The Riesz representation theorem and weak^∗ compactness of semimartingales

The space \({\mathbb{M}}_{+}(\mathbb{D})\)of nonnegative measures endowed with the weak^∗ topology is angelic. In particular, for any subset \(M\subseteq {\mathbb{M}}_{+}(\mathbb{D})\), the following are equivalent:

(i) Any sequence in \(M\)has a weak^∗ convergent subsequence in \({\mathbb{M}}_{+}(\mathbb{D})\).

(ii) The weak^∗ closure of \(M\)is weak^∗ compact in \({\mathbb{M}}(\mathbb{D})\).

Moreover, under these conditions, the weak^∗ closure of \(M\)is metrisable.

Because the underlying topological space \(\mathbb{D}\) is a regular Souslin space, it admits a continuous injective mapping to a metric space; see [19, 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 [20, Theorem 3.3]. Since the weak^∗ topology on the space \({\mathbb{M}}_{+}(\mathbb{D})\) of nonnegative measures is metrisable for a metrisable topology on the underlying space \(\mathbb{D}\), see [8, Theorem 8.3.2], and metric spaces are angelic, the space \({\mathbb{M}}_{+}(\mathbb{D})\) endowed with the weak^∗ topology is angelic under our assumption that \(\mathbb{D}\) is a regular Souslin space. By [8, Theorem 8.10.4], the weak^∗ closure of a subset \(M\) of \({\mathbb{M}}_{+}(\mathbb{D})\) satisfying (i) or (ii) is a compact metrisable subspace of \(\mathbb{M}(\mathbb{D})\); so (i) and (ii) are indeed equivalent for \(M\). □

Assume that \(M\)is a subset of \({\mathbb{M}}_{+}(\mathbb{D})\)that satisfies the equivalent conditions of Proposition 4.5. Then the sequential weak^∗ closure of \(M\)in \(\mathbb{M}_{+}(\mathbb{D})\), i.e., the set is weak^∗ closed.

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 [16, Theorem 1.6.14], the sequential closure \([M]_{\mathrm{seq}}\) coincides with the closure of \(M\). □

A subset \(M\)of \({\mathbb{M}}(\mathbb{D})\)is \(\beta _{0}\)-equicontinuous if and only if it is bounded in total variation and uniformly tight. In addition, we have the following:

(a) If \(M\)is a \(\beta _{0}\)-equicontinuous subset of \({\mathbb{M}}(\mathbb{D})\), then \(M\)is relatively compact and relatively sequentially compact in the weak^∗ topology.

(b) If \((\mu _{n})_{n\in \mathbb{N}}\)is a uniformly tight sequence in \({\mathbb{M}}(\mathbb{D})\)converging in the weak^∗ topology to \(\mu \in {\mathbb{M}}(\mathbb{D})\), then for any \(f\in \mathbb{C}(\mathbb{D})\)satisfying we have

As the underlying topological space is completely regular, the characterisation follows directly from [50, Theorem 5.1]. The compact subsets of a completely regular Souslin space are metrisable, cf. Proposition 4.5 and [8, Lemma 8.9.2]. In conjunction with the fact the space is completely regular, this verifies the assumptions for both the sequential and the nonsequential Prokhorov theorem and hence gives (a); see [8, Theorem 8.6.7]. The convergence criterion in (b) is similarly a direct consequence of the fact that the underlying space is completely regular; see [7, Lemma 3.8.7]. □

The closed convex circled hull of a \(\beta _{0}\)-equicontinuous subset of \(\mathbb{M}(\mathbb{D})\)is \(\beta _{0}\)-equicontinuous, weak^∗ compact and sequentially weak^∗ compact. In particular, the closure and the closed convex hull of a \(\beta _{0}\)-equicontinuous set are weak^∗ compact and sequentially weak^∗ compact.

