The Riesz representation theorem and weak${}^*$ compactness of semimartingales

We show that the sequential closure of a family of probability measures on the canonical space of c{\`a}dl{\`a}g paths satisfying Stricker's uniform tightness condition is a weak${}^*$ compact set of semimartingale measures in the pairing of the Riesz representation theorem under topological assumptions on the path space. Similar results are obtained for quasi- and supermartingales under analogous conditions. In particular, we give a full characterization of the strongest topology on the Skorokhod space for which these results are true.


Introduction
The Riesz representation theorem states that the operation of integration defines a one-to-one correspondence between the continuous linear functionals on the bounded continuous functions and the Radon measures on a topological space. On the Skorokhod space, it provides a locally convex way of constructing all càdlàg stochastic processes on the canonical space as tight probability measures. On conceptual level, any criteria that characterizes a certain object should give rise to some kind of compactness when applied uniformly to a family of objects. We relate Stricker's uniform tightness condition of semimartingales to the weakc ompactness in the pairing of the Riesz representation theorem on the canonical space of càdlàg paths.
Weak topologies are rich in terms of convergent subsequences and have found various applications in studying convergence of financial markets Prigent [Pri03], time series analysis in econometrics Chan and Zhang [CZ10], stochastic optimal control Kurtz and Stockbridge [KS01], Bahlali and Gherbal [BGM11], Tan and Touzi [TT13], and martingale optimal transport, as introduced by Beiglböck, Henry-Labordère and Penkner [BHLP13], and extended for continuous time parameter by Dolinsky and Soner [DS15], and Guo, Tan and Touzi [GTT17]. We aim to provide a functional analytic framework that unifies and elaborates these existing results and allows to extend the analysis beyond the convergence of sequences. In particular, the framework allows to study the non-sequential compactness of families of semimartingales. The question of compactness arises naturally in the context of convex conjugate duality on functions and measures on the càdlàg path space. Many problems in quantitative risk management can be embedded in this framework as convex risk measures and their conjugates Föllmer and Schied [FS11]. The classical example is the minimal superhedging cost of a derivative contract over a convex set hedging positions. We recall the connection between the compactness and the superhedging duality that yields model-independent price bounds for derivative contracts as originally observed by Beiglböck, Henry-Labordère and Penkner [BHLP13], and extended to the continuous time by Dolinsky and Soner [DS15], and Guo, Tan and Touzi [GTT17].
The objective of the paper is to provide a weak˚compactness result for càdlàg semimartingales under the most general topological assumption on the path space. Our main contribution is to unify the previous results on the weak convergence of semimartingales and provide an easy method for constructing weak˚compact sets of semimartingales on the canonical space of càdlàg paths. We also give examples of such sets and show that the examples are consistent with earlier results for Banach spaces of stochastic processes defined over a common probability space.
We characterize the strongest topology on the Skorokhod space for which our main result is true. A natural candidate is Jakubowski's S-topology, due to its tightness criteria. However, it is an open problem whether the S-topology is sufficiently regular. We address the problem of regularity by introducing a new weak topology on the Skorokhod space that has the same continuous functions as the S-topology, suitable compact sets and additionally satisfies a strong separation axiom. The topological space is perfectly normal (T 6 ) in comparison to the Hausdorff property (T 2 ) that has been verified for the S-topology. The topology is obtained from Jakubowski's S-topology as a result of a standard regularization method that appears already in the classical works of Alexandroff [Ale43] and Knowles [Kno65]. Our contribution is to carefully show that the important properties of the S-topology are preserved in the regularization.
The rest of the paper is organized as follows. In Section 2, we give rigorous definitions of semimartingale measures and related notions on the canonical space of càdlàg paths. We also provide a brief introduction to the aforementioned Riesz representation theorem that is the basis of our approach. The main results, examples and financial motivation are given in Section 3. The proofs of the main results and the required auxiliary results are provided in Section 4. In Section 5, we characterize the strongest topology on the Skorokhod space for which the results of the two previous sections are true. Some definitions and technical results are omitted in the main part of the article and are gathered in Appendix A.

Conventions and notations
Throughout, the comparatives 'weaker' and 'stronger' should be understood in the wide sense 'weaker or equally strong' and 'stronger or equally strong', respectively. We say that two topologies are 'comparable', if one is stronger than another.
We fix the following notations. N denotes the family of natural numbers and N 0 :" t0u Y N. R (resp. R`) denotes the family of real (resp. non-negative real) numbers and R :" R Y t˘8u.
D, DpIq and DpI; R d q denote the R d -valued càdlàg functions on I.
VpIq denotes the functions of finite variation on I. CpXq (resp. C b pXq) denotes the family of continuous (resp. bounded continuous) functions on a topological space X; e.g., X " D.
UpDq (resp. U b pDq) denotes the family of upper semi-continuous (resp. bounded upper semi-continuous) functions on D.
B b pDq (resp. B b pDq) denotes the family of Borel (resp. bounded Borel) functions on D.
B 0 pDq denotes the family of bounded Borel functions vanishing at infinity on D.
M t pDq denotes the family of Radon measures of finite variation on the Skorokhod space D.
M τ pDq denotes the family of τ -additive Borel measures of finite variation on the Skorokhod space D.
M σ pDq denotes the family of σ-additive Borel measures of finite variation on the Skorokhod space D.
If M t pDq " M τ pDq " M σ pDq, then we denote the all three families by MpDq. M`pDq denotes the family of non-negative elements of MpDq.
PpDq (resp. PpRq) denotes the family of all probability measures on the Skorokhod space D (resp. the Euclidean space R d ).
KpDq, BpDq and BapDq denote the family of compact, Borel and Baire sets on the Skorokhod space D, respectively. Similarly, BpR d q denotes the Borel sets on the Euclidean space R d .
For a process X : ΩˆI Ñ R d and a subset J Ă I, we write X J for the restriction of X on J i.e. X J : ΩˆJ Ñ R d . In the case of a singleton J " ttu, t P I, we suppress the dependence on the set J and simply write X t for the value of X at t, whence X t : Ω Ñ R.
where the supremum is taken over all finite partitions π of I. E Q rf s :" ş f dQ denotes the integral, for a measurable function f : Ω Ñ R and a probability measure Q on pΩ, F q.
}f } L p pQq :" pE Q r|f | p sq 1{p , p ě 1, denotes the L p -norm, for a measurable function f : Ω Ñ R d and a probability measure Q on pΩ, F q. }X} L p,8 pQq :" }}X} 8 } L p pQq , for X : ΩˆI Ñ R d and a probability measure Q on pΩ, F q.
EpQq denotes the family of elementary predictable processes bounded by 1, for a probability measure Q on pΩ, F q. pH ‚ Xq :" H 0 X 0`ş HdX denotes the stochastic integral, for a probability measure Q on pΩ, F q; the dependence on Q is omitted in this notation. }X} E p pQq :" sup HPEpQq }pH ‚ Xq} L p pQq , p ě 1, for some probability measure Q on pΩ, F q.
N a,b π denotes the number of upcrossings of an interval ra, bs w.r.t. π. N a,b " sup π N a,b π denotes the number of upcrossings of an interval ra, bs. ∆X :" X t´Xt´d enotes the jump of X at t. rωs t denotes the restriction of ω on r0, ts.

Terminology
We provide some frequently used terminology. Standard literature references for general topology and topological measure theory are [Eng77] and [Bog07].
Let X be a topological space. A subset Z Ă X is called: Compact, if every cover of this set by open sets contains a finite subcover.
Relatively compact, if this set is contained in a compact set. Sequentially compact, if every infinite sequence of its elements contains a subsequence converging to an element of Z.
Relatively sequentially compact, if every infinite sequence of its elements contains a subsequence converging in X.
Remark 1.1. In contrast to the case of a metric space, in a general topological space, neither does compactness imply sequential compactness nor the other way around.
The closure cl Z of Z is the set of all points x P X such that every neighborhood of x contains at least one point of Z.
The sequential closure rZs seq of Z is the set of all points x P X for which there is a sequence in Z that converges to x.
Remark 1.2. In contrast to the case of a metric space, in a general topological space, the sequential closure of a set is not necessarily a sequentially closed set.
All topological spaces considered are Hausdorff (T 2 ) and a Hausdorff space X is called: , if for every point x P X and every closed set Z in X not containing x, there exists disjoint open sets U and V such that x P U and Z Ă V .
Completely regular (T 3 1 {2 ), if for every point x P X and every closed set Z in X not containing x, there exists a continuous function f : X Ñ r0, 1s such that f pxq " 1 and f pzq " 0 for all z P Z.
Perfectly normal (T 6 ), if every closed set Z Ă X has the form Z " f´1p0q for some continuous function f on X.
Paracompact, if every open cover of X has an open refinement that is locally finite.
k-space, if the set Z Ă X is closed in X provided that the intersection of Z with any compact subspace K of the space X is closed in K.
Sequential space, if every sequentially closed set is closed.
Fréchet-Urysohn space, if every subspace is a sequential space. Polish space, if the space is homeomorphic to a complete separable metric space.
Lusin space, if the space is the image of a complete separable metric space under a continuous one-to-one mapping.
Souslin space, if the space is the image of a complete separable metric space under a continuous mapping.
Radon space, if every Borel measure on the space is a Radon measure. Perfect space, if every Borel measure on the space is perfect i.e. for every Borel measurable function f for every Borel measure Q the set f pXq contains a Borel set B for which Qrf´1pBqs " f pXq Angelic space, if every set Z Ă X with the property that every infinite sequence of its elements has a limit point in X, possesses also the following properties: Z is relatively compact and each point in the closure of Z is the limit of some sequence in Z. Remark 1.3. For closed subspaces, all these properties are hereditary, meaning that, if the space has the property, then a closed subspace endowed with the relative topology has the property as well. So, all discussion on these properties generalizes as such for relative topologies on closed sets.

