of cadlag processes 1187 is t0− ((0,t0])-measurable, where r denotes the integer part of the real number r.F Letting⊗B n we obtain that (x, t⌊) ⌋ x is ((0,t ])-measurable. The → ∞ 7→ t− Ft0− ⊗B 0 proof is similar for (x, t) x . To prove that B is measurable, let (K ) N be an 7→ t n n∈ increasing sequence of compact subsets of U such that U = n Kn. Then B = (x,s,t) D (S) [0,t ]2 x SK ∈ exp × 0 { u}s∧t≤u ⊗2 N∗ so, B t0− ([0,t0]) . To verify the last part, let us note that for n the mapping∈F from⊗BB C(U) ∈ × sign(t s) n−1 it 1 it (x,s,t,h) − h(x 0 ) s∧t≤ 0 2.2. Local Skorokhod topology. To simplify some statements, in this section we will k consider a metric d on S. To describe the convergence of a sequence (x ) N k∈ ⊂ Dloc(S) for our topology on Dloc(S), we need to introduce the following two spaces: we denote by Λ the space of increasing bijections from R+ to R+, and by Λ Λ the space of increasing bijections λ with λ and λ−1 locally Lipschitz. For λ ⊂ Λ ∈ and t R we denote e ∈ + −1 λ id t := sup λs s = λ id λt . (2.3) k − k 0≤s≤t | − | k − k ˙ ∞ For λ Λ, let λ Lloc(ds) be the density of dλ with respect to the Lebesgue measure.∈ This density∈ is non-negative and locally bounded below, and for t R ∈ + denote e λ λ dλ−1 ˙ ˙ s2 s1 s log λ t := esssup0≤s≤t log λs = sup log − = log . k k k k 0≤s 121-137 from Billingsley (1999), and are postponed to Section 4. Theorem 2.4 (Local Skorokhod topology). There exists a unique Polish topology k on Dloc(S), such that a sequence (x )k∈N converges to x for this topology if and k only if there exists a sequence (λ )k∈N in Λ such that either ξ(x) < and x ⋐ S: λk ξ(xk) for k large enough and • ∞ { s}s<ξ(x) ξ(x) ≤ k k k sup d(xs, x k ) 0, x k ∆, λ id 0, as k , λs λ ξ(x) s<ξ(x) −→ ξ(x) −→ k − k −→ → ∞ 1188 M. Gradinaru and T. Haugomat or ξ(x) = or xs s<ξ(x) ⋐ S: for all t < ξ(x), for k large enough • k k ∞ { } 6 λt < ξ(x ) and k k sup d(xs, xλk ) 0, λ id t 0, as k . s≤t s −→ k − k −→ → ∞ This topology is also described by a similar characterisation with λk Λ and λk ∈ k − id replaced, respectively, by λk Λ and log λ˙ k . Moreover, the Borel σ-algebra k ∈ k k (Dloc(S)) coincides with the σ-algebra . B F e Definition 2.5. The topology on Dloc(S) whose existence is stated in the latter theorem will be called the local Skorokhod topology. The trace topology from Dloc(S) to D(S) will be called the global Skorokhod topology. Remark 2.6. 1) We point out that these topologies do not depend on the metric d of S and this is a consequence of the fact that two metrics on a compact set are uniformly equivalent (cf. Lemma 4.3 below). 2) The convergence conditions of Theorem 2.4 may be summarised as: a sequence (xk)k converges to x for the local Skorokhod topology if and only if there exists a k sequence (λ )k in Λ satisfying that for any t R+ such that xs s We are now interested to characterise the sets of Dloc(S) which are compact and also to obtain a criterion for the tightness of a subset of probability measures in (D (S)). For x D (S), t 0, K S compact and δ > 0, define P loc ∈ exp ≥ ⊂ N N, 0= t0 < i) Consider x Dexp(S). Then x belongs to Dloc(S) if and only if ∈ ′ t 0, K S compact, ωt,K,x(δ) 0. ∀ ≥ ∀ ⊂ −→δ→0 ii) For all t 0, K S compact and δ > 0, the mapping ≥ ⊂ Dloc(S) [0, + ] x → ω′ ∞(δ) 7→ t,K,x is upper semi-continuous. Local Skorokhod topology on the space of cadlag processes 1189 Proof : Suppose that x D (S) and let t 0 be and consider a compact set ∈ loc ≥ K S. There exists T ξ(x) such that (T, xT ) [0,t] K and the limit xT − exists⊂ in S. Let ε> 0 be≤ arbitrary and consider I the6∈ set of× times s T for which there exists a subdivision ≤ 0= t < sup d(xs1 , xs2 ) ε. 0≤i We need to prove that the limit xξ(x)− exists in S. Let y1,y2 be any two limits points of x , as s ξ(x). We will prove that y = y . Let ε> 0 be arbitrary. By s → 1 2 taking t = ξ(x) and K = x in (2.4) there exists a subdivision, { s}s<ξ(x) 0= t < sup d(xs1 , xs2 ) ε. 0≤i Replacing in the latter inequality the two sub-sequences tending toward y1, y2, we can deduce that d(y1,y2) ε, and letting ε 0 we get y1 = y2. ≤ → k k We proceed with the proof of part ii). Let (x )k Dloc(S) be such that x k ⊂ converges to x Dloc(S) and let (λ )k Λ be such in Theorem 2.4. We need to prove that, ∈ ⊂ ′ e ′ lim sup ωt,K,xk (δ) ωt,K,x(δ). k→∞ ≤ We can suppose that ω′ (δ) < . Let ε > 0 be arbitrary and consider a t,K,x ∞ subdivision 0 = t < k k k We deduce that, for k large enough, we have 0 = λt0 < < λtN ξ(x ), k k k k ··· ≤ λ > λ + δ, (λ , x k ) [0,t] K, and moreover, ti+1 ti tN λ tN 6∈ × k k k sup d(x , x ) sup d(xs , xs ) + 2 sup d(xs, x k ) s1 s2 1 2 λs 0≤i Theorem 2.8 (Compact sets of Dloc(S)). For any subset D Dloc(S), D is relatively compact if and only if ⊂ ′ t 0,K S compact, sup