Journal of Probability and Statistics

Journal of Probability and Statistics / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 580292 |

Andriy Yurachkivsky, "Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale", Journal of Probability and Statistics, vol. 2011, Article ID 580292, 34 pages, 2011.

Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale

Academic Editor: Tomasz J. Kozubowski
Received24 May 2011
Accepted03 Oct 2011
Published22 Nov 2011


Let for each π‘›βˆˆβ„•π‘‹π‘› be an ℝ𝑑-valued locally square integrable martingale w.r.t. a filtration (ℱ𝑛(𝑑),π‘‘βˆˆβ„+) (probability spaces may be different for different 𝑛). It is assumed that the discontinuities of 𝑋𝑛 are in a sense asymptotically small as π‘›β†’βˆž and the relation 𝖀(𝑓(βŸ¨π‘§π‘‹π‘›βŸ©(𝑑))|ℱ𝑛(𝑠))βˆ’π‘“(βŸ¨π‘§π‘‹π‘›βŸ©(𝑑))𝖯→0 holds for all 𝑑>𝑠>0, row vectors 𝑧, and bounded uniformly continuous functions 𝑓. Under these two principal assumptions and a number of technical ones, it is proved that the 𝑋𝑛's are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes (𝑋𝑛(0),βŸ¨π‘‹π‘›βŸ©) converge in distribution to some (βˆ˜π‘‹,𝐻), then a sequence (𝑋𝑛) converges in distribution to a continuous local martingale 𝑋 with initial value βˆ˜π‘‹ and quadratic characteristic 𝐻, whose finite-dimensional distributions are explicitly expressed via those of (βˆ˜π‘‹,𝐻).

1. Introduction

The theory of functional limit theorems for martingales may appear finalized in the monographs [1, 2]. This paper focuses at two points, where the classical results can be refined.

(1) The convergence in distribution to a local martingale with 𝒒-conditional increments has been studied hitherto in the model, where the 𝜎-algebra 𝒒 enters the setting along with the prelimit processes. This assumption is worse than restrictiveβ€”it is simply unnatural when one studies the convergence in distribution, not in probability. In the present paper, conditions ensuring asymptotic conditional independence of increments for a sequence of locally square integrable martingales are formulated in terms of quadratic characteristics of the prelimit processes (Theorem 4.5). Our approach to the proving of this property is based on the idea to combine the Stone-Weierstrass theorem (actually its slight modificationβ€”Lemma 2.2) with an elementary probabilistic resultβ€”Lemma 2.4, which issues in Corollaries 2.7 and 2.8. These corollaries, as well as Lemma 2.4 itself and the cognate Lemma 2.5, will be our tools.

(2) The main object of study in [1, 2] is semimartingale. So, some specific for local martingales facts are passed by. Thus, Theorem VI.6.1 and Corollary VI.6.7 in [2] assert that under appropriate assumptions about semimartingales 𝑍𝑛, the relation 𝑍𝑛lawβŸΆπ‘,(βˆ—) where 𝑍 also is a semimartingale, entails the stronger one (Z𝑛,[𝑍𝑛])lawβ†’(𝑍,[𝑍]) (below, the notation of convergence in law will be changed). For locally square integrable martingales, one can modify the problem as follows. Let relation (*) be fulfilled. What extra assumptions ensure that 𝑍 is a continuous local martingale and 𝑍𝑛,βŸ¨π‘π‘›βŸ©ξ€Έlaw⟢(𝑍,βŸ¨π‘βŸ©)?(βˆ—βˆ—) There is neither an answer nor even the question in [1, 2]. A simple set of sufficient conditions is provided by Corollary 5.2 (weaker but not so simple conditions are given by Corollary 5.5). Recalling that the quadratic variation of a continuous local martingale coincides with its quadratic characteristic, we see that the last two relations imply together asymptotic proximity of [𝑍𝑛] and βŸ¨π‘π‘›βŸ©. Actually, this conclusion requires even less conditions than in Corollary 5.2. They are listed in Corollary 5.3.

The main results of the paper are, in a sense, converse to Corollaries 5.2 and 5.5. They deal with the problem: what conditions should be adjoined to βŸ¨π‘π‘›βŸ©law→𝐻 in order to ensure (**), where 𝑍 is a continuous local martingale with quadratic characteristic 𝐻? If the assumptions about the prelimit processes do not guarantee that 𝐻 performing as βŸ¨π‘βŸ© determines the distribution of 𝑍, then results of this kind assert existence of convergent subsequences but not convergence of the whole sequence (Theorems 5.1 and 5.4). Combining Theorems 5.4 with 4.5, we obtain Theorem 5.6 asserting that the whole sequence converges to a continuous local martingale whose finite-dimensional distributions are explicitly expressed via those of its initial value and quadratic characteristic. The expression shows that the limiting process has conditionally independent incrementsβ€”but this conclusion is nothing more than a comment to the theorem.

The proving of the main results needs a lot of preparation. Those technical results which do not deal with the notion of martingale are gathered in Section 2 (excluding Section 2.1), and the more specialized ones are placed in Section 3. The rationale in Sections 3 and 4 would be essentially simpler if we confined ourselves to quasicontinuous processes (for a locally square integrable martingale, this property is tantamount to continuity of its quadratic characteristic). To dispense with this restriction, we use a special technique sketched in Section 2.1.

All vectors are thought of, unless otherwise stated, as columns. The tensor square π‘₯π‘₯⊀ of π‘₯βˆˆβ„π‘‘ will be otherwise written as π‘₯βŠ—2. We use the Euclidean norm |β‹…| of vectors and the operator norm β€–β‹…β€– of matrices. The symbols β„π‘‘βˆ—,𝔖, and 𝔖+ signify: the space of 𝑑-dimensional row vectors, the class of all symmetric square matrices of a fixed size (in our case—𝑑) with real entries, and its subclass of nonnegative (in the spectral sense) matrices, respectively.

By Cb(𝑋), we denote the space of complex-valued bounded continuous functions on a topological space 𝑋. If 𝑋=β„π‘˜ and the dimension π‘˜ is determined by the context or does not matter, then we write simply Cb.

Our notation of classes of random processes follows [3]. In particular, β„³(𝔽) and β„³(𝔽) signify the class of all martingales with respect to a filtration (= flow of 𝜎-algebras) 𝔽≑(β„±(𝑑),π‘‘βˆˆβ„+) and its subclass of uniformly integrable martingales. An 𝔽-martingale π‘ˆ will be called: square integrable if 𝖀|π‘ˆ(𝑑)|2<∞ for all 𝑑 and uniformly square integrable if sup𝑑𝖀|π‘ˆ(𝑑)|2<∞. The classes of such processes will be denoted β„³2(𝔽) and β„³2(𝔽), respectively. The symbol 𝔽 will be suppressed if the filtration either is determined by the context or does not matter. If 𝒰 is a class of 𝔽-adapted process, then by ℓ𝒰 we denote the respective local class (see [2, Definition I.1.33], where the notation 𝒰loc is used). Members of β„“β„³, and β„“β„³2 are called local martingales and locally (better local) square integrable martingales, respectively. All processes, except quadratic variations and quadratic characteristics, are implied ℝ𝑑-valued, where 𝑑 is chosen arbitrarily and fixed.

The integral βˆ«π‘‘0πœ‘(𝑠)d𝑋(𝑠) will be written shortly (following [1, 2]) as πœ‘βˆ˜π‘‹(𝑑) if this integral is pathwise (i.e., 𝑋 is a process of locally bounded variation) or πœ‘β‹…π‘‹(𝑑) if it is stochastic. We use properties of stochastic integral and other basic facts of stochastic analysis without explanations, relegating the reader to [1–4]. The quadratic variation (see the definition in Section 2.3 [3] or Definition I.4.45 together with Theorem I.4.47 in [2]) of a random process πœ‰ and the quadratic characteristic of π‘βˆˆβ„“β„³2 will be (and already were) denoted [πœ‰] and βŸ¨π‘βŸ©, respectively. They take values in 𝔖+, which, of course, does not preclude to regard them as ℝ𝑑2-valued random processes.

2. Some Technical Results

The Stone-Weierstrass theorem (see, e.g., [5]) concerns compact spaces only. In the following two, its minor generalizations (for real-valued and complex-valued functions, resp.) both the compactness assumption and the conclusion (that the approximation is uniform on the whole space) are weakened. They are proved likewise their celebrated prototype if one argues for the restrictions of continuous functions to some compact subset fixed beforehand.

Lemma 2.1. Let 𝔄 be an algebra of real-valued bounded continuous functions on a topological space 𝑇. Suppose that 𝔄 separates points of 𝑇 and contains the module of each its member and the unity function. Then, for any real-valued bounded continuous function 𝐹, compact set π΅βŠ‚π‘‡, and positive number πœ€, there exists a function πΊβˆˆπ”„ such that β€–πΊβ€–βˆžβ‰€β€–πΉβ€–βˆž and maxπ‘₯∈𝐡|𝐹(π‘₯)βˆ’πΊ(π‘₯)|<πœ€.