Càdlàg semimartingales as linear functionals
In this preliminary section, we define the canonical space for càdlàg semimartingales, and related measures and continuous linear functionals.

Canonical space of càdlàg paths
We fix I to denote a usual time index set of a stochastic process, i.e., I :" r0, T s for 0 ă T ď 8 or I :" r0, 8q. The Skorokhod space DpI; R d q, d P N, with the domain I consists of R d -valued càdlàg functions ω on I that admit a limit ωpt´q from left, for every t ą 0, and are continuous from right, ωptq " ωpt`q, for every t ă T . The space Dpr0, 8s; R d q is regarded as a product space Dpr0, 8q; R d qˆR d ; see Appendix A.1. We write ω " pω 1 , . . . , ω d q, for ω P DpI; R d q, if ωptq " pω 1 ptq, . . . , ω d ptqq, for every t P I. We denote by X the canonical process of DpI; R d q, i.e., X t pωq " ωptq, for all pt, ωq P IˆDpI; R d q. We write X i for each coordinate processes of the canonical process X, for i ď d.
We endow the Skorokhod space DpI; R d q, d P N, with the right-continuous version F t :" Ź εą0 F o t`ε of the raw, i.e., unaugmented, canonical filtration F o t :" σpX s : s ď tq generated by the canonical process X of DpI; R d q.
Remark 2.1. The right-continuous version of the raw canonical filtration is needed in the proof of Proposition 4.15. Alternatively, we could use the universal completion of the raw canonical filtration; see Proposition 4.3 (b).
A stochastic process is understood as a probability measure on the filtered canonical space`DpI; R d q, F T , pF t q tPI˘, where F T :" Ž tPI F t and T " sup tPI t P p0, 8s. The family of all probability measures, i.e. càdlàg processes, on pDpI; R d q, F T q is denoted by PpDpI; R d q, F T q and two elements of PpDpI; R d q, F T q are identified as usual, i.e., P " Q, if (and only if) one has P rF s " QrF s, for all F P F T ; cf. Section 4.1.3.

Semimartingales on the Skorokhod space
We recall some basic concepts of semimartingale theory in the present setting. All semimartingales are assumed càdlàg. We adapt the terminology of Dolinsky and Soner [DS15], and Guo, Tan and Touzi [GTT17] and call a probability measure Q on pDpI; R d q, F T q a martingale measure, if the canonical process X is a martingale on`DpI; R d q, F T , pF t q tPI , Q˘, i.e., if X s " E Q rX t | F s s, for all s ă t in I, including t " 8 in the case that I " r0, 8s. Remark that X is a martingale on r0, 8s if and only if X is a uniformly integrable martingale on r0, 8q; see e.g. [Pha09, Theorem 1.1.2. (2)]. We say that the canonical process X is L p -bounded, for some p ě 1, on pDpI; R d q, F T , Qq, if sup tPI }X t } L p pQq ă 8.
(Special ) semimartingale and supermartingale measures are defined similarly to martingale measures. On`DpI; R d q, F T , pF t q tPI , Q˘, for a fixed probability measure Q, let EpQq denote the family of elementary predictable integrands, i.e., the family of adapted càglàd processes of the form where n P N, where The condition (UT) was introduced by Stricker in [Str85]. By the classical result of Bichteler, Dellacherie and Mokobodzki, a probability measure Q oǹ DpI; R d q, F T˘i s an pF t q tPI -semimartingale measure if and only if Q " tQu satisfies the condition (UT). The family of process (1) generates the predictable σ-algebra and the condition (UT) is sometimes called the predictable uniform tightness condition (P-UT); see e.g. [HWY92,Thm. 3.21]. Remark that, for semimartingale measures, no integrability condition is imposed on X 0 , i.e., the localization of the local martingale in a canonical semimartingale decomposition is understood in the sense of [HWY92, Def. 7.1]; cf. [JS87, Rem. 6.3]. We say that a semimartingale measure Q is of class H p , if, on`DpI; R d q, pF t q tPI , F T , Q˘, the canonical process X decomposes to a (local) martingale M and a (predictable) finite variation process A, A 0 " 0, such that X " M`A and }X} H p pQq " }}M } 8`} A} V } L p pQq ă 8.
(2) Every semimartingale of class H p , for some p ě 1, is a special semimartingale.
To obtain compact statements for quasi-and supermartingales, we introduce two conditions. The first condition is The second condition is the same condition, but the L 1 -boundedness is strengthened to the uniform integrability of the negative parts, for every t P I, i.e., Q satisfies (UB) and lim cÑ8 sup QPQ E Q rXt ½ tXt ącu s " 0, @t P I.

(UI)
The uniform integrability in (UI) yields the convergence of the first moments that preserves the supermartingale property; see Proposition 4.13. If we insist that t i n " t in (1), then the second supremum in (UB) is attained, by choosing for which the value of the integral is equal to the pF t q tPI -conditional variation of X i on r0, ts, where the supremum is taken over all partitions 0 ď t i 0 ď t i 1 ď¨¨¨ď t i n " t, n P N; see e.g. (4) The first implication is obvious. The second implication follows from Lemma 2.2.
Lemma 2.2. There exists a constant b ą 0 such that, for any Q P P pDpIq, F T q, H P EpQq and c ą 0, we have where the right-hand side is possibly infinite.
The inequality (5) is well-known, but we provide the proof for the convenience of the reader in Appendix A.2.
A family Q Ă P`DpI; R d q, F T˘i s called J 1 -tight, if it is exhausted by a sequence of J 1 -compact sets; see Appendix A.1.4. Following the classical terminology [HWY92,Definition 15.48], we say that a family Q Ă P`DpI; R d q, F Tȋ s C-tight, if it is J 1 -tight and satisfies sup QPQ Q " sup sďt |∆X s | ą c  " 0, @t P I, @c ą 0.
The paths of the canonical process X of DpI; R d q lie on CpI; R d q Q-almost surely if and only if Q " tQu is C-tight on DpI; R d q. An analogous assertion is true for the Hölder-continuity. The Markov property is not preserved by the convergence of finite dimensional distributions, but a stronger property is needed; cf. [Low09, Subsection 2.3]. Following Lowther [Low09], we say that a probability measure Q P P`DpI; R d q, F T˘i s Lipschitz-Markov, if, for every s, t P I such that s ă t, for every bounded Lipschitz continuous function g : R d Ñ R with a Lipschitz constant Lpgq ď 1, there exists a bounded Lipschitz continuous function f : R d Ñ R, with a Lipschitz constant Lpf q ď 1, such that f pX s q " E Q rgpX t q | F s s Q-a.s..
The Lipschitz-Markov property is indeed stronger than the Markov property; consider the sequence of functions g n pxq "´n _ |x|^n, n P N, on R d . The Lipschitz-Markov property was introduced by Kellerer in [Kel72].

The Riesz representation on the canonical space
The Riesz representation theorem for the laws of D-valued random variables, i.e. stochastic processes, requires topological assumptions on the canonical space.
Assumption 2.3. The Skorokhod space D is endowed with a topology, under which D is a completely regular Radon space and the Borel σ-algebra coincides with the canonical σ-algebra.
The strict topology β 0 on C b pDq is a locally convex topology generated by the family of seminorms The collection of finite intersections of the sets V g,ε :" tf P C b pDq : p g pf q ă εu, g P B 0 pDq, ε ą 0, forms a local basis at the origin for the topology β 0 . The linear space of all β 0 -continuous linear functionals on C b pDq is isomorphic to the linear space MpDq of all countable additive measures of finite total variation on D. More precisely, under Assumption 2.3, we have the following Riesz representation theorem on the Skorokhod space.
Lemma 2.4. Assume that the Skorokhod space D satisfies Assumption 2.3. Then, any µ P MpDq induces a β 0 -continuous linear functional on C b pDq by u µ pf q :" and any β 0 -continuous linear functional on C b pDq is of the form (7) for some unique µ P MpDq, and the one-to-one correspondence µ Ø u µ defined by (7) is linear.
For the elements of PpDq Ă M`pDq, we write The weak˚topology on MpDq is a locally convex topology generated by the family of seminorms We write µ α Ñ w˚µ for a net pµ α q αPA with a directed set A in MpDq converging in the weak˚topology to µ P MpDq, i.e., if ż f dµ α Ñ ż f dµ, @f P C b pDq.
In the classical case of a metric space D, the weak˚topology on the nonnegative orthant M`pDq is metrizable, i.e., the family of seminorms (8) can be replaced with a single metric and consequently it is sufficient to consider sequences pµ n q nPN in (9) that define a topology; see e.g. [Bog07,Section 8.3]. Thus, the weak˚topology coincides with the classical notion of the topology of weak convergence widely used in probability theory, that is, the convergence for probability measures pQ n q nPN and Q on a metric space D. Following Sentilles [Sen72], we write "weak˚" in the place of "weak" to distinguish the topology from the weak topology on the bounded continuous functions in the pairing of Lemma 2.4.