ωt,K,x(δ) 0. (2.5) ∀ ≥ ⊂ x∈D −→δ→0 The proof follows the strategy developed in §12 pp. 121-137 from Billingsley (1999) and it is postponed to Section 4. We conclude this section with a version of the Aldous criterion of tightness: Proposition 2.9 (Aldous criterion). Let be a subset of (Dloc(S)). If for all t 0, ε> 0, and an open subset U ⋐ S, weP have: P ≥ 1 inf sup sup P τ1 < τ2 = ξ or d(Xτ ,Xτ ) {τ <ξ} ε 0, (2.6) F ⊂P 1 2 1 P∈P\F τ1≤τ2 ≥ −→δ→0 U τ2≤(τ1+δ)∧t∧τ then is tight. Here the infimum is taken on all finite subsets F and the P ⊂ P supremum is taken on all stopping times τi. Remark 2.10. As in Billingsley (1999), Theorem 16.9, p. 177, an equivalent condi- tion for tightness can be obtained by replacing (2.6) by U 1 inf sup sup P τ < (τ + δ) τ = ξ or d(Xτ ,X(τ+δ)∧τ U ) {τ<ξ} ε 0, F ⊂P P∈P\F τ≤t∧τ U ∧ ≥ −→δ→0 by taking infimum on all finite subsets F and the supremum on all stopping times τ. ⊂ P In fact we will state and prove a version of the of the Aldous criterion, which is a necessary and sufficient condition, similarly as in Rebolledo (1979): Theorem 2.11 (Tightness for Dloc(S)). For any subset (Dloc(S)), the fol- lowing assertions are equivalent: P⊂P (1) is tight, (2) forP all t 0, ε> 0 and K a compact set we have ≥ ′ sup P ωt,K,X (δ) ε 0, P∈P ≥ −→δ→0 Local Skorokhod topology on the space of cadlag processes 1191 (3) for all t 0, ε> 0, and open subset U ⋐ S we have: ≥ α(ε,t,U,δ) := sup sup P(R ε) 0, P∈P τ1≤τ2≤τ3 ≥ −→δ→0 U τ3≤(τ1+δ)∧t∧τ where the supremum is taken on τi stopping times and with d(Xτ1 ,Xτ2 ) d(Xτ2 ,Xτ3 ) if 0 < τ1 < τ2 < τ3 < ξ, d(X ,X ∧) d(X ,X ) if 0 < τ = τ < τ < ξ, τ2− τ2 τ2 τ3 1 2 3 d(X ,X ) ∧ if 0= τ τ < ξ or 0 < τ < τ < τ = ξ, R := τ1 τ2 1 ≤ 2 1 2 3 d(Xτ −,Xτ ) if 0 < τ1 = τ2 < τ3 = ξ, 2 2 0 if τ = ξ or 0 < τ τ = τ , 1 1 ≤ 2 3 if 0= τ1 < τ2 = ξ. ∞ ∆ Remark 2.12. 1) If d is obtained from a metric on S , then if ε < d(∆,U) the expression of α(ε,t,U,δ) may be simplified as follows: α(ε,t,U,δ) = sup sup P(R ε) 0, P∈P τ1≤τ2≤τ3 ≥ −→δ→0 U τ3≤(τ1+δ)∧t∧τ e where the supremum is taken on τi stopping times and with d(X ,X ) d(X ,X ) if 0 < τ < τ , τ1 τ2 ∧ τ2 τ3 1 2 R := d(X ,X ) d(X ,X ) if 0 < τ = τ , τ2− τ2 ∧ τ2 τ3 1 2 d(Xτ1 ,Xτ2 ) if 0 = τ1. 2) It is straightforwarde to verify that a subset D D(S) is relatively compact for ⊂ D(S) if and only if D is relatively compact for Dloc(S) and t 0, x x D, s t ⋐ S. ∀ ≥ { s | ∈ ≤ } Hence we may recover the classical characterisation of compact sets of D(S) and the classical Aldous criterion. Moreover, we may obtain a version of Theorem 2.11 for D(S). 3) The difficult part of Theorem 2.11 is the implication 3 2, and its proof is adapted from the proof of Theorem 16.10 pp. 178-179 from⇒ Billingsley (1999). Roughly speaking the assertion 3 uses ′′ ωx(δ) := sup d(xs1 , xs2 ) d(xs2 , xs3 ), s1≤s2≤s3≤s1+δ ∧ while the Aldous criterion uses ωx(δ) := sup d(xs1 , xs2 ). s1≤s2≤s1+δ The term d(Xτ2−,Xτ2 ) appears because, in contrary to the deterministic case, some stopping time may not be approximate by the left. We refer to the proof of Theorem 12.4 pp. 132-133 from Billingsley (1999) for the relation between ω′′ and ω′. Proof of Theorem 2.11: 2 1 Consider (t ) a sequence of times tending to infinity and (K ) an ⇒ n n≥1 n n≥1 increasing sequence of compact subsets of S such that S = n Kn. Let η> 0 be a real number, and for n 1 define δn such that ≥ S P ′ −1 −n sup ωtn,Kn,X (δn) n 2 η . P∈P ≥ ≤ 1192 M. Gradinaru and T. Haugomat Set D := n N∗, ω′ (δ ) sup P (Dc) 2−nη = η P∈P ≤ nX≥1 so, is tight. 1 3PConsider ε, η two arbitrary positive real numbers. There exists a compact set D⇒ D (S) such that ⊂ loc sup P(Dc) η. P∈P ≤ By Theorem 2.8, there exists δ > 0 such that D ω′ (δ) <ε . ⊂{ t,U,X } Since for all τ τ τ (τ + δ) t τ U we have 1 ≤ 2 ≤ 3 ≤ 1 ∧ ∧ ω′ (δ) <ε R<ε , { t,U,X }⊂{ } we conclude that sup sup P(R ε) η . P∈P τ1≤τ2≤τ3 ≥ ≤ U τ3≤(τ1+δ)∧t∧τ 3 2 For all ε > 0, t 0 and open subset U ⋐ S, up to consider τi := τi (τ1 + δ)⇒ t τ U we have a≥ new expression of α(ε,t,U,δ): ∧ ∧ ∧ U e α(ε,t,U,δ) = sup sup P(R ε, τ3 (τ1 + δ) t τ ) 0. (2.7) P∈P τ1≤τ2≤τ3≤ξ ≥ ≤ ∧ ∧ −→δ→0 Consider ε > 0, t 0 and K a compact subset of S. We need to prove that 0 ≥ ′ inf P(ωt,K,X (δ) <ε0) 1. P∈P −→δ→0 Choose 0 <ε1 <ε0/4 such that U := y S d(y,K) <ε ⋐ S. { ∈ | 1} For n N and ε> 0, define inductively the stopping times (see Corollary 2.3) ∈ τ0 := 0, τ ε := inf s > τ d(X ,X ) d(X ,X ) ε (t + 2) τ U , n n τn s ∨ τn s− ≥ ∧ ∧ ε1 τn+1 := τn , ε It is clear that τn increases to τn+1 when ε increases to ε1. If we choose 0 <ε2 <ε1, then for all P , ∈P ε ε P ε lim sup (Xτn K, τn <ξ, d(Xτn ,Xτn ) ε2, τn t + 1) (2.8) ε→ε1 ∈ ≤ ≤ ε<ε1 ε ε P ε P lim sup Xτn K, τn <ξ, d(Xτn ,Xτn ) ε2, τn t +1 = ( )=0. ≤ ε→ε1 { ∈ ≤ ≤ } ∅ ε<ε1 Local Skorokhod topology