Lemma 2.2. Let 𝔄 be an algebra of complex-valued bounded continuous functions on a topological space 𝑇. Suppose that 𝔄 separates points of 𝑇, and contains the conjugate of each its member and the unity function. Then for any complex-valued bounded continuous function 𝐹, compact set π΅βŠ‚π‘‡, and positive number πœ€ there exists a function πΊβˆˆπ”„ such that β€–πΊβ€–βˆžβ‰€β€–πΉβ€–βˆž and maxπ‘₯∈𝐡|𝐹(π‘₯)βˆ’πΊ(π‘₯)|<πœ€.

We consider henceforth sequences of random processes or random variables given, maybe, on different probability spaces. So, for the 𝑛th member of a sequence, 𝖯 and 𝖀 should be understood as 𝖯𝑛 and 𝖀𝑛. In what follows, β€œu.i.” means β€œuniformly integrable”.

Lemma 2.3. In order that a sequence of random variables be u.i., it is necessary and sufficient that each its subsequence contain a u.i. subsequence.

Proof. Necessity is obvious; let us prove sufficiency.
Suppose that a sequence (𝛼𝑛) is not u.i. Then, there exists π‘Ž>0 such that for all 𝑁>0limπ‘›β†’βˆžπ–€||𝛼𝑛||𝐼||𝛼𝑛||ξ€Ύ>𝑁β‰₯2π‘Ž.(2.1) Consequently, there exists an increasing sequence (π‘›π‘˜) of natural numbers such that π–€π›½π‘˜πΌ{π›½π‘˜>π‘˜}β‰₯π‘Ž, where π›½π‘˜=|π›Όπ‘›π‘˜|. Then, for any infinite set π½βŠ‚β„• and 𝑁>0, we have limπ‘˜β†’βˆž,π‘˜βˆˆπ½π–€π›½π‘˜πΌξ€½π›½π‘˜ξ€Ύ>𝑁β‰₯π‘Ž,(2.2) which means that the subsequence (π›Όπ‘›π‘˜) does not contain u.i. subsequences.

Lemma 2.4. Let for each π‘›πœ‰π‘›1,…,πœ‰π‘›π‘ be random variables given on a probability space (Ω𝑛,ℱ𝑛,𝖯𝑛), and ℋ𝑛 a sub-𝜎-algebra of ℱ𝑛. Suppose that for each π‘—βˆˆ{1,…,𝑝}, π–€ξ€·πœ‰π‘›π‘—βˆ£β„‹π‘›ξ€Έβˆ’πœ‰π–―π‘›π‘—βŸΆ0asπ‘›βŸΆβˆž,(2.3) and for any π½βŠ‚{1,…,𝑝} the sequence (βˆπ‘—βˆˆπ½πœ‰π‘›π‘—,π‘›βˆˆβ„•) is u.i. Then, 𝖀𝑝𝑗=1πœ‰π‘›π‘—βˆ£β„‹π‘›ξƒͺβˆ’π‘ξ‘π‘—=1πœ‰π–―π‘›π‘—βŸΆ0.(2.4)

Proof. Denote πœ‚π‘›π‘—=𝖀(πœ‰π‘›π‘—βˆ£β„‹π‘›). By the second assumption, the sequences (πœ‰π‘›π‘—,π‘›βˆˆβ„•),(πœ‚π‘›π‘—,π‘›βˆˆβ„•),𝑗=1,…,𝑝, are stochastically bounded, which together with the first assumption yields ξ‘π‘—βˆˆπ½πœ‚π‘›π‘—βˆ’ξ‘π‘—βˆˆπ½πœ‰π–―π‘›π‘—βŸΆ0,(2.5) for any π½βŠ‚{1,…,𝑝}. Hence, writing the identity π–€ξ€·πœ‰π‘›1πœ‰π‘›2βˆ£β„‹π‘›ξ€Έπœ‰=𝖀𝑛1βˆ’πœ‚π‘›1πœ‰ξ€Έξ€·π‘›2βˆ’πœ‚π‘›2ξ€Έβˆ£β„‹π‘›ξ€Έ+πœ‚π‘›1πœ‚π‘›2,(2.6) and using both assumptions, we get (2.4) for 𝑝=2. For arbitrary 𝑝, this relation is proved by induction whose step coincides, up to notation, with the above argument.

The proof of the next statement is similar.

Lemma 2.5. Let 𝛼𝑛 and 𝛽𝑛 be random variables given on a probability space (Ω𝑛,ℱ𝑛,𝖯𝑛) and ℋ𝑛 a sub-𝜎-algebra of ℱ𝑛. Suppose that π›Όπ‘›ξ€·π›Όβˆ’π–€π‘›βˆ£β„‹π‘›ξ€Έπ–―βŸΆ0,(2.7) and the sequences (𝛼𝑛), (𝛽𝑛) and (𝛼𝑛𝛽𝑛) are u.i. Then, π–€ξ€·π›Όπ‘›π›½π‘›βˆ£β„‹π‘›ξ€Έβˆ’π›Όπ‘›π–€ξ€·π›½π‘›βˆ£β„‹π‘›ξ€Έπ–―βŸΆ0.(2.8)

Lemma 2.6. Let π‘›βˆˆβ„•Ξžπ‘› be an β„π‘˜-valued random variable given on a probability space (Ω𝑛,ℱ𝑛,𝖯𝑛), and ℋ𝑛 a sub-𝜎-algebra of ℱ𝑛. Suppose that limπ‘β†’βˆžlimπ‘›β†’βˆžπ–―ξ€½||Ξžπ‘›||ξ€Ύ>𝑁=0,(2.9) and the relation π–€ξ€·πΉξ€·Ξžπ‘›ξ€Έβˆ£β„‹π‘›ξ€Έξ€·Ξžβˆ’πΉπ‘›ξ€Έπ–―βŸΆ0(2.10) holds for all 𝐹 from some class of complex-valued bounded continuous functions on β„π‘˜ which separates points of the latter. Then, it holds for all 𝐹∈Cb.

Proof. Let 𝔄 denote the class of all complex-valued bounded continuous functions on β„π‘˜ satisfying (2.10). Obviously, it is linear. By Lemma 2.4, it contains the product of any two its members. So, 𝔄 is an algebra. By assumption, it separates points of β„π‘˜. The other two conditions of Lemma 2.2 are satisfied trivially. Thus, that lemma asserts that for any 𝐹∈Cb,𝑁>0 and πœ€>0, there exists a function πΊβˆˆπ”„ such that β€–πΊβ€–βˆžβ‰€β€–πΉβ€–βˆž and max|π‘₯|≀𝑁|𝐹(π‘₯)βˆ’πΊ(π‘₯)|<πœ€. Then, ||πΉξ€·Ξžπ‘›ξ€Έξ€·Ξžβˆ’πΊπ‘›ξ€Έ||𝐼||Ξžπ‘›||ξ€Ύ||πΉξ€·Ξžβ‰€π‘<πœ€,π‘›ξ€Έξ€·Ξžβˆ’πΊπ‘›ξ€Έ||𝐼||Ξžπ‘›||ξ€Ύ>𝑁≀2β€–πΉβ€–βˆžπΌξ€½||Ξžπ‘›||ξ€Ύ.>𝑁(2.11) By the choice of πΊπ–€ξ€·πΊξ€·Ξžπ‘›ξ€Έβˆ£β„‹π‘›ξ€Έξ€·Ξžβˆ’πΊπ‘›ξ€Έπ–―βŸΆ0,(2.12) whence by the dominated convergence theorem 𝖀||π–€ξ€·πΊξ€·Ξžπ‘›ξ€Έβˆ£β„‹π‘›ξ€Έξ€·Ξžβˆ’πΊπ‘›ξ€Έ||⟢0.(2.13) Writing the identity π–€ξ€·πΉξ€·Ξžπ‘›ξ€Έβˆ£β„‹π‘›ξ€Έξ€·Ξžβˆ’πΉπ‘›ξ€ΈπΉξ€·Ξž=π–€ξ€·ξ€·π‘›ξ€Έξ€·Ξžβˆ’πΊπ‘›πΌξ€½||Ξžξ€Έξ€Έπ‘›||ξ€Ύβ‰€π‘βˆ£β„‹π‘›ξ€ΈπΉξ€·Ξž+π–€ξ€·ξ€·π‘›ξ€Έξ€·Ξžβˆ’πΊπ‘›πΌξ€½||Ξžξ€Έξ€Έπ‘›||ξ€Ύ>π‘βˆ£β„‹π‘›ξ€Έξ€·πΊξ€·Ξž+π–€π‘›ξ€Έβˆ£β„‹π‘›ξ€Έξ€·Ξžβˆ’πΊπ‘›ξ€Έ+ξ€·πΊξ€·Ξžπ‘›ξ€Έξ€·Ξžβˆ’πΉπ‘›πΌξ€½||Ξžξ€Έξ€Έπ‘›||ξ€Ύ+ξ€·πΊξ€·Ξžβ‰€π‘π‘›ξ€Έξ€·Ξžβˆ’πΉπ‘›πΌξ€½||Ξžξ€Έξ€Έπ‘›||ξ€Ύ,>𝑁(2.14) we get from (2.11)–(2.13) limπ‘›β†’βˆžπ–€||π–€ξ€·πΉξ€·Ξžπ‘›ξ€Έβˆ£β„‹π‘›ξ€Έξ€·Ξžβˆ’πΉπ‘›ξ€Έ||≀2πœ€+4β€–πΉβ€–βˆžlimπ‘›β†’βˆžπ–―ξ€½||Ξžπ‘›||ξ€Ύ,>𝑁(2.15) which together with (2.9) and due to arbitrariness of πœ€ proves (2.10).