Background
The classical Riesz representation theorem is stated as a Banach space result for bounded continuous functions vanishing at infinity on a locally compact space. The strict topology β 0 , introduced by Buck in [Buc58] to locally compact spaces, gives up the Banach space structure, but allows to relax the assumption that the bounded continuous functions are vanishing at infinity. Further observations in the 70's by Giles [Gil71] and Hoffman-Jorgensen [HJ72] lead to a generalization of the Riesz representation theorem for completely regular spaces; locally compact spaces are completely regular. The weak˚topology was thoroughly studied by Sentilles in [Sen72] in the case of a general completely regular space. A streamlined proof for Lemma 2.4 can be found, e.g., in the book of Jarchow [Jar81]. The proof relies on the fact that on a completely regular space every continuous function admits a unique continuous extension to the Stone-Čech compactification of the space. The fact that the underlying topological space is completely regular (T 3 1 {2 ) is also necessary for the Riesz representation theorem in the sense that the separation axiom cannot be relaxed to a weaker one as there exists examples of regular (T 3 ) spaces on which every continuous function is a constant and on such space the Riesz representation theorem cannot be true; see [Her65]. However, in our setting it suffices to assume that the space is regular; see Section 4.1.

Main results and examples
A stochastic process is regarded as a probability measure on the canonical space and the family of all probability measures (processes) on the canonical space is endowed with the weak˚topology (8) of the Riesz representation theorem (7). We impose the following assumption on the canonical space.
Assumption 3.1. The Skorokhod space is endowed with a regular topology that is weaker than Jakubowski's S-topology but stronger than the Meyer-Zheng topology (M Z).
The S˚-topology, introduced in Section 5, meets the previous requirements and is arguably the strongest topology on the Skorokhod space for which the results are true. Indeed, see Theorem 5.5 and Remark 5.6.

Motivated by the analysis of a problem in finance
This work was initiated by investigations in robust pricing of derivative contracts of Guo, Tan and Touzi [GTT17, Lemma 3.7] and the current author together with Cheridito, Prömel and Soner [CKPS20, Corollary 6.7]. In a parallel work [CKPS20], we describe a general superhedging-sublinear-pricing paradigma and its relation to the compactness studied in the present work.
In our application, the canonical process X presents the value of the underlying asset that can consist of liquid options or common stocks. The objective is to determine the price for a derivative contact ξ that is a function of X i.e. ξ : D Ñ R. Every viable pricing model Q P PpDq for a class of F T -measurable derivative contracts DpF T q on the underlying asset X must satisfy where Φpξq denotes the greatest lower bound for the initial capital requirement at time t " 0 of all portfolios that produce a value greater or equal to ξ at time t " T , for every possible realization of the underlying X. Indeed, otherwise it is possible to make a sure profit by creating a portfolio that consists of a short position on a derivative contract ξ P DpF T q and a long position on a portfolio that produce a value greater or equal to ξ. On an efficient market, this should not be possible; cf. the seminal work by Black and Scholes [BS73,Abstract]. On the other hand, for ξ P DpF T q, if the least upper bound Vpξq for the expected value of ξ over all viable pricing models is taken to be the lower bound for the price of ξ, then one may ask does this lower bound coincide with the upper bound given by the superhedging price given as Φpξq, i.e., one seeks sufficient conditions for the equality Vpξq " Φpξq, @ξ P DpF T q.
In the case of an increasing, sub-linear Φ, it turns out that the necessary and sufficient condition for the two values to be equal for DpF T q " C b pDq is the lower β 0 -semicontinuity of Φ on C b pDq. The question whether the set of viable market models is weak˚compact then arises naturally. Indeed, the weak˚compactness allows to extended the duality correspondence immediately for DpF T q " U b pDq and is also a sufficient condition for the duality to hold on DpF T q " B b pDq, under another continuity assumption on Φ, namely, the upper σ-order semicontinuity of Φ on B b pDq, provided that the underlying space D is perfectly normal (T 6 ). These extension criteria are classical in measure transport; consult e.g. the seminal work by Strassen [Str65].
Though, no probabilistic assumption on the stock dynamics is made a priori, it is reasonable to restrict to the class of semimartingale measures that form the largest class of stock price models for which it is impossible to make unbounded profits (or losses) by selling and buying the stock in a non-anticipative manner. Indeed, this is the statement of the Bicheteler-Dellacherie-Mokobodzki theorem. Further, if one allows non-anticipative trading of the underlying asset without transaction costs and no interest rate, then all viable pricing models are martingale measures on the canonical space of càdlàg paths. If there is static positions available on the market, they translate to additional half-space constraints on the viable martingale measures. This is the so-called martingale optimal transport problem studied e.g. in the aforementioned works of [BHLP13], [DS15], [GTT17], [CKPS20].

Main results
The following Theorem 3.2 is our main result that together with its corollaries and an auxiliary lemma provides an easy method of constructing weaks ets of semimartingale measures. The statement regarding sequential compactness in Theorem 3.2 refines the classical results of Meyer and Zheng [MZ84], Stricker [Str85] and Jakubowski [Jak97b] for semimartingale measures, i.e., for semimartingales on the canonical space. The statement about (non-sequential) compactness is, to the best of our knowledge, a new result.
The proofs are postponed to Section 4.3.
Theorem 3.2. Let S be a family of semimartingale measures satisfying the condition (UT). Under Assumption 3.1, the set rSs seq is a weak˚compact and sequentially weak˚compact set of semimartingale measures.  By combining one or both of the assertions of the following Lemma 3.5 with Theorem 3.2, or one of its corollaries, one obtains compact sets of continuous and Markov semimartingales, quasimartingales and supermartingales.
Lemma 3.5. Let P be a family of probability measures on the Skorokhod space. Under Assumption 3.1, we have the following: (a) If the set P is C-tight, then the set rPs seq consists of continuous processes.
(b) If each measure in the set P is Lipschitz-Markov, then the set rPs seq consists of Lipschitz-Markov processes.