on the space of cadlag processes 1193 For all P , δ 1 and 0 <ε<ε1 we have using the expression (2.7) with ∈ P ≤ ε stopping times 0 τ ε τ 1 = τ ≤ 0 ≤ 0 1 ε ε P(X0 K, τ δ)= P(X0 K, Xτ ε B(X0,ε2), τ δ) ∈ 0 ≤ ∈ 0 6∈ 0 ≤ ε ε + P(X0 K, τ <ξ, d(X0,Xτ ε ) <ε2, τ δ) ∈ 0 0 0 ≤ α(ε ,t +2,U,δ) ≤ 2 ε ε + P(X0 K, τ <ξ, d(X0,Xτ ε ) <ε2, τ t + 1) ∈ 0 0 0 ≤ so, letting ε ε , since τ ε τ , by (2.8) we obtain → 1 0 ↑ 1 P(X K, τ δ) α(ε ,t +2,U,δ). (2.9) 0 ∈ 1 ≤ ≤ 2 For all P , δ 1, n N and 0 <ε<ε1 we have also using the expression (2.7) with stopping∈P times≤ τ ∈ τ ε τ and τ τ ε τ ε n ≤ n ≤ n+1 n ≤ n ≤ n+1 P(τ t, X ,X K, τ ε τ δ) n+1 ≤ τn τn+1 ∈ n+1 − n ≤ ε ε P(X K, τ <ξ, d(X ,X ε ) <ε , τ t + 1) ≤ τn ∈ n τn τn 2 n ≤ ε ε + P(Xτ K, τ <ξ, d(Xτ ,Xτ ε ) <ε2, τ t + 1) n+1 ∈ n+1 n+1 n+1 n+1 ≤ + P τ t, X ,X K, d(X ,X ε ) ε , n+1 ≤ τn τn+1 ∈ τn τn ≥ 2 ε2 d(X ε ,X ) , τ τ δ τn τn+1 ≥ 2 n+1 − n ≤ ε2 + P τn+1 t, Xτ ,Xτ K, d(Xτ ,Xτ ε ) ε2, Xτ ε B(Xτ ε , ), ≤ n n+1 ∈ n n ≥ n+1 6∈ n 2 τ ε τ δ n+1 − n ≤ ε ε P(X K, τ <ξ, d(X ,X ε ) ε , τ t + 1) ≤ τn ∈ n τn τn ≤ 2 n ≤ ε ε + P(Xτ K, τ <ξ, d(Xτ ,Xτ ε ) ε2, τ t + 1) n+1 ∈ n+1 n+1 n+1 ≤ n+1 ≤ ε +2α 2 ,t +2,U,δ 2 so, letting ε ε , since τ ε τ , by (2.8) we obtain → 1 n+1 ↑ n+2 ε P(τ t, X ,X K, τ τ δ) 2α 2 ,t +2,U,δ . (2.10) n+1 ≤ τn τn+1 ∈ n+2 − n ≤ ≤ 2 ∗ For all P , δ 1, n N and 0 <ε<ε1 we can write using the expression (2.7) with∈ stopping P ≤ times∈τ τ τ ε n ≤ n ≤ n P(τ t, X K, d(X ,X ) ε , τ ε τ δ) n ≤ τn ∈ τn− τn ≥ 2 n − n ≤ ε ε P(X K, τ <ξ, d(X ,X ε ) <ε , τ t + 1) ≤ τn ∈ n τn τn 2 n ≤ ε + P(τ t, X K, d(X ,X ) ε , X ε B(X ,ε ), τ τ δ) n ≤ τn ∈ τn− τn ≥ 2 τn 6∈ τn 2 n − n ≤ ε ε P(X K, τ <ξ, d(X ,X ε ) ε , τ t +1)+ α (ε ,t +2,U,δ) ≤ τn ∈ n τn τn ≤ 2 n ≤ 2 so, letting ε ε , since τ ε τ , by (2.8) we obtain → 1 n ↑ n+1 P(τn t, Xτn K, d(Xτn−,Xτn ) ε2, τn+1 τn δ) α (ε2,t +2,U,δ) . ≤ ∈ ≥ − ≤ ≤ (2.11) Let m 2N and 0 <δ′ 1 be such that m> 2t/δ′ and denote the event ∈ ≤ A := τ t and n m, X K . { m ≤ ∀ ≤ τn ∈ } 1194 M. Gradinaru and T. Haugomat Then for all 0 i Recalling that ε1 < 4ε0, a straightforward computation gives B ω′ (δ) <ε . m,δ ⊂ t,K,X 0 We conclude that ′ inf P(ωt,K,X (δ) <ε0) 1. P∈P −→δ→0 3. Time change and Skorokhod topologies 3.1. Definition and properties of time change. First we give the definition of the time change mapping (see also §6.1 pp. 306-311 from Ethier and Kurtz, 1986, §V.26 pp. 175-177 from Rogers and Williams, 2000). Definition 3.1 (Time Change). Let us introduce C6=0(S, R ) := g : S R g =0 is closed and g is continuous on g =0 , + { → + | { } { 6 }} and for g C6=0(S, R ), x D (S) and t [0, ] we denote ∈ + ∈ exp ∈ ∞ t du {g6=0} g g g , if t [0, τ (x)], τ (x) := inf s 0 A (x) t , where A (x) := 0 g(xu) ∈ t { ≥ | s ≥ } t otherwise. (∞R (3.1) Local Skorokhod topology on the space of cadlag processes 1195 For g C6=0(S, R ), we define a time change mapping, which is -measurable, ∈ + F g X : Dexp(S) Dexp(S) · x → g x, 7→ · as follows: for t R+ ∈ g X g if t A g , X g exists and belongs to g =0 , τ∞− τ∞ τ∞− (g X)t := ≥ { } (3.2) · X g otherwise. τt 6=0 For g C (S, R+) and P (Dexp(S)), we also define g P the pushforward of P by x∈ g x. ∈ P · 7→ · The fact that this mapping is measurable will be proved in the next section. g Remark 3.2. Let us stress that, by using Corollary 2.3, τt is a stopping time. In particular, the following stopping time will play a crucial role: τ g = τ {g6=0} := ξ inf t 0 g(X ) g(X )=0 . (3.3) ∞ ∧ { ≥ | t− ∧ t } The time of explosion of g X is given by · g if τ∞ < ξ or Xξ− exists and belongs to g =0 , ξ(g X)= ∞g { } · A otherwise. ξ g Roughly speaking, g X is given by (g X) := X g where t τ is the solution t τt t g · g· 7→ ofτ ˙ = g(X g ), on the time interval [0, τ ). t τt ∞ Proposition 3.3. (1) For U S an open subset, by identifying ⊂ C(U, R )= g C6=0(S, R ) g =0 U and g is continuous on U , + ∈ + |{ 6 }⊂ the time change mapping C(U, R+) Dexp(S) Dexp(S) (g,× x) → g x, 7→ · is measurable between (C(U, R+)) and . 6=0 B ⊗F F (2) If g1,g2 C (S, R+) and x Dexp(S), then g1 (g2 x) = (g1g2) x. ∈ ∈ 6=0 · · · (3) If g is bounded and belongs to C (S, R+), and x D(S), then g x D(S). (4) Define ∈ · ∈ C6=0(S, R ) := g C6=0(S, R ) K S compact, g(K) is bounded . + ∈ + | ∀ ⊂ 6=0 If g C (S,R ) and x Dloc(S), then g x Dloc(S). e ∈ + ∈ · ∈ Proof : The first point is straightforward by using Proposition 2.2, while the second e point is a direct consequence of the time change definition and, in particular, of the first part of (3.2). The third point can be easily deduced because, ∞ ds ∞ ds ξ(g x) = . · ≥ g(x ) ≥ g ∞ Z0 s Z0 k k To prove the fourth point we suppose that ξ(g x) < and g xs s<ξ(g·x) ⋐ S. Then, x = g x so, · ∞ { · } { s}s<ξ(x) { · s}s<ξ(g·x) ξ(x) ds ξ(x) > ξ(g x)= . ∞ · g(x ) ≥ g Z0 s k k{xs}s<ξ(x) Hence ξ(x) < and