Corollary 2.7. Let for each π‘›πœπ‘›1,…,πœπ‘›π‘ be ℝ𝑑-valued random variables given on a probability space (Ω𝑛,ℱ𝑛,𝖯𝑛) and ℋ𝑛 a sub-𝜎-algebra of ℱ𝑛. Suppose that the relations limπ‘β†’βˆžlimπ‘›β†’βˆžπ–―ξ€½||πœπ‘›π‘—||ξ€Ύπ–€ξ€·π‘”ξ€·πœ>𝑁=0,(2.16)π‘›π‘—ξ€Έβˆ£β„‹π‘›ξ€Έξ€·πœβˆ’π‘”π‘›π‘—ξ€Έπ–―βŸΆ0asπ‘›βŸΆβˆž(2.17) hold for all π‘—βˆˆ{1,…,𝑝} and 𝑔 from some class 𝔉 of complex-valued bounded continuous functions on ℝ𝑑 which separates points of the latter. Then, π–€ξ€·πΉξ€·πœπ‘›1,…,πœπ‘›π‘ξ€Έβˆ£β„‹π‘›ξ€Έξ€·πœβˆ’πΉπ‘›1,…,πœπ‘›π‘ξ€Έπ–―βŸΆ0,(2.18) for all 𝐹∈Cb(ℝ𝑝𝑑).

Proof. Denote Ξžπ‘›=(πœπ‘›1,…,πœπ‘›π‘). Condition (2.17) implies by Lemma 2.4 that relation (2.10) is valid for all 𝐹 of the kind 𝐹(π‘₯1,…,π‘₯π‘βˆ)=𝑝𝑖=1𝑔𝑖(π‘₯𝑖), where π‘”π‘–βˆˆπ”‰. Obviously, such functions separate points of ℝ𝑝𝑑. Furthermore, condition (2.16) where 𝑗 runs over {1,…,𝑝} is tantamount to (2.9). It remains to refer to Lemma 2.6.

Corollary 2.8. Let for each 𝑛𝐾𝑛 be an 𝔖-valued random process given on a probability space (Ω𝑛,ℱ𝑛,𝖯𝑛), ℋ𝑛 a sub-𝜎-algebra of ℱ𝑛, and πœπ‘›0 an ℋ𝑛-measurable β„π‘š-valued random variable. Suppose that the relations limπ‘β†’βˆžlimπ‘›β†’βˆžπ–―ξ€½β€–β€–πΎπ‘›β€–β€–ξ€Ύπ–€ξ€·π‘“ξ€·(𝑑)>𝑁=0,𝑧𝐾𝑛(𝑑)π‘§βŠ€ξ€Έβˆ£β„‹π‘›ξ€Έξ€·βˆ’π‘“π‘§πΎπ‘›(𝑑)π‘§βŠ€ξ€Έπ–―βŸΆ0,(2.19) and (2.16) hold for 𝑗=0, all 𝑑>0 and any bounded uniformly continuous function 𝑓 on ℝ. Then, for any π‘™βˆˆβ„•,𝑠𝑙>β‹―>𝑠1>0 and 𝐹∈Cb(β„π‘šΓ—π”–π‘™) the relation π–€ξ€·πΉξ€·πœπ‘›0,𝐾𝑛𝑠1ξ€Έ,…,πΎπ‘›ξ€·π‘ π‘™ξ€Έξ€Έβˆ£β„‹π‘›ξ€Έξ€·πœβˆ’πΉπ‘›0,𝐾𝑛𝑠1ξ€Έ,…,πΎπ‘›ξ€·π‘ π‘™ξ€Έξ€Έπ–―βŸΆ0(2.20) is valid.

Recall that for any π΅βˆˆπ”–β€–π΅β€–=maxπ‘₯βˆˆπ‘†π‘‘βˆ’1||π‘₯⊀||,𝐡π‘₯(2.21) where π‘†π‘‘βˆ’1 is the unit sphere in ℝ𝑑.

Lemma 2.9. For any symmetric matrices 𝐡1 and 𝐡2, maxπ‘₯βˆˆπ‘†π‘‘βˆ’1||π‘₯⊀𝐡1π‘₯π‘₯⊀𝐡2π‘₯||≀‖‖𝐡1‖‖‖‖𝐡2β€–β€–.(2.22)

Proof. It suffices to note that the left-hand side of the equality does not exceed maxπ‘₯,π‘¦βˆˆπ‘†π‘‘βˆ’1|π‘₯⊀𝐡1π‘₯π‘¦βŠ€π΅2𝑦|.

Let 𝑋,𝑋1,𝑋2… be ℝ𝑑-valued random processes with trajectories in the Skorokhod space D (= cΓ dlΓ g processes on ℝ+). We write 𝑋𝑛D→𝑋 if the induced by the processes 𝑋𝑛 measures on the Borel 𝜎-algebra in D weakly converge to the measure induced by 𝑋. If herein 𝑋 is continuous, then we write 𝑋𝑛C→𝑋. We say that a sequence (𝑋𝑛) is relatively compact (r.c.) in D (in C) if each its subsequence contains, in turn, a subsequence converging in the respective sense. The weak convergence of finite-dimensional distributions of random processes, in particular the convergence in distribution of random variables, will be denoted dβ†’. Likewise d= means equality of distributions.

Denote Ξ (𝑑,π‘Ÿ)={(𝑒,𝑣)βˆˆβ„2∢(π‘£βˆ’π‘Ÿ)+≀𝑒≀𝑣≀𝑑},Δ𝒰(𝑓;𝑑,π‘Ÿ)=sup(𝑒,𝑣)∈Π(𝑑,π‘Ÿ)||||𝑓(𝑣)βˆ’π‘“(𝑒)(π‘“βˆˆD,𝑑>0,π‘Ÿ>0).(2.23) Proposition VI.3.26 (items (i), (ii)) [2] together with VI.3.9 [2] asserts that a sequence (πœ‰π‘›) of cΓ dlΓ g random processes is r.c. in C if and only if for all positive 𝑑 and πœ€limπ‘β†’βˆžlimπ‘›β†’βˆžπ–―ξ‚»sup𝑠≀𝑑||πœ‰π‘›||ξ‚Ό(𝑠)>𝑁=0,limπ‘Ÿβ†’0limπ‘›β†’βˆžπ–―ξ€½Ξ”π’°ξ€·πœ‰π‘›ξ€Έξ€Ύ;𝑑,π‘Ÿ>πœ€=0.(2.24) Hence, two consequences are immediate.

Corollary 2.10. Let (πœ‰π‘›) and (Ξžπ‘›) be sequences of ℝ𝑑-valued and β„π‘š-valued, respectively, cΓ dlΓ g processes such that (Ξžπ‘›) is r.c. in C,|πœ‰π‘›(0)|≀|Ξžπ‘›(0)| and for any 𝑣>𝑒β‰₯0||πœ‰π‘›(𝑣)βˆ’πœ‰π‘›||≀||Ξ(𝑒)𝑛(𝑣)βˆ’Ξžπ‘›||(𝑒).(2.25) Then, the sequence (πœ‰π‘›) is also r.c. in C.

Corollary 2.11. Let (πœ‰π‘›) and (πœπ‘›) be r.c. in C sequences of cΓ dlΓ g processes taking values in ℝ𝑑 and ℝ𝑝, respectively. Suppose also that for each π‘›πœ‰π‘› and πœπ‘› are given on a common probability space. Then the sequence of ℝ𝑑+𝑝-valued processes (πœ‰π‘›,πœπ‘›) is also r.c. in C.

Lemma 2.12. Let (πœ‚π‘™π‘›,𝑙,π‘›βˆˆβ„•), (πœ‚π‘™), and (πœ‚π‘›) be sequences of cΓ dlΓ g random processes such that for any positive 𝑑 and πœ€limπ‘™β†’βˆžlimπ‘›β†’βˆžπ–―ξ‚»sup𝑠≀𝑑||πœ‚π‘™π‘›(𝑠)βˆ’πœ‚π‘›||ξ‚Ό(𝑠)>πœ€=0,(2.26) for each π‘™πœ‚π‘™π‘›DβŸΆπœ‚π‘™asπ‘›βŸΆβˆž,(2.27) the sequence (πœ‚π‘™) is r.c. in D. Then, there exists a random process πœ‚ such that πœ‚π‘™Dβ†’πœ‚.