The weak^∗ compactness of the closed convex circled hull of an equicontinuous set follows from [37, 18.5]. Closure and closed convex hull are closed subsets of the closed convex circled hull, from which the second statement follows. The \(\beta _{0}\)-equicontinuous sets are bounded in total variation, in particular, from below; so the sequential statements are true by Proposition 4.5. □

Let \((Q_{n})_{n\in \mathbb{N}}\)be a sequence converging in the weak^∗ topology to \(Q\)in \({\mathbb{P}}(\mathbb{D})\). Then there exist a subsequence \((Q_{n_{k}})_{k\in \mathbb{N}}\), a probability space \((\Omega ,\mathcal{F},P)\)and \(\mathbb{D}\)-valued random variables \((Y_{k})_{k\in \mathbb{N}}\)and \(Y\)on \((\Omega ,\mathcal{F},P)\)such that \(Q_{n_{k}}=P\circ (Y_{k})^{-1}\)for \(k\in \mathbb{N}\), \(Q=P\circ Y^{-1}\)and

The Euclidean space endowed with its usual inner product is a Hilbert space; so by [32, Theorem 1], the existence of an a.s. convergent subsequence of \(\mathbb{D}\)-valued random variables \((Y_{k})_{k\in \mathbb{N}}\) as in the assertion follows from Property 4.1. The convergence in [32, 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 (4.4); cf. (5.3) below. Moreover, modifying the \(\mathbb{D}\)-valued random variables \(Y_{k}\) and \(Y\) given by [32, Theorem 1] on a set of measure zero does not affect their weak^∗ convergence; so their almost sure convergence can be strengthened to pointwise convergence. □

(Jakubowski)

If \((Q_{n})_{n\in \mathbb{N}}\)is a sequence converging in the weak^∗ topology to \(Q\)in \({\mathbb{P}}(\mathbb{D}(I;\mathbb{R}^{d}))\), then there exist a subsequence \((Q_{n_{k}})_{k\in \mathbb{N}}\)and a set \(L\subseteq I\)of full Lebesgue measure such that \(T\in L\)if \(I=[0,T]\)and for every finite subset \(F\)of \(L\). In particular, there exists a (countable) dense set \(D\subseteq I\)such that \(T\in D\)if \(I=[0,T]\)and (4.5) is true for every finite subset \(F\)of \(D\).

By Proposition 4.9, we can find a subsequence \((Q_{n_{k}})_{k\in \mathbb{N}}\) and \(\mathbb{D}\)-valued random variables \((Y_{k})_{k\in \mathbb{N}}\) and \(Y\) on some \((\Omega ,\mathcal{F},P)\) such that \(Q_{n_{k}}=P\circ Y_{k}^{-1}\) for \(k\in \mathbb{N}\), \(Q=P\circ Y^{-1}\) and \(Y_{k}(\omega )\to _{\text{MZ}} Y(\omega )\) for every \(\omega \in \Omega \) as \(k\rightarrow \infty \); since the topology MZ is metrisable, (4.4) is equivalent to \(\to _{\text{MZ}}\). By Lemma A.10, there exist a subsequence \((Y_{k_{m}})_{m\in \mathbb{N}}\) and a set \(L\) of full Lebesgue such that \(T\in L\) if \(I=[0,T]\) and \(Y_{k_{m},t}(\omega )\rightarrow Y_{t}(\omega )\) for every \((t,\omega )\in L\times \Omega \) as \(m\rightarrow \infty \). Hence the finite-dimensional distributions of the process \((Y_{k_{m},t})_{t\in L}\) converge to those of \((Y_{t})_{t\in L}\). The complement of \(L\) is a \(\lambda \)-nullset; cf. Definition A.8, where the measure \(\lambda \) is given. Thus the set \(L\) contains a (countable) dense set \(D\) such that \(T\in D\) for \(I=[0,T]\). □

Let \((Q_{n})_{n\in \mathbb{N}}\)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.