so, g x = x exists. ∞ · ξ(g·x)− ξ(x)− 1196 M. Gradinaru and T. Haugomat 6=0 Remark 3.4. It can be proved that if g C (S, R+) and (Pa)a∈S is a strong Markov family, then (g P ) is a Markov∈ family. Furthermore, if (P ) is a · a a∈S a a∈S t+-strong Markov family, then (g Pa)a∈S is a t+-strong Markov family. We will Fnot use these properties here hence· we skip theF proofs of these statements. Another interesting fact is the following: Theorem 3.5 (Continuity of the time change). For couples (g, x) C6=0(S, R ) ∈ + × Dloc(S) consider the following two conditions: g τ∞(x)+ ds e τ g (x) < ξ(x) implies = , (3.4) ∞ g(x ) ∞ Z0 s and g A g (x) < , x g exists in S and g(x g )=0 imply x g = x g . τ∞(x) ∞ τ∞(x)− τ∞(x)− τ∞(x)− τ∞(x) (3.5) Introduce the set 6=0 B := (g, x) C (S, R ) Dloc(S) conditions (3.4) and (3.5) hold . (3.6) tc ∈ + × Then the time change e 6=0 C (S, R+) Dloc(S) Dloc(S) (g, x×) → g x 7→ · e 6=0 R is continuous on Btc when we endow respectively C (S, +) with the topology of uniform convergence on compact sets and Dloc(S) with the local Skorokhod topology. In particular e R∗ D D C(S, +) loc(S) loc(S) (g,× x) → g x 7→ · is continuous for the topologies of uniform convergence on compact sets and local R∗ R Skorokhod topology. Here and elsewhere we denote by + = + 0 the set of positive real numbers. \{ } Remark 3.6. 1) It is not difficult to prove that Btc is the continuity set. 6=0 R 2) If (g, x) Btc and h C (S, +) is such that h =0 = g =0 and h Cg for a constant∈ C R , then∈ (h, x) B . { } { } ≤ ∈ + ∈ tc 3) More generally, let B bee the set of (g, x) C6=0(S, R ) D (S) such that 0 ∈ + × loc g g g τ∞(x) < t 0, xτ∞(x)+t = xτ∞(x), ∞ ⇒ ∀ ≥ e x g exists in S and g(x g )=0 x g = x g . τ∞(x)− τ∞(x)− ⇒ τ∞(x)− τ∞(x) Then (g,g x) (g, x) C6=0(S, R ) D (S) B B . (3.7) · ∈ + × loc ⊂ 0 ⊂ tc n o To simplify the proof of the theorem we use a technical result containing a e k construction of a sequence of bi-Lipschitz bijections (λ )k useful when proving the convergence. Before stating this result let us note that, for any (g, x) 6=0 R D g ⋐ ∈ C (S, +) loc(S) and any t τ∞(x) such that xs s k N k i) there exists a sequence (λ )k Λ such that, for k large enough λ g ∈ At (x) ≤ k ξ(gk x ) and · e k k k log λ˙ 0, sup d(g x ,g x k ) 0, g x k g x g , as k . v k λ k λ g At (x) k k → g · · v → · A (x) → · → ∞ v g ii) Moreover, if τ g (x) < ξ(x) and τ∞(x)+ ds = , (λk) may be chosen ∞ 0 g(xs) k k ∞ k such that for any v 0 and k large enough λ g < ξ(gk x ) and ≥ R At (x)+v · k g lim sup sup d gk x k , xs ∞ =0. λw t≤s≤τ (x) k→∞ Ag (x)≤w≤Ag(x)+v · { } t t We postpone the proof of the lemma, and we give the proof of the continuity of time change: Proof of Theorem 3.5: We remark first that B = B B B B , tc 1 ∪ 2 ∪ 3 ∪ 4 with g g ⋐ B1 := A g (x)= or xs ∞ S , τ∞(x) ∞ { }s<τ (x) 6 g B := nτ (x)= ξ(x) < and x g o⋐ g =0 , 2 ∞ ∞ { s}s<τ∞(x) { 6 } g + τ∞(x) g ds g g g B3 := τ (x) < ξ(x), x ∞ = x ∞ , A g (x) < and = , ∞ τ (x)− τ (x) τ∞(x) ∞ g(x ) ∞ ( Z0 s ) g + τ∞(x) g ds B := τ (x) < ξ(x), g(x g ) = 0 and = . 4 ∞ τ∞(x)− 6 g(x ) ∞ ( Z0 s ) k 6=0 k Let x, x Dloc(S) and g,gk C (S, R+) be such that (gk, x ) converge to (g, x) and (g, x)∈ B. We need to prove∈ that ∈ e k Dloc(S) gk x g x, (3.8) · −−−−→k→∞ · and we will decompose the proof with respect to values of i such that (g, x) B . ∈ i g If (g, x) B1, we use the first part of Lemma 3.7 for all t < τ∞(x). We • ∈ g g obtain that At (x) < ξ(g x). Since At (x) tends to ξ(g x), when t tends to g · · τ∞(x), by a diagonal extraction procedure we deduce (3.8). If (g, x) B2, it suffices to apply the first part of Lemma 3.7 to t := ξ(x) • g ∈ and At (x)= ξ(g x). · g If (g, x) B3, let t < τ∞(x) be. Then, by the third part of Lemma 3.7 • ∈ k k there exists λ Λ such that, for any v 0, for k large enough, λ g < At (x)+v k ∈ ≥ ξ(gk x ) and · e ˙ k k g g log λ 0, lim sup sup d(g xw,gk xλk ) 2d xτ∞(x), xs t≤s≤τ∞(x) . k kk −→→∞ g · · w ≤ { } k→∞ w≤At (x)+v g Since x is continuous at τ∞(x), we conclude by a diagonal extraction pro- cedure, by letting t tends to τ g (x) and v . ∞ → ∞ 1198 M. Gradinaru and T. Haugomat g If (g, x) B4, let t = τ∞(x) be. By the second part of Lemma 3.7 there • k∈ k exists λ Λ such that, for any v 0, for k large enough λ g < ∈ ≥ At (x)+v k ξ(gk x ), and · e ˙ k k log λ 0, sup d(g xw,gk xλk ) 0. k kk −→→∞ g · · w k−→→∞ w≤At (x)+v We conclude by a diagonal extraction procedure and letting v . → ∞ Proof of Lemma 3.7: Let λk Λ be as in Theorem 2.4 and to simplify notations define, for s 0 ∈ ≥ g e e g τs := τs (x), As := As (x), k gk k k gk k τs := τs (x ), As := As (x ), ⋐ k and u := At. Since τu = t ξ(x) and xs s Using Fatou’s lemma k + k (λτ∞ ) τ∞+ ˙ τ∞+ e ds λs ds ds lim inf k = lim inf k = k→∞ k g (x ) k→∞ g (x ) ≥ g(x ) ∞ λu k s t k