Proof. Let 𝜌 be a bounded metric in D metrizing Skorokhod's π’₯-convergence (see, e.g., [2, VI.1.26]). Then, condition (2.26) with arbitrary 𝑑 and πœ€ implies that limπ‘™β†’βˆžlimπ‘›β†’βˆžξ€·πœ‚π–€πœŒπ‘™π‘›,πœ‚π‘›ξ€Έ=0.(2.28) Hence, by the triangle inequality, we have limπ‘šβ†’βˆžπ‘˜β†’βˆžlimπ‘›β†’βˆžξ€·πœ‚π–€πœŒπ‘šπ‘›,πœ‚π‘˜π‘›ξ€Έ=0.(2.29)
Let 𝐹 be a uniformly continuous with respect to 𝜌 bounded functional on D. Denote 𝐴=supπ‘₯∈D|𝐹(π‘₯)|, πœ—(π‘Ÿ)=supπ‘₯,π‘¦βˆˆD∢𝜌(π‘₯,𝑦)<π‘Ÿ|𝐹(π‘₯)βˆ’πΉ(𝑦)|. Then, πœ—(0+)=0 and for any π‘Ÿ>0𝖀||πΉξ€·πœ‚π‘šπ‘›ξ€Έξ€·πœ‚βˆ’πΉπ‘˜π‘›ξ€Έ||ξ€½πœŒξ€·πœ‚β‰€π΄π–―π‘šπ‘›,πœ‚π‘˜π‘›ξ€Έξ€Ύ>π‘Ÿ+πœ—(π‘Ÿ),(2.30) which together with (2.29) yields limπ‘šβ†’βˆžπ‘˜β†’βˆžlimπ‘›β†’βˆž||ξ€·πœ‚π–€πΉπ‘šπ‘›ξ€Έξ€·πœ‚βˆ’π–€πΉπ‘˜π‘›ξ€Έ||=0.(2.31) By condition (2.27), limπ‘›β†’βˆž||ξ€·πœ‚π–€πΉπ‘šπ‘›ξ€Έξ€·πœ‚βˆ’π–€πΉπ‘˜π‘›ξ€Έ||=||𝖀𝐹(πœ‚π‘šξ€·πœ‚)βˆ’π–€πΉπ‘˜ξ€Έ||,(2.32) which jointly with (2.31) proves fundamentality and, therefore, convergence of the sequence (𝖀𝐹(πœ‚π‘™),π‘™βˆˆβ„•). Now, the desired conclusion emerges from relative compactness of (πœ‚π‘™) in D.

Corollary 2.13. Let the conditions of Lemma 2.12 be fulfilled. Then, πœ‚π‘›Dβ†’πœ‚, where πœ‚ is the existing by Lemma 2.12 random process such that πœ‚π‘™Dβ†’πœ‚.

Proof. Repeating the derivation of (2.31) from (2.29), we derive from (2.28) the relation limπ‘™β†’βˆžlimπ‘›β†’βˆž||ξ€·πœ‚π–€πΉπ‘™π‘›ξ€Έξ€·πœ‚βˆ’π–€πΉπ‘›ξ€Έ||=0.(2.33) It remains to write |𝖀𝐹(πœ‚π‘›)βˆ’π–€πΉ(πœ‚)|≀|𝖀𝐹(πœ‚π‘›)βˆ’π–€πΉ(πœ‚π‘™π‘›)|+|𝖀𝐹(πœ‚π‘™π‘›)βˆ’π–€πΉ(πœ‚π‘™)|+|𝖀𝐹(πœ‚π‘™)βˆ’π–€πΉ(πœ‚)|.

Corollary 2.14. Let (πœ‚π‘™π‘›), (πœ‚π‘™), and (πœ‚π‘›) be sequences of cΓ dlΓ g random processes such that for any π‘‘βˆˆβ„+ and πœ€>0 (2.26) holds; for each π‘™βˆˆβ„• relation (2.27) is valid; the sequence (πœ‚π‘™) is r.c. in C. Then, there exists a random process πœ‚ such that πœ‚π‘™Cβ†’πœ‚ and πœ‚π‘›Cβ†’πœ‚.

Below, 𝒰 is the symbol of the locally uniform (i.e., uniform in every interval) convergence.

Lemma 2.15. Let 𝑋,𝑋1,𝑋2… be cΓ dlΓ g random processes such that 𝑋𝑛C→𝑋. Then, 𝐹(𝑋𝑛)d→𝐹(𝑋) for any 𝒰-continuous functional 𝐹 on D.

Proof. Lemma VI.1.33 and Corollary VI.1.43 in [2] assert completeness and separability of the metric space (D,𝜌), where 𝜌 is the metric used in the proof of Lemma 2.12. Then, it follows from the assumptions of the lemma by Skorokhod's theorem [6] that there exist given on a common probability space cΓ dlΓ g random processes 𝑋′,π‘‹ξ…ž1,π‘‹ξ…ž2… such that 𝑋′d=𝑋 (so that 𝑋′ is continuous), π‘‹ξ…žπ‘›d=𝑋𝑛 and 𝜌(π‘‹ξ…žπ‘›,𝑋′)β†’0 a.s. By the choice of 𝜌, the last relation is tantamount to π‘‹ξ…žπ‘›π’₯→𝑋′ a.s. Hence, and from continuity of 𝑋′, we get by Proposition VI.1.17 [2] π‘‹ξ…žπ‘›π’°β†’π‘‹ξ…ža.s. and, therefore, by the choice of 𝐹,𝐹(π‘‹ξ…žπ‘›)→𝐹(𝑋′) a.s. It remains to note that 𝐹(π‘‹ξ…žπ‘›)d=𝐹(𝑋𝑛),𝐹(𝑋′)d=𝐹(𝑋).

2.1. Forestopping of Random Processes

Let 𝔽 be a filtration on some probability space, 𝑆 an 𝔽-adapted random process, and 𝜏 a stopping time with respect to 𝔽. We put 𝑆(0βˆ’)=𝑆(0) and denote π‘†πœ(𝑑)=𝑆(π‘‘βˆ§πœ),πœπ‘†(𝑑)=𝑆(𝑑)𝐼[0,𝜏[(𝑑)+𝑆(πœβˆ’)𝐼[𝜏,∞[(𝑑),(2.34)πœβ„±(𝑑)=β„±(𝑑)βˆ©β„±(πœβˆ’), πœπ”½=(πœβ„±(𝑑),π‘‘βˆˆβ„+). Obviously,𝜏(π‘†πœ)=πœπ‘†,ξ€Ί(2.35)πœπ‘†ξ€»=𝜏[𝑆],(2.36) provided [𝑆] exists. In case 𝜏 is 𝔽-predictable, the operation π‘†β†¦πœπ‘† was called in [7] the forestopping. The following three statements were proved in [7].

Lemma 2.16. Let a random process π‘ˆ and a stopping time 𝜏 be 𝔽-predictable. Then, the process πœπ‘ˆ is πœπ”½-predictable.

Theorem 2.17. Let 𝑋 be an 𝔽-martingale and 𝜏 an 𝔽-predictable stopping time. Then, πœπ‘‹ is a πœπ”½-martingale. If 𝑋 is uniformly integrable, then so is πœπ‘‹.

Lemma 2.18. Let 𝑉 be an ℝ𝑑-valued right-continuous 𝔽-predictable random process and 𝐴 a closed set in ℝ𝑑. Then, the stopping time inf{π‘‘βˆΆπ‘‰(𝑑)∈𝐴} is 𝔽-predictable.

The operation of forestopping was used prior to [7] by Barlow [8] who took the assertion of Theorem 2.17 (which he did not even formulate) for granted.

We will need some subtler properties of this operation.

Lemma 2.19. Let π‘ˆ be a starting from zero locally square integrable martingale with respect to 𝔽, 𝑁 a positive number, and 𝜎 an 𝔽-predictable stopping time such that πœŽβ‰€inf{π‘‘βˆΆtrβŸ¨π‘ˆβŸ©(𝑑)β‰₯𝑁}.(2.37) Then, 𝖀supπ‘ βˆˆβ„+|πœŽπ‘ˆ(s)|2≀4𝑁.