Examples
The following Example 3.6, essentially an observation made by Guo, Tan and Touzi [GTT17, Lemma 3.7], was our original motivation to study weak˚com-pactness in the present setting. In Example 3.6, we allow an infinite time horizon i.e. the index set I " r0, T s for some T P p0, 8s.
Example 3.6. Let M u be the family of u niformly integrable (L 1 -bounded) martingale measures and let P be a weak˚compact subset of PpR d q, that is, the family of probability measures on R d . Then, the set M u P " tQ P M u : Q˝X´1 T P Pu is weak˚compact and sequentially weak˚compact.
Proof. We adapt the proof of [GTT17, Lemma 3.7]. For a ą 0, we have are weak˚compact and sequentially weak˚compact, for 1 ă p ă 8.
Proof. An increasing continuous function y Þ Ñ y p composed with a lower semicontinuous function y " }ω} 8 is lower semicontinuous, see Lemma A.12, and non-negative, so, by [Bog07, Pro. 8.9.8.], the functional }X} L p,8 pQq is lower semicontinuous in the weak˚topology. Thus, the set M p r is weak˚closed, for r ą 0 and p ą 1. The L p -boundedness, for p ą 1, implies that the set M p r satisfies the condition (UI), for r ą 0 and p ą 1, so, the set M p r is weak˚compact and sequentially weak˚compact, for r ą 0 and p ą 1; cf. Example 3.6.
Assume that a probability measure Q is fixed and p ą 1. Then the Hardy space of L p pQq-bounded (equivalence classes of indistinguishable) càdlàg martingales M p pQq :" M p pDpI; Rq, F T , pF t q tPI , Qq can be identified with the Lebesgue space L p pQq :" L p pDpI; Rq, F T , Qq. Indeed, there exists a linear one-to-one correspondence between the (uniformly integrable) L p pQq-bounded martingales on r0, T s and the random variables X T of L p pQq, p ą 1, as each X T P L p pQq defines a Q-a.s. unique càdlàg L p pQq-bounded martingale on r0, T s via and vice versa, each such càdlàg martingale has Q-a.s. unique terminal value X T P L p pQq. By Doob's L p -maximal inequality, the L p,8 pQq-norm on M p pQq and the L p pQq-norm on L p pQq are equivalent, for p ą 1. In the case p " 1, there is no one-to-one correspondence, but we only have M 1 pQq Ă L 1 pQq, for I " r0, 8q. Further discussion on this one-to-one correspondence can be found e.g. in [DM78, B.VII.64]. Let us mention [Str81, Theorem 3], where the one-to-one correspondence is used in the construction of semimartingale decomposition for quasimartingales, that is a central step in the classical proofs for the Bichteler-Dellacherie-Mokobodzki theorem.
Due to the one-to-one correspondence, the compactness follows from the classical results for Banach spaces. For p ą 1, the space L p pQq is a reflexive Banach space, so, the (sequential) weak˚compactness of the sets (10) follows from the Banach-Alaoglu theorem in conjunction with the Eberlein-Šmulian theorem; see [Str85]. The Dunford-Pettis theorem states that a uniformly integrable subset of a non-reflexive Banach space L 1 pQq is relatively sequentially compact in the weak topology, but the random variables of L 1 pQq are not in one-to-one correspondence with neither the family of L 1,8 pQq-bounded, nor L 1 pQq-bounded martingales, for I " r0, 8q.
Example 3.8. Let H p denote the family of H p -semimartingale measures. Then, the sets S p r :" tQ P H p : are weak˚compact and sequentially weak˚compact, for 1 ď p ă 8.
Proof. The sets S p r , r ą 0, p ě 1, satisfy the condition (UB), so, by Corollary 3.3, the sets rS p r s seq are weak˚compact and sequentially weak˚compact sets of quasimartingales. Moreover, for any sequence pQ n q nPN in S p r converging in the weak˚topology to some Q, we have . Now, let Q P rS p r s seq and assume that we are given an elementary predictable integrand H " pH i q d i"1 , i.e., an element of EpQq, see (1), such that each H i functions is homeomorphic to a closed subset of DpI; R d q; cf. Corollary A.11. Thus, by Lusin's theorem in conjunction with Tietze's extension theorem, for as n Ñ 8; see e.g. [Fel81] and [Eng77, 2.1.8.]. Moreover, càglàd step functions can be approximated from right with smooth functions (and vice versa), so, there exists a sequence pA n q nPN , A n " pA i,n q d i"1 , of elements of A such that }A n } V ď }H} V Q-a.s., for all n P N, and pA n ‚ Xq T Ñ pH ‚ Xq T Q-a.s., as n Ñ 8. Integrating by parts, we get where c :" 2}H} V ă 8 Q-a.s. and }X} 8 P L p pQq, by (12). Thus, by the dominated convergence, for any Q P rS p r s seq , we have The elements of A can similarly be approximated with the elements of EpQq. Moreover, due to the uniform bound (14), for any sequence of integrands bounded in total variation, by the right-continuity X " pX 1 , X 2 , . . . , X d q, the F o for every ω P DpI; R d q, the function |pA ‚ Xq T | p is continuous, see (55), and non-negative, so, by Proposition 4.5 and [Bog07, Proposition 8.9.8.], the functional }pA ‚ Xq T } L p pQq is weak˚lower semicontinuous on S p r , for every A P A. Consequently, the functional }X} A p pQq is weak˚lower semicontinuous on S p r , which in conjunction with the weak˚lower semicontinuity (12) of the functional }X} L p,8 pQq , yields the weak˚closedness of the sets S p r in E p . Indeed, for any r ą 0 and p ě 1, for any sequence pQ n q nPN in S p r , converging in the weakt opology to some Q, we have i.e., Q P S p r . Thus, we have S p r " rS p r s seq Ă H p , i.e., the sets S p r are weak˚compact, for r ą 0 and p ě 1. Every element in S p r is indeed an H p -semimartingale measure; cf. (15)-(16).
The pseudonorm in Example 3.8, given by the sum of the L p,8 -norm and the Emery pseudonorm is equivalent to the (maximal) E p -norm In contrast to the classical Banach space pairing, in the pairing of the Riesz representation theorem it is straightforward to construct compact sets of continuous and Markov processes by invoking the classical stability criteria for the almost sure (Hölder) continuity and the (Lipschitz) Markov property.
Example 3.9. Let S be a set of semimartingale measures satisfying the condition (UT).
(a) Assume that there exist constants a, b, c ą 0 such that sup QPS E Q r|X t´Xs | a s ď b|s´t| 1`c , @s, t P I.
Then, the set rSs seq is a weak˚compact set of continuous semimartingales.
(b) Let Q be the standard Wiener measure on the Skorokhod space DpR`; Rq. Assume that, for every Q α in P, there exists a function σ α : R`ˆR Ñ R that is continuous, continuously differentiable in the first component and that Q α is the law of a (unique strong) solution X α of Then, the set rSs seq is a weak˚compact set of Markov semimartingales. Hence, X α " pX α t q tě0 satisfies the Lipschitz-Markov property on the space pDpR`; Rq, F 8 , pF t q tě0 , Qq, for every α, i.e., the law Q α of X α is a Lipschitz-Markov measure on the space pDpR`; Rq, F 8 , pF t q tě0 q, for every α. Thus, by Theorem 3.2 in conjunction with Lemma 3.5 (b), every element of rPs seq is a (Lipschitz) Markov semimartingale measure.
The statements for quasimartingale measures and supermartingale measures are obtained by replacing Theorem 3.2 above with Corollary 3.3 and Corollary 3.4, respectively.
4 Auxiliary results and the proofs for Section 3.2 The purpose of this section is to provide the proofs for Theorem 3.2 and its corollaries that we omitted in Section 3.2 and the required auxiliary results leading to the proofs. In Section 4.1, we establish three basic results for the weak˚topology deployed in the proof of Theorem 3.2. The required stability and tightness results for the weak˚topology are covered in Section 4.2. Finally, the proofs for the results of Section 3.2 are provided in Section 4.3.
In Section 4.1, in addition to that the underlying topological space is regular and Souslin, we assume the space posses the following property.
Property 4.1. There exists a countable family of real-valued continuous functions f k , k P N, such that, for all x, y P X, we have Remark 4.2. A topological space satisfying Property 4.1 is submetrizable i.e. there exists a weaker topology that is metrizable. Indeed, such topology is given e.g. by the metric dpx, yq :" where f k 's are given by (19). The author would like to thank Professor Jakubowski for pointing out this fact.
Property 4.1 was extensively studied in [Jak95]; see also Section 5.3. In particular, we notice that a Skorokhod space satisfying Assumption 3.1 has this property. Indeed, for a Skorokhod space DpI; R d q, a countable family of separating continuous functions is given e.g. by the family of functions

Weak˚topology
The results of this section are established under the assumption that the space D is a regular Souslin space satisfying Property 4.1. Under the assumption, we obtain a stronger separation axiom than the required T 3 1 {2 ; cf. Section 2.3. Indeed, combining the fact that the topological space is regular (T 3 ) with the fact that the space is a Souslin space, it follows, from a result of Fernique [Fer67, Proposition I.6.1], that the space D is perfectly normal (T 6 ). The families M t pDq, M τ pDq and M σ pDq are defined at the end of Section 1.  ż f dp µ, @f P C b pDq.
Since any measure of finite variation is a linear combination of two probability measures, it suffices to observe that the mapping µ Þ Ñ p µ is a bijection; see e.g.  We use the equality (21) without mentioning it when we apply the results from the book of Bogachev [Bog07].

The Eberlein-Šmulian properties
In this section, we show that the non-negative orthant M`pDq endowed with the weak˚topology is angelic. In angelic spaces, the properties of compactness and sequential compactness coincide. In general, one does not imply another; see e.g. [ Moreover, under these conditions, the weak˚closure of M is metrizable.
Proof. By the assumption that the underlying topological space D is a regular Souslin space, it admits a continuous injective mapping to a metric space; see [FGT12, Theorem 2.25 (i)]. It is also known that if a regular space can be continuously injected into an angelic space, then this regular space is also angelic; see [Flo06,Theorem 3.3]. Since the weak˚topology on the space of non-negative measures M`pDq is metrizable for a metrizable topology on the underlying space D, see [Bog07, Theorem 8.3.2], and metric spaces are angelic, the space M`pDq endowed with the weak˚topology is angelic under our assumption that the underlying topological space D is a regular Souslin space. By It is immediate from Proposition 4.5 that the properties of weak˚compactness and sequential weak˚compactness are equivalent for the subsets of M`pDq. In fact, a stronger statement is true. In angelic space, the closure of a relatively compact set is completely exhausted by the limits of sequences of points in this set.
Corollary 4.6. Assume that M is a subset of M`pDq that satisfies the equivalent conditions of Proposition 4.5. Then the sequential weak˚closure of M in M`pDq, i.e., the set rM s seq " tµ P M`pDq : Dpµ n q nPN Ă M s.t. µ n Ñ w˚µ u, is weak˚closed.
Proof. By Proposition 4.5, the closure of M endowed with the relative topology is a first countable space. In particular, the space is a Fréchet-Urysohn space. By [Eng77, Theorem 1.6.14.], the sequential closure rM s seq coincides with the closure of M .
Various criteria that guarantee tightness and stability of a family of processes are not preserved in the weak˚convergence, so, the previous results are crucial for constructing weak˚compact sets of stochastic processes.