The proof is essentially a combination of [34, Lemmas 1.1 and 1.3]. By Lemma 4.10, there exist a subsequence \((Q_{n_{k}})_{k\in \mathbb{N}}\) and a countable dense set \(D\subseteq I\) such that \(T\in D\) if \(I=[0,T]\) and \((Q_{n_{k}})_{k\in \mathbb{N}}\) converges to \(Q\) in finite-dimensional distributions on the set \(D\). For every finite collection \(t_{1}<\cdots <t_{j}\) in \(D\), let \(\mathcal{A}_{t_{1},\dots ,t_{j}}\) denote the family of continuity sets of the marginal law of \(Q\) on \(t_{1}<\cdots <t_{j}\), i.e., \(\mathcal{A}_{t_{1},\dots ,t_{j}}\) consists of Borel sets \(B\in \bigotimes _{i\leq j}{\mathcal{B}}(\mathbb{R}^{d})\) for which \(Q\circ X^{-1}_{t_{1},\dots ,t_{j}}[\partial B]=0\), where \(\partial B\) denotes the (Euclidean) topological boundary of \(B\) on \(\mathbb{R}^{d\times j}\). Following [34], we introduce an auxiliary class \(\mathcal{J}(D)\) of integrands, determined by the weak^∗ limit \(Q\) and the dense set \(D\), that are of the form where every \(t^{i}_{0}< t^{i}_{1}<\cdots <t^{i}_{n(i)}\) 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 \(s_{1}<\cdots <s_{j}\leq t^{i}_{k-1}\), \(s_{1},\dots ,s_{j}\in D\), embedded in \(\mathbb{D}(I;\mathbb{R}^{d})\) and bounded by 1 in absolute value, i.e., \(|J_{t^{i}_{k-1}}|\leq 1\) and each \(J_{t^{i}_{k-1}}\) is of the form for some elements \(s_{1}<\cdots <s_{j}\) of \(D\) and \(j\) and \(p\) finite, for every \(i\leq d\) and every \(k\leq n\). Now, since \((Q_{n_{k}})_{k\in \mathbb{N}}\) is converging to \(Q\) in finite-dimensional distributions on the set \(D\), by the vectorial Portemanteau lemma, see e.g. [57, Lemma 2.2], we have where \(J\in {\mathcal{J}}(D)\subseteq {\mathcal{E}}(Q^{n_{k}})\) for all \(k\in \mathbb{N}\). Due to the condition (UT), for every \(t\in D\), the last term in (4.6) tends to zero uniformly over \(\mathcal{J}(D)\) as \(c\rightarrow \infty \), i.e., the family \(\{(J\bullet X)_{t}:J\in {\mathcal{J}}(D)\}\) is \(Q\)-tight, i.e., bounded in \(Q\)-probability, for every \(t\in D\). The topology of convergence in probability is metrisable; so for every \(t\in D\), sets contained in the (sequential) closure of \(\{(J\bullet X)_{t}:J\in {\mathcal{J}}(D)\}\) are bounded, which in particular entails that for every \(t\in I\), the set \(\{(H\bullet X)_{t}:H\in {\mathcal{E}}(Q)\}\) is bounded in \(Q\)-probability. Indeed, by adapting a sequence of approximation arguments from [34], we show that for every \(t\in I\), the set \(\{(H\bullet X)_{t}:H\in {\mathcal{E}}(Q)\}\) is contained in the closure of \(\{(J\bullet X)_{s}:J\in {\mathcal{J}}(D)\}\), for any \(s\geq t\), \(s\in D\). First, fix \(s\geq t\), \(s\in D\). Now, since \(D\) is dense in \(I\) and contains \(T\), for every \(t_{0}< t_{1}<\cdots < t_{n}=t\) in \(I\), \(n\in \mathbb{N}\), there exist \(t_{0}^{k}\leq t_{1}^{k}\leq \cdots \leq t_{n}^{k}\leq s\) in \(D\), \(k\in \mathbb{N}\), such that \(t_{j}\leq t_{j}^{k}\) for every \(j\leq