Proof. Predictability of 𝜎 implies by Theorem 2.1.13 [3] that there exists a sequence (πœŽπ‘›) of stopping times such that ξ€½πœŽ{𝜎>0}βŠ‚π‘›ξ€ΎπœŽ<𝜎,(2.38)π‘›β†—πœŽa.s.(2.39)
By the choice of π‘ˆ there exists a sequence (πœπ‘˜) of stopping times such that πœπ‘˜π‘ˆβ†—βˆža.s,(2.40)π‘˜β‰‘π‘ˆπœπ‘˜βˆˆβ„³2(𝔽).(2.41) Then, supπ‘ β‰€πœŽπ‘›||||π‘ˆ(𝑠)=limπ‘˜β†’βˆžsupπ‘ β‰€πœŽπ‘›βˆ§πœπ‘˜||||.π‘ˆ(𝑠)(2.42) Herein, obviously, supπ‘ β‰€πœŽπ‘›βˆ§πœπ‘˜||||π‘ˆ(𝑠)=supπ‘ β‰€πœŽπ‘›||π‘ˆπ‘˜||.(𝑠)(2.43)
From (2.41) we have by Doob's inequality 𝖀supπ‘ β‰€πœŽπ‘›||π‘ˆπ‘˜||(𝑠)2||π‘ˆβ‰€4π–€π‘˜ξ€·πœŽπ‘›ξ€Έ||2.(2.44) Noting that: (1) for any π‘₯βˆˆβ„π‘‘|π‘₯|2=trπ‘₯π‘₯⊀, (2) π–€π‘ˆπ‘˜(πœŽπ‘›)π‘ˆπ‘˜(πœŽπ‘›)⊀=π–€βŸ¨π‘ˆπ‘˜βŸ©(πœŽπ‘›), we may rewrite the last inequality in the form 𝖀supπ‘ β‰€πœŽπ‘›||π‘ˆπ‘˜||(𝑠)2≀4𝖀trβŸ¨π‘ˆπ‘˜βŸ©ξ€·πœŽπ‘›ξ€Έ.(2.45) Writing βŸ¨π‘ˆπ‘˜βŸ©ξ€·πœŽπ‘›ξ€Έ=βŸ¨π‘ˆπœπ‘˜βŸ©ξ€·πœŽπ‘›ξ€Έ=βŸ¨π‘ˆβŸ©πœπ‘˜ξ€·πœŽπ‘›ξ€Έξ€·πœ=βŸ¨π‘ˆβŸ©π‘˜βˆ§πœŽπ‘›ξ€Έ,(2.46) we get from (2.37) and (2.38) trβŸ¨π‘ˆπ‘˜βŸ©(πœŽπ‘›)<𝑁, which together with (2.45) results in 𝖀supπ‘ β‰€πœŽπ‘›|π‘ˆπ‘˜(𝑠)|2<4𝑁. Then, from (2.42) and (2.43), we have by Fatou's theorem 𝖀supπ‘ β‰€πœŽπ‘›||||π‘ˆ(𝑠)2≀4𝑁.(2.47) The assumption π‘ˆ(0)=0 yields 𝖀sup𝑠<𝜎||||π‘ˆ(𝑠)2=𝖀sup𝑠<𝜎||||π‘ˆ(𝑠)2𝐼{𝜎>0}.(2.48) Relations (2.38) and (2.39) imply that sup𝑠<𝜎||||π‘ˆ(𝑠)2𝐼{𝜎>0}=limπ‘›β†’βˆžsupπ‘ β‰€πœŽπ‘›||||π‘ˆ(𝑠)2𝐼{𝜎>0},(2.49) which together with (2.47) yields by Fatou's theorem 𝖀sup𝑠<𝜎|π‘ˆ(𝑠)|2𝐼{𝜎>0}≀4𝑁. It remains to note that supπ‘ βˆˆβ„+|πœŽπ‘ˆ(𝑠)|=sup𝑠<𝜎|π‘ˆ(𝑠)|𝐼{𝜎>0} in view of (2.34).

Lemma 2.20. Let π‘ˆ be a locally square integrable martingale with respect to 𝔽 such that 𝖀||||π‘ˆ(0)2<∞,(2.50) and for any 𝑑𝖀max𝑠≀𝑑||||Ξ”π‘ˆ(𝑠)2<∞.(2.51) Let, further, 𝑁 be a positive number and 𝜎 a predictable time satisfying condition (2.37). Then, π‘ˆπœŽβˆˆβ„³2(𝔽).

Proof. In view of (2.50) it suffices to show that (π‘ˆβˆ’π‘ˆ(0))πœŽβˆˆβ„³2(𝔽). In other words, we may consider that π‘ˆ(0)=0. Then condition (2.51) and the evident inequality sup𝑠≀𝑑||π‘ˆπœŽ||(𝑠)≀sup𝑠≀𝑑||πœŽπ‘ˆ||(𝑠)+max𝑠≀𝑑||||Ξ”π‘ˆ(𝑠)(2.52) imply by Lemma 2.19 that for any 𝑑𝖀sup𝑠≀𝑑||π‘ˆπœŽ||(𝑠)2<∞.(2.53) It remains to prove that for all 𝑑2>𝑑1β‰₯0, π–€ξ€·π‘ˆπœŽξ€·π‘‘2ξ€Έξ€·π‘‘βˆ£β„±1ξ€Έξ€Έ=π‘ˆπœŽξ€·π‘‘1ξ€Έ.(2.54)
Taking a sequence (πœπ‘›) with properties (2.40) and (2.41), we write π–€ξ€·π‘ˆξ€·π‘‘2βˆ§πœŽβˆ§πœπ‘˜ξ€Έξ€·π‘‘βˆ£β„±1ξ€Έξ€Έ=π‘ˆπœŽξ€·π‘‘1βˆ§πœŽβˆ§πœπ‘˜ξ€Έ.(2.55) To deduce (2.54) from this inequality and (2.40), it suffices to note that ||π‘ˆξ€·π‘‘βˆ§πœŽβˆ§πœπ‘˜ξ€Έ||≀sup𝑠≀𝑑||π‘ˆπœŽ||,(𝑠)(2.56) so that (2.53) provides uniform integrability of the sequence (π‘ˆ(𝑑2βˆ§πœŽβˆ§πœπ‘˜),π‘˜βˆˆβ„•).

Corollary 2.21. Under the conditions of Lemma 2.20β€‰β€‰πœŽπ‘ˆβˆˆβ„³2(πœŽπ”½).

Theorem 2.22. Let π‘ˆ be a locally square integrable martingale with respect to 𝔽 satisfying conditions (2.50) and (2.51), 𝑁 a positive number, and 𝜎 a predictable time satisfying condition (2.37). Then, βŸ¨πœŽπ‘ˆβŸ©=πœŽβŸ¨π‘ˆβŸ©.

Proof. Denote 𝑋=(π‘ˆβŠ—2βˆ’βŸ¨π‘ˆβŸ©)πœŽβ‰‘(π‘ˆπœŽ)βŠ—2βˆ’βŸ¨π‘ˆπœŽβŸ©,π‘Œ=πœŽπ‘ˆβŠ—2βˆ’πœŽβŸ¨π‘ˆβŸ©β‰‘πœŽ(π‘ˆβŠ—2βˆ’βŸ¨π‘ˆβŸ©). It suffices to show that π‘Œ is a πœŽπ”½-martingale. To deduce this fact from Theorem 2.17, we note that, firstly, π‘‹βˆˆβ„³(𝔽) by construction and Lemma 2.20, and, secondly, π‘Œ=πœŽπ‘‹ by construction of both processes and because of (2.35).

3. Martingale Preliminaries

The next statement is obvious.

Lemma 3.1. Let (𝑀𝑙) be a sequence of martingales such that 𝑀𝑙dβŸΆπ‘€,(3.1) and for any 𝑑 the sequence (|𝑀𝑙(𝑑)|) is uniformly integrable. Then, 𝑀 is a martingale.

Lemma 3.2. Let (𝑀𝑙) be a sequence of martingales such that (3.1) holds and supπ‘™βˆˆβ„•,π‘‘βˆˆβ„+𝑀𝖀tr𝑙(𝑑)<∞.(3.2) Then, sup𝑑𝖀|𝑀(𝑑)|2<∞.

Proof. By condition (3.2) and the definition of quadratic characteristic, there exists a constant 𝐢 such that 𝖀|𝑀𝑙(𝑑)|2≀𝐢 for all 𝑑 and 𝑙. Hence, and from (3.1), we have by Fatou's theorem (applicable due to the above-mentioned Skorokhod's principle of common probability space) 𝖀|𝑀(𝑑)|2≀𝐢.

Corollary 3.3. Let a sequence (𝑀𝑙) of square integrable martingales satisfy conditions (3.1) and (3.2). Then, 𝑀 is a uniformly integrable martingale.

Lemma 3.4. Let π‘Œ be a local martingale and 𝐾 be an 𝔖-valued random process. Suppose that they are given on a common probability space and (π‘Œ,𝐾)d=(π‘Œ,[π‘Œ]). Then for any 𝑑𝐾(𝑑)=[π‘Œ](𝑑) a.s.

Proof. By assumption, 𝑛𝑖=1ξ€·π‘Œξ€·π‘‘π‘–ξ€Έξ€·π‘‘βˆ’π‘Œπ‘–βˆ’1ξ€Έξ€ΈβŠ—2βˆ’πΎ(𝑑)d=𝑛𝑖=1ξ€·π‘Œξ€·π‘‘π‘–ξ€Έξ€·π‘‘βˆ’π‘Œπ‘–βˆ’1ξ€Έξ€ΈβŠ—2βˆ’[π‘Œ](𝑑),(3.3) for all 𝑛,𝑑 and 𝑑0<𝑑1<⋯𝑑𝑛. Hence, recalling the definition of quadratic variation, we get [π‘Œ](𝑑)βˆ’πΎ(𝑑)d=0.