Prokhorov's theorem
We say that a subset M of MpDq is β 0 -equicontinuous if and only if the respective family of linear functionals " f Þ Ñ u µ pf q :" ż f dµ : µ P MpDq * is equicontinuous in the β 0 -topology on C b pDq, i.e., if, for every ε ą 0, there exists a β 0 -neighbourhood V in C b pDq such that |u µ pf q| ă ε, for all pf, µq P VˆM . A measure µ P MpDq is called tight, if there exists an exhausting net of compact sets pK ε q εą0 for µ, i.e., |µ|pDzK ε q ă ε, for every ε ą 0, where |µ| is the total variation of µ. A subset M of MpDq is called uniformly tight, if there exists a net of compact sets pK ε q εą0 which is uniformly exhausting for the total variation of M , i.e., sup µPM |µ|pDzK ε q ă ε, for every ε ą 0; cf. (6). Moreover, we have the following useful convergence criteria.
(b) If pµ n q nPN is an uniformly tight sequence in MpDq converging in the weakt opology to µ P MpDq, then, for any f P CpDq satisfying lim cÑ8 sup n ż |f |½ t|f |ěcu dµ n " 0, we have ż f dµ n Ñ ż f dµ, as n Ñ 8.
Proof. The underlying topological space is completely regular, and the characterization follows directly from [Sen72, The characterization of the relative compactness in terms of the property of β 0 -equicontinuity yields also a criteria for compactness of closures (of convex (circled) hulls).
Corollary 4.8. The closed convex circled hull of a β 0 -equicontinuous subset of MpDq is β 0 -equicontinuous, weak˚compact and sequentially weak˚compact. In particular, the closure and the closed convex hull of a β 0 -equicontinuous set are weak˚compact and sequentially weak˚compact.
Proof. The weak˚compactness of the closed convex circled hull of an equicontinuous set follows from [KN76,18.5]. Closure and closed convex hull are closed subsets of closed convex circled hull, from which the second statement follows. The β 0 -equicontinuous sets are bounded in total variation, in particular, from below, so, the sequential statements are true, by Proposition 4.5.

Skorokhod's representation theorem
Jakubowski's fundamental observation was that Property 4.1 yields a subsequential Skorokhod representation theorem.
Proposition 4.9. Let pQ n q nPN be a sequence converging in the weak˚topology to Q in PpDq. Then there exists a subsequence pQ n k q kPN , a probability space pΩ, F , P q and D-valued random variables pY k q kPN and Y on pΩ, F , P q such that Q n k " P˝pY k q´1, k P N, Q " P˝Y´1 and f pY k pωqqÑf pY pωqq, @ω P Ω, @f P C b pDq.
Proof. The Euclidean space endowed with its usual inner product is a Hilbert space, so, by [Jak97a, Theorem 1], the existence of an a.s. convergent subsequence follows from Property 4.1. The convergence in [Jak97a, Theorem 1] is the pointwise convergence in the topology of the underlying space, which, for a sequence in a completely regular space, is equivalent to the convergence (22); cf.
(48). Moreover, modifying the D-valued random variables Y k and Y , given by [Jak97a, Theorem 1], on a set of measure zero does not affect on their weak˚convergence, so, their almost sure convergence can be strengthened to the pointwise convergence.
In particular, by Proposition 4.9, every element of PpDq can be regarded as a law of some D-valued random variable. Complementarily, any such random variable induces a probability measure on D.

Stability and tightness
In this section, we cover the required stability and tightness results. We present the required multi-dimensional infinite horizon extensions of the stability results of Meyer and Zheng [MZ84], Jakubowski, Mémin and Pages [JMP89], and Lowther [Low09], for the right-continuous version of the raw canonical filtration.