λk t s Z Z e s Z k k k k k so, lim supk→∞ τλk λτ∞ 0. Moreover, for k large enough, τλk τλk = λt , u+v − ≤ u+v ≥ u k k so, λ < ξ(gk x ) and u+v · e e k lim sup sup d gk xλk , xs t≤s≤τ∞ =0. k→∞ u≤w≤u+v · w { } Local Skorokhod topology on the space of cadlag processes 1199 3.2. Connection between local and global Skorokhod topologies. Generally to take into account the explosion, one considers processes in D(S∆), the set of cadlag processes described in Definition 2.1, associated to the space S∆, and endowed with the global Skorokhod topology (see Definition 2.5). More precisely, the set of cadlag paths with values in S∆ is given by t 0, x = lim x , and D ∆ ∆ R+ t s↓t s (S )= x (S ) ∀ ≥ ∆ . ∈ t> 0, xt− := lims↑t xs exists in S ∀ k ∆ A sequence (x )k in D(S ) converges to x for the global Skorokhod topology if and k only if there exists a sequence (λ )k of increasing homeomorphisms on R+ such that k k t 0, lim sup d(xs, xλk ) = 0 and lim λ id t =0. ∀ ≥ k→∞ s≤t s k→∞ k − k In this section we give the connection between D(S∆) with the global Skorokhod topology and Dloc(S) with the local Skorokhod topology. We first identify these two measurable subspaces D (S) D(S∆)= x D (S) 0 < ξ(x) < x exist in S∆ loc ∩ ∈ loc ∞ ⇒ ξ(x)− D ∆ S = x (S ) t τ , xt = ∆ . ∈ ∀ ≥ We can summarise our trajectories spaces by D(S) D (S) D(S∆) D (S) D (S). ⊂ loc ∩ ⊂ loc ⊂ exp D(S∩∆) ∆ Hence Dloc(S) D(S ) will be endowed with two topologies, the local topology ∩ ∆ from Dloc(S) and the global topology from D(S ). ∆ Remark 3.8. 1) On Dloc(S) D(S ) the trace topology from Dloc(S) is weaker than the trace topology from D(S∩∆). Eventually, these two topologies coincide on D(S). Indeed, this is clear using a metric d on S∆ and the characterisations of topologies given in Remark 2.6. The result in Corollary 3.10 below is a converse sentence of the present remark. 2) If x D (S) D(S∆) then g x is well-defined in D (S) D(S∆) for ∈ loc ∩ · loc ∩ g C (S, R∗ ) g C6=0(S∆, R ) g(∆) = 0 . ∈ b + ⊂ ∈ + | We deduce from Theorem 3.5 and the third point of Remark 3.6 that the mapping R∗ D D ∆ D D ∆ Cb(S, +) loc(S) (S ) loc(S) (S ) ×(g, x) ∩ → g ∩x 7→ · is continuous between the topology of the uniform convergence and the global Sko- rokhod topology. The following result is stated in a very general form because it will be useful when studying, for instance, the martingale problems. ∆ Proposition 3.9 (Connection between Dloc(S) and D(S )). Let E be an arbitrary locally compact Hausdorff space with countable base and consider P : E (Dloc(S)) a → P P 7→ a 1200 M. Gradinaru and T. Haugomat a weakly continuous mapping for the local Skorokhod topology. Then for any open subset U of S, there exists g C(S, R ) such that g =0 = U, for all a E ∈ + { 6 } ∈ g P (0 <ξ< X exists in U)=1, · a ∞ ⇒ ξ− and the application g P : E ( 0 <ξ< Xξ− exists in U ) · a → P { ∞ ⇒g P } 7→ · a is weakly continuous for the global Skorokhod topology from D(S∆). Before giving the proof of Proposition 3.9 we point out a direct application: we take E := N , U = S and a sequence of Dirac probability measures P = δ k , ∪{∞} k x P∞ = δx. Then we deduce from Proposition 3.9 the following: 1 2 Corollary 3.10 (Another description of Dloc(S)). Let x, x , x ,... Dloc(S). k ∈ Then the sequence x converges to x in Dloc(S), as k , if and only if there exists g C(S, R∗ ) such that g x, g x1,g x2,... D(S→∆), ∞ and g xk converges to ∈ + · · · ∈ · g x in D(S∆), as k . · → ∞ We proceed with the proof of Proposition 3.9 and, firstly we state a important result which will be our main tool: Lemma 3.11. Let D be a compact subset of Dloc(S) and U be an open subset of S. There exists g C(S, R ) such that: ∈ + i) g =0 = U, ii) {for6 all }x D, (g, x) is in the set B given by (3.6) in Theorem 3.5 and ∈ tc g x 0 <ξ< X exists in U . · ∈ ∞ ⇒ ξ− ∆ iii) the trace topologies of Dloc(S) and D(S ) coincide on g x x D . { · | ∈ } Furthermore, if g C(S, R+) satisfies i)-iii) and if h C(S, R+) is such that h = 0 = U and ∈h Cg with a non-negative constant∈C, then h also satisfies {i)-iii).6 } ≤ Proof of Proposition 3.9: Let (Kn)n∈N∗ be an increasing sequence of compact sub- ∗ set of E such that E = Kn, then Pa is tight, for all n N . So, there n { }a∈Kn ∈ exist subsets D D (S) whiche are compactse for the topology of D (S), and n locS loc such that ⊂ e 1 sup P (Dc ) . a n ≤ n a∈Kn e For any n N∗, consider g satisfying i)-iii) of Lemma 3.11 associated to the ∈ n compact set Dn. It is no difficult to see that there exists g C(S, R+) such that ∗ ∈ g =0 = U and for all n N , g Cngn for non-negative constants Cn. Hence, g{ satisfies6 } i)-iii) for all D , ∈n N∗.