We shall identify indistinguishable processes, writing simply πœ‰=πœ‚ if 𝖯{βˆ€π‘‘βˆˆβ„+,πœ‰(𝑑)=πœ‚(𝑑)}=1. Theorem 2.3.5 [3] asserts that[π‘Œ]=βŸ¨π‘ŒβŸ©,(3.4) for a continuous local martingale π‘Œ. Hence, and from Lemma 3.4, we have

Corollary 3.5. Let π‘Œ be a continuous local martingale and 𝐾 an 𝔖-valued random process. Suppose that they are given on a common probability space and (π‘Œ,𝐾)d=(π‘Œ,βŸ¨π‘ŒβŸ©). Then, 𝐾=βŸ¨π‘ŒβŸ©.

Proof. Lemma 3.4 and formula (3.4) yield 𝖯{βˆ€π‘‘βˆˆβ„š+,𝐾(𝑑)=βŸ¨π‘ŒβŸ©(𝑑)}=1. Continuity of both processes enables us to substitute β„š+ by ℝ+.

Lemma 3.6. Let π‘ˆ be a locally square integrable martingale. Then, β€–βŸ¨π‘ˆβŸ©β€– is an increasing process.

Proof. For any π‘₯βˆˆβ„π‘‘, the process π‘₯βŠ€π‘ˆ is a numeral locally square integrable martingale and, therefore, the process ⟨π‘₯βŠ€π‘ˆβŸ© increases. It remains to note that ⟨π‘₯βŠ€π‘ˆβŸ©=π‘₯βŠ€βŸ¨π‘ˆβŸ©π‘₯ and to recall formula (2.21).

Lemma 3.7. Let 𝑍1 and 𝑍2 be locally square integrable martingales with respect to a common filtration. Then, β€–β€–βŸ¨π‘1,𝑍2⟩+βŸ¨π‘2,𝑍1βŸ©β€–β€–βˆšβ‰€2β€–β€–βŸ¨π‘1βŸ©β€–β€–β€–β€–βŸ¨π‘2βŸ©β€–β€–.(3.5)

Proof. For 𝑑=1 (then βŸ¨π‘2,𝑍1⟩=βŸ¨π‘1,𝑍2⟩), this is the Kunita-Watanabe inequality [3, page 118]. In the general case, we take an arbitrary vector π‘₯βˆˆπ‘†π‘‘βˆ’1 and write π‘₯βŠ€ξ€·βŸ¨π‘1,𝑍2⟩+βŸ¨π‘2,𝑍1βŸ©ξ€Έξ«π‘₯π‘₯=2βŠ€π‘1,π‘₯βŠ€π‘2≀2⟨π‘₯βŠ€π‘1⟩⟨π‘₯βŠ€π‘2ξ”βŸ©=2π‘₯βŠ€βŸ¨π‘1⟩π‘₯π‘₯βŠ€βŸ¨π‘2⟩π‘₯,(3.6) hereupon the required conclusion ensues from (2.21) and Lemma 2.9.

Lemma 3.8. Let π‘ˆ1 and π‘ˆ2 be locally square integrable martingales with respect to some common filtration. Then, for any 𝑑>0supπ‘ β‰€π‘‘β€–β€–βŸ¨π‘ˆ1⟩(𝑠)βˆ’βŸ¨π‘ˆ2β€–β€–β‰€β€–β€–βŸ©(𝑠)βŸ¨π‘ˆ1βˆ’π‘ˆ2β€–β€–βˆšβŸ©(𝑑)+2β€–β€–βŸ¨π‘ˆ1βˆ’π‘ˆ2β€–β€–βˆšβŸ©(𝑑)β€–β€–βŸ¨π‘ˆ2β€–β€–.⟩(𝑑)(3.7)

Proof. Writing the identities βŸ¨π‘ˆ1⟩=βŸ¨π‘ˆ1βˆ’π‘ˆ2+π‘ˆ2⟩=βŸ¨π‘ˆ1βˆ’π‘ˆ2⟩+βŸ¨π‘ˆ1βˆ’π‘ˆ2,π‘ˆ2⟩+βŸ¨π‘ˆ2,π‘ˆ1βˆ’π‘ˆ2⟩+βŸ¨π‘ˆ2⟩,(3.8) we deduce from Lemma 3.7 that β€–β€–βŸ¨π‘ˆ1⟩(𝑠)βˆ’βŸ¨π‘ˆ2β€–β€–β‰€β€–β€–βŸ©(𝑠)βŸ¨π‘ˆ1βˆ’π‘ˆ2β€–β€–βˆšβŸ©(𝑠)+2β€–β€–βŸ¨π‘ˆ2βˆ’π‘ˆ1β€–β€–βˆšβŸ©(𝑠)β€–β€–βŸ¨π‘ˆ2β€–β€–.⟩(𝑠)(3.9) It remains to note that the right-hand side increases in 𝑠 by Lemma 3.6.

For a function π‘“βˆˆD we denote Δ𝑓(𝑑)=𝑓(𝑑)βˆ’π‘“(π‘‘βˆ’).

Let us introduce the conditions:(RC) The sequence (trβŸ¨π‘Œπ‘›βŸ©) is r.c. in C.(UI1) The sequence (|π‘Œπ‘›(𝑑)βˆ’π‘Œπ‘›(0)|2) is u.i.(UI2) For any π‘§βˆˆβ„π‘‘βˆ— the sequence (trβŸ¨π‘§π‘Œπ‘›βŸ©(𝑑)) is u.i.(UI3) The sequence (sup𝑠≀𝑑|π‘Œπ‘›(𝑠)βˆ’π‘Œπ‘›(0)|2) is u.i.

Lemma 3.9. Let (π‘Œπ‘›) be a sequence of local square integrable martingales satisfying the conditions: (RC), limπΏβ†’βˆžlimπ‘›β†’βˆžπ–―ξ€½||π‘Œπ‘›||ξ€Ύ(0)>𝐿=0,(3.10) and, for each 𝑑>0, the condition max𝑠≀𝑑||Ξ”π‘Œπ‘›||(𝑠)π–―βŸΆ0.(3.11) Then, (π‘Œπ‘›) is r.c. in C.

Proof. It follows from (RC) and (3.10) by Rebolledo's theorem [2, VI.4.13] that (π‘Œπ‘›) is r.c. in D. Hereon, the desired conclusion follows from Proposition VI.3.26 (items (i) and (iii)) [2] with account of VI.3.9 [2].

Combining Lemma 3.9 with Corollary 2.11, we get

Corollary 3.10. Under the assumptions of Lemma 3.9, the sequence of compound processes (π‘Œπ‘›,βŸ¨π‘Œπ‘›βŸ©) is r.c. in C.

Some statements below deal with random processes on [0,𝑑], not on ℝ+. In this case, the time variable is denoted by 𝑠 and C means C[0,𝑑] instead of C(ℝ+).

Lemma 3.11. Let (π‘Œπ‘›) be a r.c. in C and satisfying condition (UI1) sequence of martingales on [0,𝑑]. Then, for any π‘§βˆˆβ„π‘‘βˆ—, the sequence ([π‘§π‘Œπ‘›](𝑑)) is u.i.

Proof. The obvious equality [π‘‰βˆ’π‘‰(0)]=[𝑉] allows us to consider that π‘Œπ‘›(0)=0. Then, condition (UI1) together with Doob's inequality yields sup𝑛𝖀sup𝑠≀𝑑|π‘Œπ‘›(𝑠)|2<∞,(3.12) whence sup𝑛𝖀max𝑠≀𝑑||Ξ”π‘Œπ‘›||(𝑠)<∞.(3.13)
By assumption, for any infinite set 𝐽0βŠ‚β„•, there exist an infinite subset π½βŠ‚π½0 and a random process π‘Œ such that π‘Œπ‘›CβŸΆπ‘Œasπ‘›βŸΆβˆž,π‘›βˆˆπ½.(3.14) Condition (UI1) implies that π‘Œ is a square integrable martingale and for any π‘§βˆˆβ„π‘‘βˆ—, π–€ξ€·π‘§π‘Œπ‘›ξ€Έ(𝑑)2βŸΆπ–€(π‘§π‘Œ(𝑑))2asπ‘›βŸΆβˆž,π‘›βˆˆπ½.(3.15)
From (3.14) and (3.13), we have by Corollary VI.6.7 [2] ξ€Ίπ‘§π‘Œπ‘›ξ€»C⟢[]π‘§π‘Œasπ‘›βŸΆβˆž,π‘›βˆˆπ½.(3.16) Hence, and from (3.15), recalling that for any ℝ-valued π‘€βˆˆβ„³2 one has 𝖀(π‘€βˆ’π‘€(0))2=𝖀[𝑀][3, Theorem 2.2.4], we get π–€ξ€Ίπ‘§π‘Œπ‘›ξ€»[](𝑑)βŸΆπ–€π‘§π‘Œ(𝑑)asπ‘›βŸΆβˆž,π‘›βˆˆπ½.(3.17) Comparing this relation with (3.16), we conclude that the sequence ([π‘§π‘Œπ‘›](𝑑),π‘›βˆˆπ½) is uniformly integrable. Hence, in view of arbitrariness of 𝐽0, we deduce by Lemma 2.3 uniform integrability of ([π‘§π‘Œπ‘›](𝑑),π‘›βˆˆβ„•).