Stability
Under Assumption 3.1, it suffices to establish the required stability results for the Meyer-Zheng topology; see Appendix A.1.3. The required stability results are classical and thoroughly studied in the aforementioned works [MZ84], [JMP89] and [Jak97b] for scalar-valued processes. We demonstrate that, after some slight modifications, they are true in the present setting. We utilize the following multi-dimensional extension of [MZ84, Threorem 5], provided by Jakubowski's subsequential Skorokhod's representation theorem.
Lemma 4.10. (Jakubowski) If pQ n q nPN is a sequence converging in the weakt opology to Q in PpDpI; R d qq, then there exists a subsequence pQ n k q kPN and a set L Ă I of full Lebesgue measure such that T P L, if I " r0, T s, and for every finite subset F of L. In particular, there exists a (countable) dense set D Ă I such that T P D, if I " r0, T s, and (23) is true, for every finite subset F of D.
Proof. By Proposition 4.9, we find a subsequence pQ n k q, k P N, and D-valued random variables pY k q kPN and Y on some pΩ, F , P q such that Q n k " P˝Y´1 k , k P N, Q " P˝Y´1 and Y k pωq Ñ MZ Y pωq, for every ω P Ω, as k Ñ 8; since the topology M Z is metrizable, (22) is equivalent to Ñ MZ . By Lemma A.10, there exists a subsequence Y km pωq, m P N, and a set L of full Lebesgue such that T P L, if I " r0, T s, and Y km,t pωq Ñ Y t pωq, for every pt, ωq P LˆΩ, as m Ñ 8. Hence, the finite dimensional distributions of the process pY km,t q tPL converge to those of pY t q tPL . The complement of the set L is a λ-null set. Thus, the set L contains a (countable) dense set D such that T P D, for I " r0, T s; cf. Definition A.8.
In Proposition 4.11, we show that the required part of [JMP89, Theorem 2.1], which is an extension of [Str81, Theorem 2] for a right-continuous canonical filtration, is true on a multi-dimensional Skorokhod space.
Proposition 4.11. Let pQ n q nPN be a sequence of semimartingale measures satisfying the condition (UT) and converging in the weak˚topology to Q. Then, the weak˚limit Q is a semimartingale measure.
Proof. The proof is essentially a combination of [JMP89, Lemma 1.1 and 1.3]. By Lemma 4.10, there exists a subsequence pQ n k q kPN and a countable dense set D Ă I such that T P D, if I " r0, T s, and pQ n k q kPN converges to Q in finite dimensional distributions on the set D. For every finite collection t 1 ă¨¨¨ă t j in D, let A t1,...,tj denote the family of continuity sets of the marginal law of Q on t 1 ă¨¨¨ă t j , i.e., A t1,...,tj consists of Borel sets B P b iďj BpR d q for which Q˝X´1 t1,...,tj rBBs " 0, where BB denotes the (Euclidean) topological boundary of B on R dˆj . Following [JMP89], we introduce an auxiliary class J pDq of integrands, determined by the weak˚limit Q and the dense set D, that are of the form where every t i 0 ă t i 1 ă¨¨¨ă t i npiq is a finite collection of elements of D and every J t i k´1 is a finite linear combination of indicator functions of the continuity sets of the marginal law of Q on s 1 ă¨¨¨ă s j ď t i k´1 , s 1 , . . . , s j P D, embedded on DpI; R d q and bounded by 1 in absolute value, i.e., |J t i k´1 | ď 1 and each J t i k´1 is of the form for some elements s 1 ă¨¨¨ă s j of D and j and p finite, for every i ď d, for every k ď n. Now, since pQ n k q kPN is converging to Q in finite dimensional distributions on the set D, by the vectorial Portmanteau's lemma, see e.g. [Vaa98, Lemma 2.2], we have where J P J pDq Ă EpQ n k q, for all k P N. Due to the condition (UT), for every t P I, the term on the last line, in (24), tends to zero, uniformly over J pDq, as c Ñ 8, i.e., the family tpJ ‚ Xq t : J P J pDqu, is Q-tight, i.e, bounded in probability Q, for every t P I. The topology of the convergence in probability is metrizable, so, for every t P I, the sets, that are contained in the (sequential) closure of tpJ ‚ Xq t : J P J pDqu, are bounded, which, in particular, entails that the set tpH ‚ Xq t : H P EpQqu is bounded in probability Q. Indeed, we will show this, by adapting a sequence of approximation arguments from [JMP89]. First, since D is dense in I and contains T , for every t 0 ă t 1 ă¨¨¨ă t n in I, n P N, there exists t k 0 ď t k 1 ď¨¨¨ď t k n in D, k P N, such that t j ă t k j , for every j ď n, for every k ě 1, and t j Ó t k j , for every j ď n, as k Ñ 8; we allow t k j " T , if t j " T . Since d and n are finite, by the right-continuity of X " pX 1 , . . . , X d q, we have uniformly over i " 1, . . . , d and j " 1, . . . , n, as k Ñ 8.
Secondly, for every i ď d, for every j ď n, for every t j ă T , any F tj -measurable measurable, for all k ě 1, and can therefore be expressed as an uniform limit of simple F o t k j´-measurable functions bounded by 1 in absolute value, for every k ě 1, i.e., for every i ď d, for every j ď n, for every k ě 1, and we have Further, since each A t1,...,tj is an algebra generating b iďj BpR d q on R dˆj , and the finite unions of the cylindrical sets X´1 t1,...,tj pb iďj B`R d q˘form an algebra generating the canonical σ-algebra on DpI; R d q, for every 0 ă t P I, the family Finally, by combining the approximations (25) and (27), and in (27), invoking the approximation (28) in the sums (26), we conclude that, for every H P EpQq, there exists a sequence pJ n q nPN , n " npk, ℓ, mq, of elements in J pQq such that, for every t P I, we have Thus, the family of simple integrals tpH ‚ Xq t : H P EpQqu is contained in the closure of tpJ ‚ Xq t : J P J pDqu, for every t P I, and, by (24), the weak2 limit Q is an pF t q tPI -semimartingale measure; and, consequently, an pF o t q tPIsemimartingale measure; see e.g. [Pro05,Theorem II.4].
Proposition 4.12. Let pQ n q nPN be a sequence of quasimartingale measures satisfying the condition (UB) and converging in the weak˚topology to Q. Then, the weak˚limit Q is a quasimartingale measure.
Proof. Let i ď d be fixed. We adapt the proof of [MZ84, Theorem 4] and show that the coordinate process X i is a quasimartingale under Q on DpI; R d q. We have for every t P I, for every ε ą 0, where b i :" lim inf nÑ8 sup tPI E Qn r|X i t |s ă 8, by the condition (UB). Thus, by Fatou's lemma, we get for every t P I. The truncated coordinate process, that is, is a difference of two convex 1-Lipschitz functions of X i , for every c ą 0, so, we have Var Qn t pX i,c q ď 4Var Qn t pX i q, n P N, t P I; (30) see e.g. [Str79]. Let 0 " t 0 ă t 1 ă¨¨¨ă t k " t and |f j | ď 1, j ă k, be continuous F o tj´-measurable functions. By (30), we have so, by Fubini's theorem, for every n P N, for every ε ą 0, we get The mappings are lower semicontinuous and bounded from below, see (55) and Lemma A.12, so, we have where v i :" lim inf nÑ8 sup tPI Var Qn t pX i q ă 8, by the assumption (UB). Due to (29), letting ε Ñ 0 and then c Ñ 8 in (32), by the right-continuity and the monotone convergence, respectively, we get for all F o tj´-measurable continuous functions |f j | ď 1, j ă k. Furthermore, by choosing f j pXq " f pX tj`u q preceding (31), for a continuous function |f | ď 1 on R d , we conclude that the inequality (33) is true for a family of continuous functions that, for every j ă k, generates the σ-algebra F o tj ; cf. Corollary A.11. Thus, by the standard L 1 -approximation via Lusin's theorem and Tietze's extension theorem, for any F o tj -measurable |H tj | ď 1 in L 8 pQq, for every j ă k, the exists a sequence pf n j q nPN , |f n j | ď 1, of functions satisfying (33) such that f n j Ñ H tj in L 1 pQq, as n Ñ 8; see e.g. [Fel81] and [Eng77, 2.1.8.]. Thus, the inequality (33) is true for all F o tj -measurable functions |H tj | ď 1 in L 8 pQq, so, the process X is an pF o t q tPI -quasimartingale on pDpIq, F T , Qq, see [DM78, B. App. II (3.5)], which, by Rao's decomposition theorem, is a necessary and sufficient condition for the process X to be decomposable to a difference X " Y´Z of two càdlàg pF o t q tPI -supermartingales Y and Z on pDpIq, F T , Qq; see [HWY92,Theorem 8.13]. On the other hand, by Föllmer's lemma, Y and Z are pF t q tPI -supermartingales, see [HWY92, Theorem 2.46], so, by Rao's decomposition theorem, the process X is an pF t q tPI -quasimartingale on pDpIq, F T , Qq.
Proposition 4.13. Let pQ n q nPN be a sequence of supermartingale measures satisfying the condition (UI) and converging in the weak˚topology to Q. Then, the weak˚limit Q is a supermartingale measure.
Proof. We adapt the proof of [MZ84, Theorem 11] and show that each coordinate process X i , i ď d, is a supermartingale under Q on DpI; R d q. We have E Q r|X t |s ă 8, for every t P I; cf. (29). Moreover, by Lemma 4.10, there exists a subsequence pQ n k q kPN and a countable dense set D Ă I such that T P D, if I " r0, T s, and pQ n k q kPN converges to Q in finite dimensional distributions on the set D. Let X i,c denote the coordinate process X i truncated from above at c ą 0, i.e., X i,c :" X i^c , c ą 0.
By the condition (UI) and the fact that each Q n k is a supermartingale measure for X i,c , for every c ą 0, we have where f pXq :" f 1 pX t1 qf 2 pX t2 q¨¨¨f n pX tn q, t j P D, f j P C b pR d q, j ď n; for every s ă t in D and F P F o s´. Letting c Ñ 8 in (34), by the monotone convergence theorem, we get the same inequality for the coordinate process X i . By Föllmer's lemma, the inequality extends immediately to the whole I, and further, for F P F o s`. Indeed, we have for every F P Proposition 4.14. Let pQ n q nPN be a sequence of probability measures converging in the weak˚topology to Q. Then, we have the following: (a) If the sequence pQ n q nPN is C-tight, then the limit Q is C-tight.
(b) If each Q n is Lipschitz-Markov, then the limit Q is Lipschitz-Markov.
Proof. (a) By [HWY92,Lemma 15.49], the C-tighness of the sequence pQ n q nPN implies the convergence along a subsequence in the weak˚topology of the Skorokhod's J 1 -topology, and a fortiori in any weaker topology, to a law of a continuous process, which is the limit Q; cf. Proposition 5.4.
(b) Let s ă t in I be fixed and take a bounded Lipschitz continuous function g : R d Ñ R with a Lipschitz constant Lpgq ď 1. For each n P N, there exists bounded Lipschitz continuous function f n : Further, we can take it granted that Lpf n q ď 1 and }f n } 8 ď }g} 8 ă 8, for all n P N. Thus, by the Arzelá-Ascoli theorem, there exists a subsequence pf n k q kPN and a bounded Lipschitz continuous function f : R d Ñ R with Lpf q ď 1 such that f n k converges to f uniformly on compacta, as k Ñ 8. Further, by Lemma 4.10, there exists a further subsequence of pQ n k q kPN , that again we denote by pQ n k q kPN , converging in the finite-dimensional distributions to Q on a dense set D Ă I such that T P D, if I " r0, T s. Let i P N and 0 ď s 1 ă s 2 ă¨¨¨ă s i ď s ă t in D and take a bounded continuous compactly supported α : R d Ñ R and a bounded continuous β : R dˆi Ñ R. By (36), we have E Qn k rpf n k pX s q´gpX t qqαpX s qβpX s1 , . . . , X si qs " 0, @k P N.
Since f n k converges uniformly to f in the compact support of each α and Q n k converges to Q in the finite dimensional distributions on the set D, as k Ñ 8, from (37), by the vectorial Portmanteau's lemma, see e.g. [Vaa98, Lemma 2.2], we get E Q rpf pX s q´gpX t qqαpX s qβpX s1 , . . . , X si qs " 0.
The equality (38) holds for all bounded continuous α with compact support, which yields E Q rpf pX s q´gpX t qqβpX s1 , . . . , X si qs " 0, and further, the equality (39) holds for all bounded continuous β, which yields for every bounded F o s -measurable function h, for every s ă t in D. Since D is a dense subset of I and s Þ Ñ f pX s q is bounded and right-continuous on I, by the bounded convergence theorem, the equality (40) extends to the whole I, and further, for every bounded F s -measurable function h; cf. (35).