≤ Hence, for all a E n ∈ ∈ g P (0 <ξ< X exists in U) P D =1. · a ∞ ⇒ ξ− ≥ a n N∗ n[∈ Let ak,a E such that ak a. For n large enough ak k Kn. Then, if F ∈ k−→→∞ { } ⊂ is a subset of 0 <ξ< Xξ− exists in U which is closed for the topology of { ∞ ⇒ } e Local Skorokhod topology on the space of cadlag processes 1201 D(S∆), lim sup g Pak (F ) g Pa(F ) k→∞ · − · 1 lim sup Pak (X Dn, g X F ) Pa(X Dn, g X F )+ . ≤ k→∞ ∈ · ∈ − ∈ · ∈ n But thanks to iii) in Lemma 3.11, X D , g X F is a subset of D (S) which { ∈ n · ∈ } loc is closed for the topology of Dloc(S). Hence, by using the Portmanteau theorem (see for instance Theorem 2.1 from Billingsley, 1999, p. 16) lim sup Pak (X Dn, g X F ) Pa(X Dn, g X F ) k→∞ ∈ · ∈ ≤ ∈ · ∈ and so, letting n , → ∞ lim sup g Pak (F ) g Pa(F ). k→∞ · ≤ · By using the Portmanteau theorem, the proof of the proposition is complete, except for the proof of Lemma 3.11. Proof of Lemma 3.11: Let d be a metric on S∆ and denote K := a S d(a,S∆ U) 2−n . n ∈ \ ≥ By using Theorem 2.8,there exists a sequence (η ) (0, 1)N decreasing to 0 such n n that ∈ ′ −n−2 sup ω2n,B(∆,2−n−2)c,x(ηn) < 2 . (3.9) x∈D ∆ −n c Moreover, there exists g C(S , [0, 1]) such that g =0 = U and g|Kn 2 ηn. For instance, we can define∈ { 6 } ≤ −n ∆ ∆ g(a) := sup (2 ηn) d(a,S Kn) , a S . n≥0 ∧ \ ∈ Let x D be. We consider the following two situations: ∈ g If τ (x) < and x g is not a compact of U, take m N such • ∞ ∞ { s}s<τ∞(x) ∈ that 2m τ g (x), denote ≥ ∞ ◦ t := min s 0 x K < τ g (x) ≥ s 6∈ m+1 ∞ ◦ and let n m be such that xt Kn+2 Kn+1. Using (3.9) there exist R ≥ ∈ g \ t1,t2 + such that t1 t g g g N If τ∞(x) < ξ(x) and g(xτ∞(x)−) = 0, then g(xτ∞(x)) = 0. Let m • m g 6 −m−2 c ∈ be such that 2 τ (x) and x g B(∆, 2 ) . Using (3.9), ≥ ∞ { s}s≤τ∞(x) ⊂ 1202 M. Gradinaru and T. Haugomat R g there exist t1,t2 + such that t1 τ∞(x) g τ∞(x)+ ds = . g(x ) ∞ Z0 s Hence, we obtain that (g, x) Btc and g x 0 <ξ< Xξ− exists in U and ii) is verified. ∈ · ∈ { ∞ ⇒ } We proceed by proving iii). Thanks to Remark 3.8, to get the equivalence of the topologies it is enough to prove that if xk, x D are such that g xk g x for the topology from D (S) and ξ(g x) < , then∈ the convergence· also→ holds· for the loc · ∞ topology from D(S∆). Let λk Λ be such that ∈ k ˙ k sup d(g xs,g x k ) 0, log λ 0, as k . λs e ξ(g·x) s≤ξ(g·x) · · −→ k k −→ → ∞ ˙ k k We may suppose that λs = 0, for s ξ(g x). Denote tk := λξ(g·x) and choose ≥ ◦ · m m N be such that g xs s<ξ(g·x) ⋐ Km and ξ(g x) < 2 . Then, for k large ∈ { ◦ · } · enough g xk ⋐ K , g xk K and t < 2m. { · s }s k k −m−2 sup d(g xs,g x k ) sup d(g xs,g x k )+2 , λs λs s≤ξ(g·x)+2m · · ≤ s≤ξ(g·x) · · so letting m goes to the infinity we obtain that g xk converge to g x for the global Skorokhod topology from D(S∆). Hence, the proof· of iii) is done. · Finally, to prove the last part of the lemma let g C(S, R+) be such that i)-iii) are satisfied and let h C(S, R ) be such that h∈= 0 = U and h Cg with ∈ + { 6 } ≤ a non-negative constant C. Thanks to Remark 3.6, (h, x) belongs to the set Btc given by (3.6), it is also clear that h x 0 <ξ< X exists in U . We · ∈ { ∞ ⇒ ξ− } have that h C (U, R∗ ), so using (3.7) for S and S∆, the bijection g ∈ b + g x x D h x x D { · | ∈ } → { · | ∈ } x h x 7→ g · ∆ is continuous for the topology of Dloc(S), but also of D(S ). But since g x x D is compact, this application is bi-continuous, and we obtain the result{ .· Now| ∈ the} proof of lemma is complete. Local Skorokhod topology on the space of cadlag processes 1203 4. Proofs of main results on local Skorokhod metrics In this section we will prove Theorem 2.4 and Theorem 2.8, by following the strategy developed in §12, pp. 121-137from Billingsley (1999). To construct metrics on Dloc(S), we will consider a metric d on S. To begin with, we define two families of pseudo-metrics: Lemma 4.1 (Skorokhod metrics). For 0 t < and K S a compact subset, ≤ ∞ ⊂ the following two expressions on Dexp(S): 1 2 1 2 ρt,K(x , x ) := inf sup d(x , x ) λ id t i s λs 1 ti≤ξ(x ) s Proof : Let us perform the proof for ρt,K , the proof being similar for ρt,K . The 1 2 3 non-trivial part is the triangle inequality. Let x , x , x Dexp(S) and ε > 0 be 1 2 3 1∈ 2 then there are t1 ξ(x ), t2, t2 ξ(x e), t3 ξ(x ) and λ Λ, λ Λ such that ≤ ≤ ≤ ∈ ∈ 1 2 1 2 ˙ 1 1 ρt,K(x , x )+ ε sup d(x , x 1 ) log λ t λ id t sb λs b 1 1 e e ≥ s e 2 3 2 3 2 2 b b b ˙ ρt,K(x , x )+ ε sup d(x , x 2 ) log λ λ id ≥ s λs ∨k kt2 ∨k − kt2 s max d(x ,K ) (t ti)+ i . b ti ti<ξ(x ) b e b ∨ i∈{2,3} ∧ − b b 1 −1 2 2 1 Define t2 := t2 t2, t1 := (λ ) , t3 := λ and λ := λ λ . Thenb t2 t2 b ∧ ◦ 1 3 1 2 2 3 1 2 b b sup d(x , x ) sup d(bx , x ) + sup d(x , x ), s λs b s λs s λs b b ≤ s b b b b1 2 b 1 2 log λ˙ log λ˙ t + log λ˙ , λ id λ id t + λ id . k kt1 ≤k k 1 k kt2 k − kt1 ≤k − k 1 k − kt2 b b b b b Moreover,b for instance, if t = t , then t Lemma 4.2. Take x, y Dloc(S), t 0 and a