n\) and every \(k\geq 1\), and \(t^{k}_{j}\downarrow t_{j}\) for every \(j\leq n\) as \(k\rightarrow \infty \); we allow \(t_{n}^{k}=T\) if \(t_{n}=T\). Since \(d\) and \(n\) are finite, the right-continuity of \(X=(X^{1},\dots ,X^{d})\) yields Secondly, for every \(i\leq d\), every \(j\leq n\) and every \(t_{j}< T\), any \(\mathcal{F}_{t_{j}}\)-measurable \(|H^{i}_{t_{j}}|\leq 1\) is \(\mathcal{F}^{0}_{t_{j}^{k}-}\)-measurable for all \(k\geq 1\) and can therefore be expressed as a uniform limit of simple \(\mathcal{F}^{0}_{t_{j}^{k}-}\)-measurable functions bounded by 1 in absolute value for every \(k\geq 1\), i.e., for every \(i\leq d\), every \(j\leq n\) and every \(k\geq 1\), there exist functions \(|S^{i,\ell }_{t^{k}_{j}}|\leq 1\), \(\ell \in \mathbb{N}\), such that each \(S^{i,\ell }_{t^{k}_{j}}\) is of the form and we have Further, since each \(\mathcal{A}_{t_{1},\dots ,t_{j}}\) is an algebra generating \(\bigotimes _{i\leq j}{\mathcal{B}}(\mathbb{R}^{d})\) on \(\mathbb{R}^{d\times j}\) and the finite unions of the cylindrical sets \(X^{-1} _{t_{1},\dots ,t_{j}}(\bigotimes _{i\leq j}{\mathcal{B}}( \mathbb{R}^{d}) )\) form an algebra generating the canonical \(\sigma \)-algebra on \(\mathbb{D}(I;\mathbb{R}^{d})\), for every \(0< t\in I\), the family is an algebra generating \(\mathcal{F}^{0}_{t-}=\sigma (X_{u}:u< t)\) on \(\mathbb{D} (I;\mathbb{R}^{d})\); cf. Corollary A.11. Thus for every \(F^{h,\ell }_{i,j,k}\in {\mathcal{F}}^{0}_{t_{j}^{k}-}\) in (4.8), there exists a sequence \((A^{h,\ell ,m}_{i,j,k})_{m\in \mathbb{N}}\) in \(\mathcal{A}^{o}_{t_{j}^{k}-}\) with Finally, by combining the approximations (4.7) and (4.9), and in (4.9) invoking the approximation (4.10) in the sums (4.8), we conclude that for every \(t\in I\) and \(s\geq t\), \(s\in D\), for every \(H\in {\mathcal{E}}(Q)\), there exist a sequence \((J_{n})_{n\in \mathbb{N}}\), \(n=n(k,\ell ,m)\), of elements in \(\mathcal{J}(Q)\) and a sequence \((t_{n})_{n\in \mathbb{N}}\) in \(D\cap [t,s]\) such that \(t_{n}\downarrow t\) and we have Thus for every \(t\in I\) and for any \(s\geq t\), \(s\in D\), the family of simple integrals \(\{(H\bullet X)_{t}:H\in {\mathcal{E}}(Q)\}\) is contained in the closure of \(\{(J\bullet X)_{s}:J\in {\mathcal{J}}(D)\}\) that by (4.6) is bounded in \(Q\)-probability. More precisely, (4.6) is immediate only for \(t \in D\). However, this is sufficient: because \(D\) is dense in \(I\), the set of simple integrals up to \(t \in I\) is the intersection of the sets of simple integrals up to \(s \in D\) for \(s > t\), and an intersection of bounded sets is bounded. Hence for every \(t\in I\), the family of simple integrals \(\{(H \bullet X)_{t}:H\in {\mathcal{E}}(Q)\}\) is bounded in \(Q\)-probability, i.e., the weak^∗ limit \(Q\) is an \((\mathcal{F}_{t})_{t\in I}\)-semimartingale measure and consequently an \((\mathcal{F}^{0}_{t})_{t\in I}\)-semimartingale measure; see e.g. [47, Theorem II.4]. □