Tightness
We say that a family Q of probability measures on`DpI; R d q, F T˘s atisfies Jakubowski's uniform tightness criteria, if we have lim cÑ8 sup QPQ Qr}X i,t } 8 ą cs " 0 and lim cÑ8 sup QPQ QrN a,b pX i,t q ą cs " 0, @a ă b, (US) for every finite t P I, for every i ď d, where X i,t denotes the coordinate process X i restricted on r0, ts; cf. Corollary 5.12. It was shown in [Jak97b] that a family of probability measures on pDpr0, T s; Rq, F T q, T ă 8, satisfies the condition (US) if and only if it is uniformly S-tight. In particular, we have the following hierarchy, cf. (4), where (US˚) stands for the uniform tightness in the S˚-topology; see Section 5. The second implication in (41) is immediate from the definition of the S˚topology; see Proposition 5.7 (i). The first implication in (41) follows from Proposition 4.15, that is essentially the result of Stricker [Str85, Theorem 2], which states that a sequence satisfying the condition (UT) admits a convergent subsequence and the limit law is a law of a semimartingale. Analogous results were obtained for the S-topology by Jakubowski in [Jak97b, Theorem 4.1]; see also [Jak97b, Proposition 3.1]. Proof. Let X i,t denote the coordinate processes X i restricted on r0, ts, for i ď d and t ă 8. Following Stricker [Str81, Theorem 2], we define a family of stopping times τ i,c " infts P I : |X i,t s | ą cu, i ď d, c ą 0, and each τ i,c is approximated from right with the sequence of the simple stopping times τ i,c n " mintm{n : m P N, τ i,c ď m{nu, n P N.
Since we are assuming a right-continuous filtration pF t q tPI and X i is rightcontinuous, the hitting times τ i,c , and consequently, their approximations τ i,c n are indeed stopping times. Moreover, since each τ i,c n takes only finitely many values on r0, ts, every process |H n | ď 1 of the form is an elementary predictable integrand; see (1). Now, due to the right-continuity of X i , by the bounded convergence theorem, for every Q P Q, we have for every t P I, for every c ą 0. By the condition (UT), the left-hand side in (44) tends to 0, uniformly over Q P Q, for every i ď d, for every t P I, as c Ñ 8. Similarly, for a ă b, we define, recursively, for all k P N 0 , the stopping times and the respective decreasing sequences pσ i,a k,n q nPN and pτ i,b k,n q nPN of approximative stopping times, taking only finitely many values on finite intervals; cf. (42). The processes |H m,n | ď 1, m, n P N, of the form are finite linear combinations of processes of the form (43), so, each process |H m,n | ď 1 is an elementary predictable integrand. Moreover, we have By the condition (UT), for every a ă b, the right-hand side of (45) tends to zero, uniformly over Q P Q, as c Ñ 0; cf. (24)-(25). Thus, by Corollary 5.12, the family Q satisfies the condition (US).

The proofs of the main results
In this section we provide the proofs for the results of Section 3.2 that we omitted there. We begin by proving Theorem 3.2 by invoking the results of Section 4.1 in conjunction with the stability and tightness result on the semimartingale property of Section 4.2. Then, the rest of the results of Section 3.2 follow from the respective stability results of Section 4.2.
Proof of Theorem 3.2. The condition (UT) is stronger than the condition (US˚); cf. (41). So, by Proposition 4.7, the family S is β 0 -equicontinuous. Thus, by Corollary 4.8, the closure of S is compact and sequentially compact in the weakt opology; see Proposition 4.5. By Corollary 4.6, the closure of S coincides with the sequential closure of S. It remains to show that, every element in the sequential closure rSs seq is a semimartingale measure. This particular fact is the statement of Proposition 4.11.
Proof of Corollary 3.3. The condition (UB) is weaker than the condition (UT); see (4). By Proposition 4.12, the class of quasimartingale measures is stable in the weak˚convergence under the property (UB). Thus, Corollary 3.3 follows from Theorem 3.2.
Proof of Corollary 3.4. The condition (UI) is weaker than the condition (UB); see (4). By Proposition 4.13, the class of supermartingale measures is stable in the weak˚convergence under the condition (UI). Thus, Corollary 3.4 follows from Corollary 3.3.
Proof of Lemma 3.5. Since every sequence in the set is C-tight, by Proposition 4.14 (a), every limit point in the sequential closure is C-tight. Thus, Lemma 3.5 (a) is true. Likewise, Lemma 3.5 (b) is a direct consequence of the stability of the (Lipschitz) Markov property in the weak˚convergence; see Proposition 4.14 (b).

The weak S-topology
We introduce the notion of weak S-topology and study its properties and relation to other topologies on the Skorokhod space.

Definition
A possibility of defining a completely regular (non-sequential) S-topology is discussed already by Jakubowski in [Jak97b]. Indeed, see the page 18 in [Jak97b]. We describe a general method for regularizing any given topology. Our approach is inspired by the seminal work of Alexandroff [Ale43]. Let X " pX, τ q be an arbitrary topological space and V an arbitrary subbase for the Euclidean topology on R, then the family is a subbase for a (unique) topology on X. Indeed, the topology generated by the subbase (46) on X is independent of the choice of the subbase V on R; see e.g. [GJ60,3.4]. The topology is generated by the family of pseudometrics where ρ f1,f2,...,f k px, yq :" maxt|f 1 pxq´f 1 pyq|, |f 2 pxq´f 2 pyq|, . . . , |f k pxq´f k pyq|u for x, y P X, and thus, the convergence of a net px α q to an element x in this topology is equivalent to that

Relation to other topologies
The definitions of Jakubowski's Σ-topology, Jakubowski's S-topology, the Meyer-Zheng topology (M Z) and Skorokhod's J 1 -topology are given in Appendix A.1. The S˚-topology is related to these topologies as follows. Since the topology S˚is the strongest completely regular weaker than S, we have Σ Ă S˚Ă S. The final inclusion S Ă J 1 is proved in [Jak97b] for a finite compact interval, and extends immediately for the infinite interval due to (59); cf. (54).
The Skorokhod space endowed with the J 1 -topology is a Polish space, so, the space is a Lusin space for any topology that is weaker than the J 1 -topology. The following Theorem 5.5 states that the S˚-topology, which is the strongest (completely) regular topology that is weaker than the S-topology, is the strongest (completely) regular Souslin topology on the Skorokhod space for which the sets (52) are compact, and consequently, Jakubowski's uniform tightness criteria (US) is a sufficient tightness criteria; cf. Section 4.2.2.
Theorem 5.5. Let T be a completely regular Souslin topology on the Skorokhod space, comparable to S, and KpT q " KpSq. Then Proof. Assume that S Ă T and let T s denote the sequential topology generated by T . Since the compact sets of a completely regular Souslin space are metrizable, we have KpT q Ă KpT s q; see e.g. [Bog07,p. 218]. Consequently, we have KpSq " KpT q " KpT s q, where and S and T s are sequential; see Appendix A.1.1. By [Eng77, Theorem 3.3.20.], the Skorokhod space is a (Hausdorff) k-space for S and T s , so, by (49) and (50), we have S " T .
Remark 5.6. For the Riesz representation theorem given in Lemma 2.4 and the required auxiliary results given in Section 4 it is necessary that the underlying topological space is completely regular and Souslin. Among all such topologies, the topology S˚is the strongest one that is weaker than S. Recall that, in addition to Lemma 2.4 and results of Section 4, the proof of the main result Theorem 3.2 goes through the property (US), while the underlying relative compactness criteria (52) that gives arise for the tightness (US) is both necessary and sufficient for S; see also Remark 5.10. On the other hand, as shown in Theorem 5.5 above, any topology with the previously cited properties and the appropriate compact closed sets (52) cannot be strictly stronger than S.

Properties
Recall Property 4.1 from Section 4. A topological space satisfying Property 4.1 is submetrizable, from which various useful properties follow; see (20) and [Jak95].
In fact, all key properties of the S-topology follow immediately from Property 4.1, and Property 4.1 is preserved in the regularization (46).
Proposition 5.7. The S-topology has the following properties: (a) S is Hausdorff, where q and q`r run over the rationals in I and i over the spatial dimensions 1, . . . , d, constitute a countable family of continuous functions that separates the Skorokhod space; cf. Example 5.13. (d) We prove the claim for I " r0, T s. The proof is completely similar for I " r0, 8q. Fix a coordinate i ď d. For all 0 ď t ă T , we have i.e., the mapping ω Þ Ñ ω i ptq is a limit of elements of CpSq, for every t in I, while for t " T , the mapping ω Þ Ñ ω i ptq is an element of CpSq. Consequently, we have σpX u : u P Iq Ă BpS˚q and since S˚is weaker than S, we have BpS˚q Ă BpSq. On the other hand, by Proposition 5.4, S is weaker than J 1 , so, we have BpSq Ă BpJ 1 q, where BpJ 1 q " σpX u : u P Iq. Thus, BpS˚q " BpSq " σpX u : u P Iq. By Proposition 5.4, we have S˚Ă S Ă J 1 and the Skorokhod space endowed with J 1 is a Polish space, so, the Skorokhod space endowed with S or S˚is a Lusin space. Thus, we have (e

Compact sets and continuous functions
In this section, we recall the compactness and continuity criteria for the Stopology from [Jak97b] and [Jak18].

Compactness criteria
The necessity and sufficiency of the condition (52) for the relative (sequential) compactness in the S-topology was proved in [Jak97b], for I " r0, T s, T ă 8, and the multi-dimensional infinite horizon extension was provided in [Jak18].
2. The mappings of ω in Proposition 5.9 are lower semicontinuous in the S˚-topology, so, their lower level sets are closed in the S˚-topology; cf. Example 5.14.
3. A Cartesian product set in a multi-dimensional Skorokhod space is relatively sequentially S-compact if and only if each set in the product is relatively sequentially S-compact; cf. Definition A.1.
Combining the previous facts we obtain the following compactness criteria.
Corollary 5.12. Let K " K 1ˆ¨¨¨ˆK d be a Cartesian product set on the Skorokhod space DpI; R d q endowed with S or S˚. Then the set K is compact, if, for each i ď d, there exists a (non-decreasing) function C i q,r : I Ñ R`, for all q ă r in Q, and a (non-decreasing) function M i : I Ñ R`such that where the intersection is taken over all rationals q ă r and rω i s t denotes the restriction of ω i on r0, ts.
Remark that, for I " r0, T s, T ă 8, it suffices to consider constant C i q,r and M i in Corollary 5.12.