Lemma 3.12. Let (π‘Œπ‘›) be a sequence of martingales on [0,𝑑] satisfying condition (UI1) and (UI2). Suppose that there exists an ℝ𝑑×𝔖+-valued random process (π‘Œ,𝐾) such that ξ€·π‘Œπ‘›,βŸ¨π‘Œπ‘›βŸ©ξ€ΈC⟢(π‘Œ,𝐾).(3.18) Then, firstly, ξ€Ίπ‘Œπ‘›ξ€»βˆ’βŸ¨π‘Œπ‘›βŸ©CβŸΆπ‘‚,(3.19) where 𝑂 is the null matrix, π‘Œ is a continuous martingale, and, secondly, (π‘Œ,𝐾)d=(π‘Œ,βŸ¨π‘ŒβŸ©).(3.20)

Proof. For the same reason as in the proof of Lemma 3.11, we may consider that π‘Œπ‘›(0)=0. Then, as was shown above, condition (UI1) implies (3.13). Combining the latter with π‘Œπ‘›CβŸΆπ‘Œ,(3.21) (a part of (3.18)), we get by Corollary VI.6.7 [2] that ξ€·π‘Œπ‘›,ξ€Ίπ‘Œπ‘›ξ€»ξ€ΈC[π‘Œ]⟢(π‘Œ,).(3.22) From (3.18) and (3.22), we get by Corollary 2.11 that for any infinite set 𝐽0βŠ‚β„•, there exist an infinite subset π½βŠ‚π½0 and an 𝔖+×𝔖+-valued random process (𝑄𝐽,𝑅𝐽) such that π‘Œξ€·ξ€Ίπ‘›ξ€»,βŸ¨π‘Œπ‘›βŸ©ξ€ΈβŸΆξ€·π‘„π½,𝑅𝐽asπ‘›βŸΆβˆž,π‘›βˆˆπ½.(3.23) (Of course 𝑄𝐽d=𝐾,𝑅𝐽d=[π‘Œ].)
Denote 𝑍𝑛=[π‘Œπ‘›]βˆ’βŸ¨π‘Œπ‘›βŸ©. This is a martingale by Lemma 10.4 in [4]. Relation (3.23) implies that 𝑍𝑛CβŸΆπ‘„π½βˆ’π‘…π½asπ‘›βŸΆβˆž,π‘›βˆˆπ½.(3.24) For any π‘§βˆˆβ„π‘‘βˆ—, the sequence (𝑧𝑍𝑛(𝑑)π‘§βŠ€) is, by Lemma 3.11 and condition (UI2), u.i. So, relation (3.24) implies by Lemma 3.1 that 𝑧(π‘„π½βˆ’π‘…π½)π‘§βŠ€ is a martingale. Also, it implies its continuity. Relation (3.23) shows that the processes π‘§π‘„π½π‘§βŠ€ and π‘§π‘…π½π‘§βŠ€ increase and start from zero. So, 𝑧(π‘„π½βˆ’π‘…π½)π‘§βŠ€ starts from zero and has finite variation in [0,𝑑]. These four properties of π‘„π½βˆ’π‘…π½ imply together that 𝑄𝐽(𝑠)βˆ’π‘…π½(𝑠)=𝑂 for all π‘ βˆˆ[0,𝑑]. Thus, any subsequence (𝑍𝑛,π‘›βˆˆπ½0) contains, in turn, a subsequence (𝑍𝑛,π‘›βˆˆπ½) such that 𝑍𝑛→𝑂asπ‘›β†’βˆž,π‘›βˆˆπ½. This proves (3.19).
From (3.22) and (3.19), we have (π‘Œπ‘›,βŸ¨π‘Œπ‘›βŸ©)Cβ†’(π‘Œ,[π‘Œ]). And this is, in view of (3.4), tantamount to ξ€·π‘Œπ‘›,βŸ¨π‘Œπ‘›βŸ©ξ€ΈC⟢(π‘Œ,βŸ¨π‘ŒβŸ©).(3.25) Comparing this relation with (3.18), we arrive at (3.20).

Remark 3.13. The second conclusion of Lemma 3.12 implies by Corollary 3.5 that 𝐾=βŸ¨π‘ŒβŸ©.

Corollary 3.14. Let a sequence (π‘Œπ‘›) of martingales on [0,𝑑] satisfy conditions (RC), (3.11), (UI1), and (UI2). Then, relation (3.19) holds.

Proof. By Corollary 3.10 for any infinite set 𝐽0βŠ‚β„•, there exist an infinite set π½βŠ‚π½0 and an ℝ𝑑×𝔖+-valued random process (π‘Œ,𝐾) such that relation (3.18) holds as π‘›β†’βˆž,π‘›βˆˆπ½. Then, by Lemma 3.12, so does (as π‘›β†’βˆž,π‘›βˆˆπ½) (3.19). Due to arbitrariness of 𝐽0 this relation holds when 𝑛 ranges over β„•, too.

Corollary 3.15. Let (π‘Œπ‘›) be a sequence of martingales on ℝ+ satisfying conditions (RC) and for all 𝑑>0, conditions (3.11), (UI1), (UI2). Then, relation (3.19) holds.

Lemma 3.16. Let a sequence (π‘Œπ‘›) of martingales on ℝ+ satisfy conditions (RC) and, for any 𝑑>0, (3.11) and (UI3). Then, relation (3.19) holds.

Proof. Denote πœŽπ‘π‘›=inf{π‘ βˆΆtrβŸ¨π‘Œπ‘›βŸ©(𝑠)β‰₯𝑁},π‘Œπ‘π‘›=πœŽπ‘π‘›π‘Œπ‘›. Obviously, ξ€½πœŽπ‘π‘›ξ€ΎβŠ‚ξ€½<𝑑trβŸ¨π‘Œπ‘›βŸ©ξ€Ύ(𝑑)β‰₯𝑁.(3.26) By Corollary 2.21β€‰β€‰π‘Œπ‘π‘› is a square integrable martingale with respect to πœŽπ‘π‘›π”½π‘›. By Theorem 2.22, ξ«π‘Œπ‘π‘›ξ¬=πœŽπ‘π‘›βŸ¨π‘Œπ‘›βŸ©.(3.27) Condition (RC) implies relative compactness of the sequence (βŸ¨π‘Œπ‘π‘›βŸ©,π‘›βˆˆβ„•). By construction |π‘Œπ‘π‘›(𝑑)βˆ’π‘Œπ‘π‘›(0)|≀sup𝑠≀𝑑|π‘Œπ‘›(𝑑)βˆ’π‘Œπ‘›(0)|. So, condition (UI3) implies that for any 𝑑 and 𝑁 the sequence (|π‘Œπ‘π‘›(𝑑)βˆ’π‘Œπ‘π‘›(0)|2,π‘›βˆˆβ„•) is u.i. Thus, Corollary 3.15 asserts that for any π‘ξ€Ίπ‘Œπ‘π‘›ξ€»βˆ’ξ«π‘Œπ‘π‘›ξ¬CβŸΆπ‘‚asπ‘›βŸΆβˆž.(3.28) Equalities (3.27) and (2.36) yield the relation ξ‚»supπ‘ β‰€π‘‘ξ€·β€–β€–ξ€Ίπ‘Œπ‘π‘›ξ€»ξ€Ίπ‘Œ(𝑠)βˆ’π‘›ξ€»β€–β€–+β€–β€–ξ«π‘Œ(𝑠)𝑁𝑛(𝑠)βˆ’βŸ¨π‘Œπ‘›β€–β€–ξ€Έξ‚ΌβŠ‚ξ€½πœŽβŸ©(𝑠)>0𝑁𝑛,<𝑑(3.29) which together with (3.26), (3.28) and (RC) entails (3.19).

Corollary 3.15 and Lemma 3.16 are only the steps towards the final result about asymptotic proximity of quadratic variations and quadratic characteristicsβ€”Corollary 5.3.

Corollary 3.17. Let a sequence (π‘Œπ‘›) of martingales on ℝ+ satisfy conditions (RC) and for any 𝑑>0, (3.11) and (UI3). Suppose also that there exists an ℝ𝑑×𝔖+-valued random process (π‘Œ,𝐾) such that relation (3.18) is valid. Then, π‘Œ is a continuous martingale, and (3.20) holds.

Proof. Lemma 3.16 asserts (3.19). The implications ((3.21) and (UI1) β‡’ (3.22)); ((3.22) and (3.19) β‡’ (3.25)), were established in the proof of Lemma 3.12.