Let \((Q_{n})_{n\in \mathbb{N}}\)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.

Let \(i\leq d\) be fixed. We adapt the proof of [44, Theorem 4] and show that the coordinate process \(X^{i}\) is a quasimartingale under \(Q\) on \(\mathbb{D}(I;\mathbb{R}^{d})\). We have for every \(t\in I\) and every \(\varepsilon >0\), where \(b^{i}:=\liminf _{n\rightarrow \infty }\sup _{t\in I}E_{Q_{n}}[|X^{i}_{t}|]< \infty \) by the condition (UB). Thus by Fatou’s lemma, we get for every \(t\in I\). The truncated coordinate process, that is, is a difference of two convex 1-Lipschitz functions of \(X^{i}\) for every \(a,b>0\); so we have see e.g. [53]. Let \(0=t_{0}< t_{1}<\cdots <t_{k}=t\) and \(|f_{j}|\leq 1\), \(j< k\), be continuous \(\mathcal{F}^{0}_{t_{j}-}\)-measurable functions. By (4.12), we have so by Fubini’s theorem, for every \(n\in \mathbb{N}\) and every \(\varepsilon >0\), we get The mappings are lower semicontinuous and bounded from below, see (A.2) and Lemma A.12, so that we have for all \(\varepsilon >0\) that where \(v^{i}:=\liminf _{n\rightarrow \infty }\sup _{t\in I}\text{Var}^{Q_{n}}_{t}(X^{i})< \infty \) by the assumption (UB). Due to (4.11), letting \(\varepsilon \rightarrow 0\), then \(a\rightarrow \infty \) and finally \(b\rightarrow \infty \) in (4.14), right-continuity and applying the monotone convergence theorem twice yield for all \(\mathcal{F}^{0}_{t_{j}-}\)-measurable continuous functions \(|f_{j}|\leq 1\), \(j< k\). Furthermore, by choosing \(f_{j}(X)=f(X_{t_{j}+u})\) before (4.13) for a continuous function \(|f|\leq 1\) on \(\mathbb{R}^{d}\), we conclude that the inequality (4.15) is true for a family of continuous functions that for every \(j< k\) generates the \(\sigma \)-algebra \(\mathcal{F}^{0}_{t_{j}}\); cf. Corollary A.11. Thus by the standard \(L^{1}\)-approximation via Lusin’s theorem and Tietze’s extension theorem, for any \(\mathcal{F}^{0}_{t_{j}}\)-measurable \(|H_{t_{j}}|\leq 1\), for every \(j< k\), the exists a sequence \((f^{n}_{j})_{n\in \mathbb{N}}\), \(|f^{n}_{j}|\leq 1\), of functions satisfying (4.15) such that \(f^{n}_{j}\rightarrow H_{t_{j}}\) in \(L^{1}(Q)\) as \(n\rightarrow \infty \); see e.g. [17] and [16, 2.1.8]. Thus the inequality (4.15) is true for all \(\mathcal{F}^{0}_{t_{j}}\)-measurable functions \(|H_{t_{j}}|\leq 1\), and so the process \(X\) is an \((\mathcal{F}^{0}_{t})_{t\in I}\)-quasimartingale on \((\mathbb{D}(I),\mathcal{F}_{T},Q)\); see [14, Appendix 2, (3.5)]. By Rao’s decomposition theorem, this is a necessary and sufficient condition for \(X\) to be decomposable into a difference \(X=Y-Z\) of two càdlàg \((\mathcal{F}^{0}_{t})_{t\in I}\)-supermartingales \(Y\) and \(Z\) on \((\mathbb{D}(I),\mathcal{F}_{T},Q)\); see [25, Theorem 8.13]. On the other hand, by Föllmer’s lemma, \(Y\) and \(Z\) are \(({ \mathcal{F}}_{t})_{t\in I}\)-supermartingales, see [25, Theorem 2.46]; so again by Rao’s decomposition theorem, the process \(X\) is an \((\mathcal{F}_{t})_{t\in I}\)-quasimartingale on \((\mathbb{D}(I),\mathcal{F}_{T},Q)\). □

Let \((Q_{n})_{n\in \mathbb{N}}\)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.

We adapt the proof of [44, Theorem 11] and show that each coordinate process \(X^{i}\), \(i\leq d\), is a supermartingale under \(Q\). We have \(E_{Q}[|X_{t}|]<\infty \) for every \(t\in I\); cf. (4.11). Moreover, by Lemma 4.10, there exist a subsequence \((Q_{n_{k}})_{k\in \mathbb{N}}\) and a countable dense set \(D\subseteq I\) such that \(T\in D\) if \(I=[0,T]\) and \((Q_{n_{k}})_{k\in \mathbb{N}}\) 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 \(c>0\), i.e., By the condition (UI) and the fact that each \(Q_{n_{k}}\) is a supermartingale measure for \(X^{i,b}\), we have for every \(0< a< b\) that where see e.g. [57, Theorem 2.20]. Consequently, by Corollary A.11, we have for every \(s< t\) in \(D\) and \(F\in {\mathcal{F}}^{0}_{s-}\). Letting \(b\rightarrow \infty \) and then \(a\rightarrow \infty \) in (4.16), applying monotone convergence twice yields the same inequality for the coordinate process \(X^{i}\). By Föllmer’s lemma, the inequality extends immediately to the whole set \(I\), and further to \(F\in {\mathcal{F}}^{0}_{s+}\). Indeed, we have for every \(F\in {\mathcal{F}}^{0}_{s+}\); cf. [25, Theorem 2.44]. □

Let \((Q_{n})_{n\in \mathbb{N}}\)be a sequence of probability measures converging in the weak^∗ topology to \(Q\). Then we have the following:

(a) By [25, Lemma 15.49], the \(C\)-tightness of the sequence \((Q_{n})_{n\in \mathbb{N}}\) implies the convergence along a subsequence in the weak^∗ topology of the Skorokhod \(J^{1}\)-topology, and a fortiori in any weaker topology, to the law of a continuous process, which is the limit \(Q\); cf. Proposition 5.12 below.