Examples of (semi-)continuous functions
By Proposition 5.7 (h), S˚-continuous functions are precisely the S-continuous ones. In particular, we would like to emphasize that the evaluation mapping at t is not continuous for any t ă T ; see [Jak97b,p.11]. By Proposition 5.4, the S˚-topology is stronger than the Meyer-Zheng topology (M Z), so, the uniform norm and the number of upcrossings of an interval ra, bs are lower semicontinuous functions; see Lemma A.12.
Example 5.14. The functions ω Þ Ñ }ω} 8 and ω Þ Ñ N a,b pω i q, a ă b, i ď d, are lower semicontinuous in the S˚-topology on DpI; R d q.

A Appendix
The appendix collects the definitions of topologies and auxiliary results used in the main part of the article.

A.1 Topologies on the Skorokhod space
We recall the definitions of Jakubowski's S-topology and Σ-topology, the Meyer-Zheng pseudo-path topology and Skorokhod's J 1 -metric topology. We define each topology separately on Dpr0, T s; R d q, for T ă 8, and Dpr0, 8q; R d q. In particular, we use a formal definition of the Meyer-Zheng pseudo-path topology (M Z) that takes into account the fluctuations of the terminal value in the case of a finite time horizon. The space Dpr0, 8s; R d q is regarded as a product space Dpr0, 8s; R d q " Dpr0, 8q; R d qˆR d , where the space R d is endowed with the Euclidean topology.

A.1.1 The S-topology
Jakubowski's S-topology, introduced in [Jak97b], is a sequential topology. The following definition of the S-convergence on Dpr0, T s; Rq is taken from [Jak97b]; the multi-dimensional version can be found in [Jak18].
Definition A.1. On Dpr0, T s; R d q, we write ω n Ñ S ω 0 , if, for every i ď d, for every ε ą 0, one can find pν i,ε n q nPN0 Ă Vpr0, T sq such that }ω i n´ν i,ε n } 8 ď ε, @n P N 0 , and ν i,ε n Ñ w˚ν i,ε 0 , as n Ñ 8, where the convergence "Ñ w˚" is in the weak˚topology on Vpr0, T sq, which can be identified with the Banach dual of Cpr0, T sq, under the uniform norm.
The following definition of the S-convergence on Dpr0, 8q; R d q is taken from [Jak18].
Definition A.2. On Dpr0, 8q; R d q we write ω n Ñ S ω 0 , if, for every i ď d, one can find a sequence of positive real numbers pT r q rPN , increasing to 8, such that rω i n s T r Ñ S rω i 0 s T r , for every r P N, where rω i s T r denotes the restriction of a path ω i P Dpr0, 8q; Rq on Dpr0, T r s; Rq.
A topological convergence is obtained by requiring that every subsequence admits a further S-convergent subsequence; see [Jak18, Theorem 6.3]. The following definition for the S-topology on the Skorokhod space Dpr0, T s; R d q and Dpr0, 8q; R d q are taken from [Jak97b] and [Jak18], respectively.
Definition A.3. The S-topology is the topology generated on the Skorokhod space by the subsequential S-convergence.
The Skorokhod space endowed with the S-topology is known to be a Hausdorff (T 2 ) space and a stronger separation axiom is an open problem. A weak separation axiom is a well-known issue for topologies defined via the subsequential convergence (KVPK recipe); see [Jak18,Appendix] for elaboration. The difficulties encountered in establishing the regularity of the S-topology are explained in [Jak97b, Rem. 3.12].

A.1.2 The Σ-topology
Jakubowski's Σ-topology was introduced in [Jak18]. The Skorokhod space endowed with Σ is a locally convex vector space. Following [Jak18], we start by defining an auxiliary mode of convergence Ñ τ on the space of (bounded) continuous functions of finite variation Apr0, T s; Rq :" Cpr0, T s; Rq X Vpr0, T s; Rq. Namely, for a sequence pA n q nPN0 Ă Apr0, T s; Rq, we write A n Ñ τ A 0 , if }A n´A0 } 8 Ñ 0, as n Ñ 8, and sup nPN0 }A n } V ă 8, where }¨} V denotes the total variation norm.
Definition A.4. The topology Σ on Dpr0, T q; R d q is the topology generated by the seminorms where A ranges over relatively τ -compact subsets of Apr0, T q; Rq.
Definition A.5. The topology Σ on Dpr0, T s; R d q is the topology generated by the seminorm ρ i T pωq " |ω i pT q|, i ď d, and the seminorms where A ranges over relatively τ -compact subsets of Apr0, T s; Rq.
The topology Σ was defined on the Skorokhod space Dpr0, T s; Rq for T " 1 in [Jak18]. The following properties were shown to be true for Σ on Dpr0, T s; Rq.
Proof. The proof is adapted from [MZ84]. Let i ď d and ω i n Ñ MZ ω i with sup n }ω i n } ď c. If }ω i } 8 ą c, then there exists s ă t such that ω i puq ą c, for all u P rs, tq, or we have ω i pT q ą c, either way, there exists an M Z-continuous function F for which lim nÑ8 F pω i n q ă F pω i q, cf. (55) and (56), so, this is a contradiction. Thus, the mapping ω Þ Ñ }ω} 8 :" }ω 1 } 8 _¨¨¨_ }ω d } 8 is lower M Z-semicontinuous. Similarly, one can show that the sets of the form tω : Du P rs, tq s.t. ω i puq ą bu and tω : Du P rs, tq s.t. ω i puq ă au, s ă t, a ă b are open in the M Z-topology, from which the lower M Z-semicontinuity of the mappings N a,b , a ă b, follows. Indeed, let a ă b be fixed and consider a finite partition π :" tt 0 ă t 1 ă¨¨¨ă t n u of r0, t n s. We write N a,b π pω i q ě k, if one can find l 1 ă m 1 ď l 2 ă m 2 ď¨¨¨ă l k ă m k ď n such that, for all j ă k, ω i psq ă a, for some s P rt lj´1 , t lj q, and ω i ptq ą b, for some t P rt mj´1 , t mj q (or, for t " T , if j " k and m k " n). The partition π is finite, so, the sets tω : N a,b π pω i q ě ku " tω : N a,b π pω i q ą k´1u, k P N, are open in the M Z-topology. Consequently, the mapping ω Þ Ñ N a,b π pω i q and the mapping N a,b pω i q :" sup π N a,b π pω i q are lower M Z-semicontinuous, for every i ď d.
We refer the reader to the book by Dellacherie and Meyer [DM78, A, IV] and the paper [MZ84] by Meyer and Zheng for details on pseudo-paths and the Meyer-Zheng topology, respectively.
A.1.4 The Skorokhod's J 1 -topology The following complete metric, generating a topology called Skorokhod's J 1topology, was introduced by Kolmogorov in [Kol56]. The metric introduced by Skorokhod himself in [Sko56] was not yet complete despite generating an equivalent topology.
Definition A.13. The Skorokhod's J 1 -topology on Dpr0, T s; R d q is the topology generated by the complete metric where Λ denotes the class of strictly increasing, continuous mappings of r0, T s onto itself and i is the identity map on r0, T s.
Let us recall the criteria for the relative compactness in the J 1 -topology from [Bil68]. Let δ ą 0 and denote m δ pωq :" inf tiPπ max iďn sup s,tPsti´1,tis |ωpsq´ωptq|, ω P Dpr0, T s; R d q, where the infimum is take over all finite partitions 0 " t 0 ă t 1 ă¨¨¨ă t n " T of r0, T s with the mesh size t i´ti´1 ą δ, for all i ď n.
Lemma A.15. A subset K of Dpr0, T s; R d q, T ă 8, is relatively J 1 -compact, if it is bounded and lim δÑ0 sup ωPK m δ pωq " 0.
A subset K of Dpr0, 8q; R d q is relatively J 1 -compact, if the restriction of K on Dpr0, T s; R d q is relatively J 1 -compact, for every T ă 8.
We refer the reader to [Bil68, Section 12, 16] for details on the Skorokhod's J 1 -metric on Dpr0, T s; R d q and Dpr0, 8q; R d q, respectively.

A.2 The proof of Lemma 2.2
Recall that we claimed that there exists a uniform constant b ą 0 such that, for any Q P P`DpI; R d q, F T˘, H P EpQq and c ą 0, we have Qr|pH ‚ Xq t | ą cs ď b c « E Q r|X t |s`sup HPEpQq E Q rpH ‚ Xq t s ff , t P I.
Proof. The inequality (60) is a generalizations of Burkholder's inequality, which states that there exists a uniform constant a ą 0 such that, for any H P EpQq, Q-martingale M and c ą 0, we have Qr|H ‚ M | t ą cs ď a c E Q r|M t |s, t P I; Var Q t pX i q, t P I.
Let us fix i ď d and assume that E Q r|X i