4. Sequences of Martingales with Asymptotically Conditionally Independent Increments

Lemma 4.1. Let for each 𝑛𝑀𝑛 be an ℝ-valued square integrable martingale on [0,𝑑] with respect to a flow (𝒒𝑛(𝑠),π‘ βˆˆ[0,𝑑]) and ℋ𝑛 a sub-𝜎-algebra of 𝒒𝑛(0). Suppose that conditions (UI1) and (RC) are fulfilled for π‘Œπ‘›=𝑀𝑛, 𝑀𝑛(0)=0,(4.1)max𝑠≀𝑑||Δ𝑀𝑛||(𝑠)π–―βŸΆ0(4.2) and there exists a nonrandom number 𝑁 such that for all π‘›βŸ¨π‘€π‘›βŸ©(𝑑)≀𝑁.(4.3) Then, 𝖀𝑒𝑖𝑀𝑛(𝑑)+βŸ¨π‘€π‘›βŸ©(𝑑)/2βˆ£β„‹π‘›ξ€Έπ–―βŸΆ1.(4.4)

Proof. Conditions (RC) (π‘Œπ‘›=𝑀𝑛), (4.1), and (4.2) entail, by Lemma 3.9, relative compactness of (𝑀𝑛) in C.
Denote 𝑇𝑛=βŸ¨π‘€π‘›βŸ©/2, πœ‰π‘›=𝑒𝑖𝑀𝑛+𝑇𝑛, 𝑋𝑛=([𝑀𝑛]βˆ’βŸ¨π‘€π‘›βŸ©)/2, 𝛾𝑛=πœ‰βˆ’π‘›βˆ˜π‘‹π‘›, πœ‚π‘›=ξ“π‘ β‰€π‘‘πœ‰π‘›ξ‚€π‘’(π‘ βˆ’)Δ𝑇𝑛(𝑠)+𝑖Δ𝑀𝑛(𝑠)βˆ’1βˆ’Ξ”π‘‡π‘›(𝑠)βˆ’π‘–Ξ”π‘€π‘›1(𝑠)+2Δ𝑀𝑛(𝑠)2.(4.5) Condition (RC) (π‘Œπ‘›=𝑀𝑛) implies that max𝑠≀𝑑Δ𝑇𝑛(𝑠)π–―βŸΆ0.(4.6) In view of (4.1) πœ‰π‘›(0)=1. Then, by ItΓ΄'s formula πœ‰π‘›(𝑑)=1+π‘–πœ‰βˆ’π‘›β‹…π‘€π‘›(𝑑)+πœ‰βˆ’π‘›βˆ˜π‘‡π‘›1(𝑑)βˆ’2πœ‰βˆ’π‘›βˆ˜βŸ¨π‘€π‘π‘›+ξ“βŸ©(𝑑)π‘ β‰€π‘‘πœ‰π‘›ξ€·π‘’(π‘ βˆ’)𝑖Δ𝑀𝑛(𝑠)+Δ𝑇𝑛(𝑠)βˆ’1βˆ’π‘–Ξ”π‘€π‘›(𝑠)βˆ’Ξ”π‘‡π‘›ξ€Έ.(𝑠)(4.7) Hence, recalling that βŸ¨π‘€π‘π‘›βŸ©(𝑑)=[π‘€π‘›βˆ‘](𝑑)βˆ’π‘ β‰€π‘‘(Δ𝑀𝑛(𝑠))2, we get πœ‰π‘›(𝑑)=1+π‘–πœ‰βˆ’π‘›β‹…π‘€π‘›(𝑑)βˆ’πœ‰βˆ’π‘›βˆ˜π‘‹π‘›(𝑑)+πœ‚π‘›.(4.8)
By the definition of πœ‰π‘› and by condition (4.3), sup𝑠≀𝑑||πœ‰π‘›||(𝑠)≀𝑒𝑁/2.(4.9) Consequently, π–€ξ€·πœ‰βˆ’π‘›β‹…π‘€π‘›(𝑑)βˆ£β„‹π‘›ξ€Έ=0,(4.10) and 𝖀|πœ‰βˆ’π‘›β‹…π‘€π‘›(𝑑)|2=𝖀(|πœ‰βˆ’π‘›|2βˆ˜βŸ¨π‘€π‘›βŸ©(𝑑)). The right-hand side of the last equality being less than 𝑒𝑁𝑁, the sequence (πœ‰βˆ’π‘›β‹…π‘€π‘›(𝑑)) is u.i., and so is ([𝑀𝑛](𝑑)) by Lemma 3.11 whose conditions (those not postulated) we have verified. This together with (4.9) and (4.3) implies uniform integrability of (πœ‰βˆ’π‘›βˆ˜π‘‹π‘›(𝑑)). Now, (4.8) and inequality (4.9) show that (πœ‚π‘›) has this property, too.
By construction and Lemma 10.4 in [4], 𝑋𝑛 is a martingale. Then, it follows from (4.9) that 𝖀(πœ‰βˆ’π‘›βˆ˜π‘‹π‘›(𝑑)βˆ£β„‹π‘›)=0, which together with (4.10) and (4.8) yields π–€ξ€·πœ‰π‘›(𝑑)βˆ£β„‹π‘›ξ€Έξ€·πœ‚=1+π–€π‘›βˆ£β„‹π‘›ξ€Έ.(4.11) So, it suffices to show that πœ‚π‘›π–―βŸΆ0.(4.12)
Obviously, for any real π‘Ž and π‘π‘’π‘Ž+π‘π‘–βˆ’π‘Ž=(π‘’π‘Žβˆ’1βˆ’π‘Ž)𝑒𝑏𝑖𝑒+π‘Žπ‘π‘–ξ€Έβˆ’1+𝑒𝑏𝑖,||π‘’π‘Ž||βˆ’1βˆ’π‘Žβ‰€|π‘Ž|2𝑒|π‘Ž|,||𝑒𝑏𝑖||≀||𝑏||,||||π‘’βˆ’1π‘π‘–π‘βˆ’1βˆ’π‘π‘–+22||||≀||𝑏||3.(4.13) Hence, from (4.5), (4.9), we get ||πœ‚π‘›||≀𝑒𝑁𝑒𝑒𝑁𝑠≀𝑑Δ𝑇𝑛(𝑠)2+max𝑠≀𝑑||Δ𝑀𝑛||(𝑠)𝑠≀𝑑Δ𝑇𝑛(𝑠)+𝑠≀𝑑Δ𝑀𝑛(𝑠)2ξƒͺξƒͺ.(4.14) Now, (4.12) ensues from (4.6), (4.3), (4.2), and stochastic boundedness of the sequence ([𝑀𝑛](𝑑)).

Lemma 4.2. Let for each 𝑛,𝑀𝑛 be an ℝ-valued starting from zero locally square integrable martingale with respect to some flow (𝒒𝑛(𝑑),π‘‘βˆˆβ„+) and ℋ𝑛 a sub-𝜎-algebra of 𝒒𝑛(0). Suppose that condition (RC) is fulfilled for π‘Œπ‘›=𝑀𝑛; 𝖀max𝑠≀𝑑Δ𝑀𝑛(𝑠)2⟢0,(4.15) for any 𝑑>0; there exists a nonrandom function π‘žβˆΆβ„+→ℝ+ such that βŸ¨π‘€π‘›βŸ©(𝑑)β‰€π‘ž(𝑑),(4.16) for all 𝑛 and 𝑑. Then, for any 𝑑, relation (4.4) holds.

Proof. Let us denote, only in this proof, πœπ‘π‘›=inf{π‘‘βˆΆ|𝑀𝑛(𝑑)|β‰₯𝑁}, 𝑀𝑁𝑛(𝑑)=𝑀𝑛(π‘‘βˆ§πœπ‘π‘›), 𝑇𝑁𝑛(𝑑)=𝑇𝑛(π‘‘βˆ§πœπ‘π‘›) (so that π‘€π‘π‘›βˆˆβ„“β„³2 and βŸ¨π‘€π‘π‘›βŸ©=2𝑇𝑁𝑛),πœ‰π‘π‘›=𝑒𝑖𝑀𝑁𝑛+𝑇𝑁𝑛. The evident inequality sup𝑠≀𝑑||𝑀𝑁𝑛||(𝑠)≀𝑁+max𝑠≀𝑑||Δ𝑀𝑛||,(𝑠)(4.17) and condition (4.15) show us that for any positive 𝑑 and 𝑁, the sequence (𝑀𝑁𝑛(𝑑)2,π‘›βˆˆβ„•) is u.i.
By assumption, there exists a sequence (πœŽπ‘˜) of stopping times such that πœŽπ‘˜β†—βˆž a.s. and for each π‘˜,π‘€πœŽπ‘˜βˆˆβ„³2. Then, for any 𝑑>𝑠β‰₯0, 𝑁>0 and 𝑛, π‘˜βˆˆβ„•, π–€ξ€·π‘€π‘π‘›ξ€·π‘‘βˆ§πœŽπ‘˜ξ€Έβˆ£π’’π‘›ξ€Έξ€·π‘€(𝑠)=π–€πœŽπ‘˜ξ€·π‘‘βˆ§πœπ‘π‘›ξ€Έβˆ£π’’π‘›ξ€Έ(𝑠)=π‘€πœŽπ‘˜ξ€·π‘ βˆ§πœπ‘π‘›ξ€Έ=π‘€π‘π‘›ξ€·π‘ βˆ§πœŽπ‘˜ξ€Έ.(4.18) Writing |𝑀𝑁𝑛(π‘‘βˆ§πœŽπ‘˜)|≀sup𝑠≀𝑑|𝑀𝑁𝑛(𝑠)|, we deduce from (4.17) and (4.15) uniform integrability of the sequence (𝑀𝑁𝑛(π‘‘βˆ§