(b) Let \(s< t\) in \(I\) be fixed and take a bounded Lipschitz-continuous function \({g:\mathbb{R}^{d}\rightarrow \mathbb{R}}\) with a Lipschitz constant \(L(g)\leq 1\). For each \(n\in \mathbb{N}\), there exists a bounded Lipschitz-continuous function \(f_{n}:\mathbb{R}^{d}\rightarrow \mathbb{R}\) such that Further, we can assume that \(L(f_{n})\leq 1\) and \(\|f_{n}\|_{\infty }\leq \|g\|_{\infty }<\infty \) for all \(n\in \mathbb{N}\). Thus by the Arzelà–Ascoli theorem, there exist a subsequence \((f_{n_{k}})_{k\in \mathbb{N}}\) and a bounded Lipschitz-continuous function \(f:\mathbb{R}^{d}\rightarrow \mathbb{R}\) with \(L(f)\leq 1\) such that \((f_{n_{k}})\) converges to \(f\) uniformly on compacts as \(k\rightarrow \infty \). Further, by Lemma 4.10, there exists a further subsequence of \((Q_{n_{k}})_{k\in \mathbb{N}}\), that we again denote by \((Q_{n_{k}})_{k\in \mathbb{N}}\), converging in finite-dimensional distributions to \(Q\) on a dense set \(D\subseteq I\) such that \(T\in D\) if \(I=[0,T]\). Let \(i\in \mathbb{N}\) and \(0\leq s_{1}< s_{2}<\cdots < {s_{i}}\leq s<t\) in \(D\) and take a bounded continuous compactly supported \(\alpha :\mathbb{R}^{d}\rightarrow \mathbb{R}\) and a bounded continuous \(\beta :\mathbb{R}^{d\times i}\rightarrow \mathbb{R}\). By (4.18), we have Since \((f_{n_{k}})_{k\in\mathbb{N}}\) converges uniformly to \(f\) on the compact support of each \(\alpha\) and \((Q_{n_{k}})_{k\in\mathbb{N}}\) converges to \(Q\) in finite-dimensional distributions on the set \(D\) as \(k\rightarrow \infty \), from (4.19), by the vectorial Portemanteau lemma, see e.g. [57, Lemma 2.2], we get Because (4.20) holds for all bounded continuous \(\alpha \) with compact support, we get and as (4.21) holds for all bounded continuous \(\beta \), we obtain for every bounded \(\mathcal{F}^{0}_{s}\)-measurable function \(h\), for every \(s< t\) in \(D\). Since \(D\) is a dense subset of \(I\) and \(s\mapsto f(X_{s})\) is bounded and right-continuous on \(I\), (4.22) extends by dominated convergence to the whole set \(I\), and further to every bounded \(\mathcal{F}_{s}\)-measurable function \(h\); cf. (4.17). □

A family of semimartingale measures satisfying the condition (UT) satisfies the condition (US).

Following Stricker [54, Theorem 2], we define the family of stopping times and approximate each \(\tau ^{i,c}\) from the right with the sequence of simple stopping times Since we are assuming a right-continuous filtration \((\mathcal{F}_{t})_{t\in I}\) and \(X^{i}\) is right-continuous, the hitting times \(\tau ^{i,c}\) and consequently their approximations \(\tau ^{i,c}_{n}\) are indeed stopping times. Moreover, since each \(\tau ^{i,c}_{n}\) takes only finitely many values in \([0,t]\), every process \(|H^{n}|\leq 1\) of the form is an elementary predictable integrand; see (2.1). Now due to the right-continuity of \(X^{i}\), dominated convergence gives for every \(Q\in {\mathcal{Q}}\) that for every \(t\in I\) and every \(c>0\). By the condition (UT), the left-hand side in (4.26) tends to 0 uniformly over \(Q\in {\mathcal{Q}}\) for every \(i\leq d\) and every \(t\in I\) as \(c\rightarrow \infty \). Similarly, for \(a< b\), we define recursively for all \(k\in \mathbb{N}_{0}\) the stopping times \(\sigma ^{i,a}_{0}=\tau ^{i,b}_{0}=0\) and and the respective decreasing sequences \((\sigma ^{i,a}_{k,n})_{n\in \mathbb{N}}\) and \((\tau ^{i,b}_{k,n})_{n\in \mathbb{N}}\) of approximating stopping times taking only finitely many values on finite intervals; cf. (4.24). The processes \(|H^{m,n}|\leq 1\), \(m,n\in \mathbb{N}\), of the form are finite linear combinations of processes of the form (4.25) so that each process \(|H^{m,n}|\leq 1\) is an elementary predictable integrand. Moreover, we have By the condition (UT), for every \(a< b\), the right-hand side of (4.27) tends to zero uniformly over \(Q\in {\mathcal{Q}}\) as \(c\rightarrow 0\); cf. (4.6) and (4.7). Thus by Corollary 5.9 below, the family \(\mathcal{Q}\) satisfies the condition (US). □