compact subset K S, if ρ (x, y) 1 then ∈ ≥ ⊂ t,K ≤ 9 ρ (x, y) 6 ρ (x, y) ω′ ρ (x, y) . t,K ≤ · t,K ∨ t,K,x t,K q q e 1204 M. Gradinaru and T. Haugomat Proof : Let ε > 0 be arbitrary. There exist µ Λ and T 0 such that T ξ(x), µ ξ(y) and ∈ ≥ ≤ T ≤ sup d(xs,yµs ) µ id T ρt,K (x, y)+ ε, s r log(1 + r) | | for r < 1. | |≤ 1 r | | − | | we deduce µt µt 2ρ (x, y)+2ε log λ˙ = sup log i+1 − i t,K . k k t t ≤ δ 2ρ (x, y) 2ε 0≤i Finally, by taking δ := ρ (x, y) we have for ρ (x, y) 1 t,K t,K ≤ 9 2ρ (x, y) ρ (x, y) 2ρ (px, y)+ ω′ (δ) (2δ) t,K t,K ≤ t,K t,K,x ∨ ∨ δ 2ρ (x, y) − t,K 2 e ρ (x, y)+ ω′ ρ (x, y) 2 ρ (x, y) 6 ρ (x, y) ≤ 3 t,K t,K,x t,K ∨ t,K ∨ t,K q q q q 6 ρ (x, y) ω′ ρ (x, y) . ≤ · t,K ∨ t,K,x t,K At thisq level it can be pointed q out that we obtain the definition of the local Skorokhod topology. Indeed, by using Proposition 2.7, Lemma 4.2 and the fact that ρ ρ , the two families of pseudo-metrics (ρ ) and (ρ ) define t,K ≤ t,K t,K t,K t,K t,K the same topology on Dloc(S), the local Skorokhod topology. If (Kn)n∈eN is an exhaustive sequence of compact sets of S, then thee mapping D (S)2 R loc + (4.1) (x, y) → 2−nρ (x, y) 1 7→ n∈N n,Kn ∧ is a metric for the local Skorokhod topology.P By using a diagonal extraction pro- k cedure, it is not difficult to prove that a sequencee (x )k converges to x for this k topology if and only if there exists a sequence (λ )k in Λ such that ⋐ k k either ξ(x) < and xs s<ξ(x) S: for k large enough λξ(x) ξ(x ) and • ∞ { } e ≤ k k ˙ k sup d(xs, x k ) 0, x k ∆, log λ 0, as k , λs λ ξ(x) s<ξ(x) −→ ξ(x) −→ k k −→ → ∞ or ξ(x) = or xs s<ξ(x) ⋐ S: for all t < ξ(x), for k large enough • k k ∞ { } 6 λt < ξ(x ) and k ˙ k sup d(xs, xλk ) 0, log λ t 0, as k . s≤t s −→ k k −→ → ∞ The local Skorokhod topology can be described by a similar characterisation with λk Λ replaced by λk Λ and respectively, log λ˙ k replaced by λ id . The fact∈ that the local Skorokhod∈ topology doesk not dependk on the disk tance− kd is a consequencee of the following lemma, which states essentially that two metrics on a compact set are uniformly equivalent: Lemma 4.3. Let T be a set and x, xk ST be such that x ⋐ S, then ∈ { t}t∈T k sup d(xt, xt ) 0, as k , t∈T −→ → ∞ if and only if U S2 open subset containing (y,y) , k k k , t T, (x , xk) U. ∀ ⊂ { }y∈S ∃ 0 ∀ ≥ 0 ∀ ∈ t t ∈ T So the topology of the uniform convergence on x S xt t∈T ⋐ S depends only of the topology of S. ∈ { } Proof : Suppose that sup d(x , xk) 0 as k and take an open subset t∈T t t −→ → ∞ U S2 containing (y,y) . By compactness there exists ε> 0 such that ⊂ { }y∈S (y ,y ) S2 y x , d(y ,y ) <ε U, 1 2 ∈ | 1 ∈{ t}t 1 2 ⊂ n k o so, for k large enough and for all t, (xt, xt ) U. To get the converse property it ∈ 2 suffices to consider, for each ε> 0, the open set Uε := (y1,y2) S d(y1,y2) <ε which clearly contains (y,y) . ∈ | { }y∈S 1206 M. Gradinaru and T. Haugomat In the next lemma we discuss the completeness : k Lemma 4.4. Suppose that (S, d) is complete. Then any sequence (x )k N ∈ (Dloc(S)) satisfying k1 k2 t 0, K S compact, ρt,K (x , x ) 0, ∀ ≥ ∀ ⊂ k1,k−→2→∞ admits a limit for the local Skorokhod topology.e The proof of this lemma follows from the same reasoning as the proof of the triangular inequality, the proof of Theorem 12.2 pp. 128-129 from Billingsley (1999) and the proof of Theorem 5.6 pp. 121-122 from Ethier and Kurtz (1986). k Proof : It suffices to prove that (x )k have a converging subsequence. By taking, possibly, a subsequence we can suppose that t 0, K S compact, ρ (xk, xk+1) < . (4.2) ∀ ≥ ∀ ⊂ t,K ∞ kX≥0 We split our proof in five steps. e k k Step 1: we construct a sequence (λ )k Λ. Let µ Λ and tk 0 be such that for all t 0 and K S compact, we have⊂ for k large∈ enough ≥ ≥ ⊂ e k e k e k+1 tk ξ(x ), µ ξ(x ), ≤ tk ≤ k k+1 k k e k k+1 sup d(xs , x k ) logµ ˙ t µ id t 2ρt,K (x , x ), µs ∨k k k ∨k e − k k ≤ s µℓ id < , logµ ˙ ℓ < , k − ktℓ ∞ k ktℓ ∞ Xℓ≥0 e Xℓ≥0 e Local Skorokhod topology on the space of cadlag processes 1207 k+i−1 k so, the restriction to [0,tk] of continuous functions µ µ converges uniformly to a continuous function. Set ◦···◦ λk := lim µk+i−1 µk(s), if s t , s i→∞ ◦···◦ ≤ k λ˙ k := 1, if s t . s ≥ k Clearly for s t , λk = λk+1 µk(s). We have ≤ k s ◦ k k+i−1 k ℓ λ id sup µ µ id tk µ id tℓ < , (4.5) k − k≤ i≥0 k ◦···◦ − k ≤ k − k ∞ Xℓ≥k e λk λk d ˙ k s2 s1 k+i−1 k log λ = sup log − sup log (µ µ (s)) tk k k 0≤s1 logµ ˙ ℓ < , (4.6) ≤ k k tℓ ∞ Xℓ≥k e so, λk Λ. ∈ D k k+1 k+1 Step 2: we construct a path x exp(S). For all k 0: λ = λ k λ and tk µ tk+1 ∈ ≥ tk ≤ moreovere for all 0 k k ≤ 1 ≤ 2 k2−1 k1 k2 k1 k2 ℓ ℓ+1 sup d x − , x − = sup d x , x k − k sup d(x , x ℓ ). k1 1 k2 1 s µ 1 1◦···◦µ 1 (s) s µ k (λ )s (λ )s s It is clear that sℓ >tk1 and sℓ is a decreasing sequence converging to tk1 so, the set ℓ> 0 sℓ k1+ℓ−1 k1 c k1 k1+ℓ k1 +ℓ c i i+1 d(xs ,K ) d(xs , x )+ d(x ,K ) < sup d(xs, x i )+ ε< 2ε. ℓ ℓ tk +ℓ tk +ℓ µs ≤ 1 1 s m−1 e k c ℓ ℓ+1 ℓ 1 k d(x ,K ) (t tk)+ tk<ξ(x ) sup d(x , x ℓ ) µ id tk s µs tℓ ∧ − ≤ s It is clear that λk = ξ(x) if and only if M = . If M < tk ∞ ∞ c k c M k M d(xλ ,K ) (t λtk )+ = d(xλ ,K ) (t λtM )+ tk ∧ − tM ∧ − d(xM+1,Kc) (t µM ) + sup d(xℓ , xℓ+1) µℓ id µM tM + s µℓ tℓ ≤ tM ∧ − s ∨k − k ℓ>M s k ℓ ℓ+1 ρt,K(x , x) 2 ρt,K (x , x ) 0. ≤ k−→→∞ Xℓ≥k e e Step 5: we prove that x Dloc(x). Suppose that ξ(x) < and that x ⋐ S. ∈ ∞ { s}s<ξ(x) Let ε > 0 be such that K := y S d(y, x ) ε is compact and set ∈ { s}s<ξ(x) ≤ t = ξ(x)+ ε. By using (4.9) we have, for k large enough, c k ℓ ℓ+1 k 1 k d(xλ ,K ) (t λtk )+ λ <ξ(x) 2 ρt,K (x , x ) < ε. tk ∧ − tk ≤ Xℓ≥k k e Then ξ(x)= λtk and we deduce that k ℓ ℓ+1 sup d x(λk )−1s, xs 2 ρt,K (x , x ) 0 ≤ k−→→∞ s<ξ(x) ℓ≥k X e k and that the limit xξ(x)− exists in S. Therefore, x Dloc(S) and x converges to x for the local Skorokhod topology. ∈ To prove the separability and the criterion of compactness we will use the fol- lowing technical result: Lemma 4.5. Let R S, δ > 0 and N N be. Define ⊂ ∈ := x Dloc(S) ξ(x) Nδ, k N, x is a constant in R ∆ ER,δ,N ∈ ≤ ∀ ∈ ∪{ } on [kδ, (k + 1)δ) . Then for any x Dloc(S) ∈ ′ ρNδ,K(x, R,δ,N ) sup d(a, R)+ ωNδ,K,x(δ) δ. E ≤ a∈K ∨ Proof : For arbitrary ε> 0 , there exist 0= t < such that for all 0 i ρNδ,K(x, R,δ,N ) ρNδ,K(x, x) sup d(xs, xλs ) λ id E ≤ ≤ s sup d(a,e R)+ ω′ (δ)+2e ε δ, ≤ t,K,x ∨ a∈K so, letting ε 0 we obtain the result. → The separability is an easy consequence: Lemma 4.6. The local Skorokhod topology on Dloc(S) is separable. Proof : Let R be a countable dense part of S and introduce the countable set E := R, 1 ,N . E n N∗ n,N[∈ Consider x Dloc(S), t 0, K S a compact set and let ε > 0 be. We choose n N∗ such∈ that n−1 ε≥and ω′⊂ (n−1) ε and set N := nt . We can write ∈ ≤ t+1,K,x ≤ ⌈ ⌉ ′ 1 1 ρt,K (x, R, 1 ,N ) ρ N ,K(x, R, 1 ,N ) sup d(a, R)+ ω N ,K,x( ) E n ≤ n E n ≤ n n ∨ n a∈K 1 1 ω′ ( ) ε. ≤ t+1,K,x n ∨ n ≤ We deduce that E is dense, hence Dloc(S) is separable. We have now all the ingredients to prove the characterisation of the compactness: Proof of Theorem 2.8: First, notice that, similarly as in the proof of Lemma 4.3, the condition (2.5) is equivalent to: for all t 0, all compact subset K S and all open subset U S2 containing the diagonal≥ (y,y) y S , there exists⊂ δ > 0 such that for all x⊂ D there exist 0 = t < sup d(a, R) + sup ω′ (δ) δ η ≤ Nδ,K,x ∨ ≤ a∈K x∈D and by using Lemma 4.2, ′ sup ρt,K (x, R,δ,N ) 6 sup ρt,K (x, R,δ,N ) ωt,K,x ρt,K (x, R,δ,N ) ε. x∈D E ≤ x∈D E ∨ E ≤ q q Sincee R,δ,N is finite we can conclude that D is relatively compact. ToE prove the converse sentence, thanks to the first part of Proposition 2.7 it is k k enough to prove that if x , x Dloc(S) with x converging to x, then for all t 0 and all compact subset K S∈, ≥ ⊂ ′ lim sup ωt,K,xk (δ) 0. k→∞ −→δ→0 This is a direct consequence of Proposition 2.7. Let us stress that although we cite Theorem 2.4 in the proof of Proposition 2.7, in reality we only need the sequential characterisation of the convergence. We close this section by the study of the Borel σ-algebra (D (S)). B loc Lemma 4.7. Borel σ-algebra (Dloc(S)) coincides with . B F ∆ k Proof : Let f C(S ) and 0 a Conversely, since the space is separable, it is enough to prove that for each 0 x Dloc(S), t 0, K S compact and ε > 0 there exists V Dloc(S), - measurable,∈ such≥ that ⊂ ⊂ F x D (S) ρ (x, x0) ε V x D (S) ρ (x, x0) 3ε . (4.10) ∈ loc t,K ≤ ⊂ ⊂ ∈ loc t,K ≤ Proposition 2.7 allows to get the existence of 0 = t0 < 0= t0 tM ξ(x),M N such that: ∃ ≤···≤ ≤ 0 ≤ 0 i M, ti ti ε ∀ ≤ ≤ | − |≤ 0 V := x D (S) 0 i < M, t [ti,ti+1), d(xt, xt0 ) 2ε , loc ∀ ≤ ∀ ∈ i ≤ ∈ 0 c 0 1 d(xt0 ,K ) (t tM )+ t0 <ξ(x0) 2ε M ∧ − M ≤ d(x ,Kc) d(x ,Kc) (t t ) 1 3ε tM − tM M + tM <ξ(x) ∧ ∧ − ≤ it is straightforward to obtain (4.10). Since δ > 0, 0= q0 qM < ξ(x) δ, M N such that: ∀ ∃ ≤···≤ 0 − ≤ 0 i M, qi ti ε + δ ∀ ≤ ≤ | − |≤ 0 V = x D (S) 0 i < M, q [qi + δ, qi+1 δ], d(xq, xt0 ) 2ε , loc ∀ ≤ ∀ ∈ − i ≤ ∈ 0 c 0 1 d(xt0 ,K ) (t tM )+ t0 <ξ(x0) 2ε M ∧ − M ≤ d(x ,Kc) (t q ) 1 3ε + δ qM M + qM <ξ(x) ∧ − ≤ where q, q and δ are chosen to be rational, V belongs . The proof is now i complete. F Acknowledgements The authors are grateful to the anonymous referee for his/her careful reading of the manuscript and useful suggestions. References P. Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition (1999). ISBN 0-471-19745-9. MR1700749. S. N. Ethier and T. G. Kurtz. Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York (1986). ISBN 0-471-08186-8. MR838085. M. Gradinaru and T. Haugomat. 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