A subset \(K\)of \(\mathbb{D}([0,T];\mathbb{R})\), \(T<\infty \), is relatively sequentially \(S\)-compact if and only if it satisfies the conditions

A right-continuous function \(\omega :[0,T]\rightarrow \mathbb{R}\) is càdlàg if and only if it satisfies the conditions

A subset \(K\)of \(\mathbb{D}([0,\infty );\mathbb{R})\)is relatively sequentially \(S\)-compact if and only if the set \(K\)restricted to \([0,t]\)satisfies (5.4) for every \(0< t<\infty \).

Let \(K=K^{1}\times \cdots \times K^{d}\)be a Cartesian product set in the Skorokhod space \(\mathbb{D}(I;\mathbb{R}^{d})\)endowed with \(S\)or \(S^{*}\). Then the set \(K\)is compact if for each \(i\leq d\), there exist a (non-decreasing) function \(C^{i}_{q,r}:I\rightarrow \mathbb{R}_{+}\)for all \(q< r\)in ℚ and a (non-decreasing) function \(M^{i}:I\rightarrow \mathbb{R}_{+}\)such that where the intersection is taken over all rationals \(q< r\)and \([\omega ^{i}]^{t}\)denotes the restriction of \(\omega ^{i}\)to \([0,t]\).

The product set \(K=K^{1}\times \cdots \times K^{d}\) is compact in the product topology if each set \(K^{i}\) in the product is \(S\)-compact; cf. Proposition 5.4 (i). For any two real numbers \(a< b\), one can find rationals \(r< q\) with \(a< r< q< b\), and so it is sufficient to let \(a< b\) range through rationals in Proposition 5.6. Given that for each \(t<\infty \) and every \(i\leq d\), one has \(N^{q,r} ([\omega ^{i}]^{t} )\leq C^{i}_{q,r}(t)\) and \(\|[\omega ^{i}]^{t}\|_{\infty }\leq M^{i}(t)\) for some constants \(C^{i}_{q,r}(t)<\infty \) and \(M^{i}(t)<\infty \), by Proposition 5.6, each set \(K^{i}\) is relatively \(S\)-compact. By Lemma A.12, the mappings \(N^{q,r}(\cdot )\) and \(\|\cdot \|_{\infty }\) are lower semicontinuous in the MZ-topology, and by Proposition 5.12 below in the \(S\)-topology as well; so the sets \(K^{i}\), \(i\leq d\), are \(S\)-closed. Thus the sets \(K^{i}\) are \(S\)-compact, and hence the product set \(K\) is compact in the product topology. □

(a) The mappings are \(S^{*}\)-continuous on \(\mathbb{D}(I;\mathbb{R}^{d})\) whenever \(G\) is measurable as a mapping of \((t,x)\), continuous as a mapping of \(x\) for every \(t\in I\) and such that and \(\mu \) is a diffusive (i.e., atomless) measure on \(I\); see [31, Corollary 2.11].

(b) The mapping is \(S^{*}\)-continuous on \(\mathbb{D}([0,T];\mathbb{R}^{d})\); see [31, Remark 2.4].

By Proposition 5.12 below, the \(S^{*}\)-topology is stronger than the Meyer–Zheng topology MZ; so the functions are lower semicontinuous in the \(S^{*}\)-topology on \(\mathbb{D}(I;\mathbb{R}^{d})\). See Lemma A.12.