# Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale

Authors: Andriy Yurachkivsky
Journal: Journal of Probability and Statistics
 (2011)

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## Abstract

Let for eachn∈ℕXn be an ℝd-valued locally square integrable martingale w.r.t. a filtration (ℱn(t),t∈ℝ+) (probability spaces may be different for different n). It is assumed that the discontinuities of Xn are in a sense asymptotically small as n→∞ and the relation Ef(〈zXn〉(t))|ℱn(s)-f〈zXn〉(t)→P0 holds for all t>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),〈Xn〉) converge in distribution to some (X∘,H), then a sequence (Xn) converges in distribution to a continuous local martingale X with initial value X∘ and quadratic characteristic H, whose finite-dimensional distributions are explicitly expressed via those of (X∘,H).


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## Body

## 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 with𝒢-conditional increments has been studied hitherto in the model, where the σ-algebra 𝒢 enters the setting along with the prelimit processes. This assumption is worse than restrictive—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 modification—Lemma 2.2) with an elementary probabilistic result—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
(*)Zn⟶lawZ,
where Z also is a semimartingale, entails the stronger one (Zn,[Zn])→law(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,〈Zn〉)⟶law(Z,〈Z〉)?
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 〈Zn〉. 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 Corollaries5.2 and 5.5. They deal with the problem: what conditions should be adjoined to 〈Zn〉→lawH 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 〈Z〉 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 increments—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 Section2 (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 squarexx⊤ of x∈ℝd will be otherwise written as x⊗2. We use the Euclidean norm |·| of vectors and the operator norm ∥·∥ of matrices. The symbols ℝd*,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.ByCb(X), we denote the space of complex-valued bounded continuous functions on a topological space X. If X=ℝk and the dimension k 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, ℳ¯(𝔽) and ℳ(𝔽) signify the class of all martingales with respect to a filtration (= flow of σ-algebras) 𝔽≡(ℱ(t),t∈ℝ+) and its subclass of uniformly integrable martingales. An 𝔽-martingale U will be called: square integrable if E|U(t)|2<∞ for all t anduniformly square integrable if suptE|U(t)|2<∞. The classes of such processes will be denoted ℳ¯2(𝔽) and ℳ2(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 ℓℳ, and ℓℳ2 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∫0tφ(s)dX(s) will be written shortly (following [1, 2]) as φ∘X(t) if this integral is pathwise (i.e., X is a process of locally bounded variation) or φ·X(t) 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 ξ and the quadratic characteristic of Z∈ℓℳ2 will be (and already were) denoted [ξ] and 〈Z〉, 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.
Let𝔄 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 B⊂T, and positive number ɛ, there exists a function G∈𝔄 such that ∥G∥∞≤∥F∥∞ and maxx∈B|F(x)-G(x)|<ɛ.Lemma 2.2.
Let𝔄 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 B⊂T, and positive number ɛ there exists a function G∈𝔄 such that ∥G∥∞≤∥F∥∞ and maxx∈B|F(x)-G(x)|<ɛ.We consider henceforth sequences of random processes or random variables given, maybe, on different probability spaces. So, for thenth member of a sequence, P and E should be understood as Pn and En. In what follows, “u.i.” means “uniformly 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 a>0 such that for all N>0(2.1)lim⁡¯n→∞E|αn|I{|αn|>N}≥2a.
Consequently, there exists an increasing sequence (nk) of natural numbers such that EβkI{βk>k}≥a, where βk=|αnk|. Then, for any infinite set J⊂ℕ and N>0, we have
(2.2)lim⁡̲k→∞,k∈JEβkI{βk>N}≥a,
which means that the subsequence (αnk) does not contain u.i. subsequences.Lemma 2.4.
Let for eachnξn1,…,ξnp be random variables given on a probability space (Ωn,ℱn,Pn), and ℋn a sub-σ-algebra of ℱn. Suppose that for each j∈{1,…,p},
(2.3)E(ξnj∣Hn)-ξnj⟶P0asn⟶∞,
and for any J⊂{1,…,p} the sequence (∏j∈Jξnj,n∈ℕ) is u.i. Then,
(2.4)E(∏j=1pξnj∣Hn)-∏j=1pξnj⟶P0.Proof.
Denoteηnj=E(ξnj∣ℋn). By the second assumption, the sequences (ξnj,n∈ℕ),(ηnj,n∈ℕ),j=1,…,p, are stochastically bounded, which together with the first assumption yields
(2.5)∏j∈Jηnj-∏j∈Jξnj⟶P0,
for any J⊂{1,…,p}. Hence, writing the identity
(2.6)E(ξn1ξn2∣Hn)=E((ξn1-ηn1)(ξn2-ηn2)∣Hn)+ηn1ηn2,
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.
Letαn and βn be random variables given on a probability space (Ωn,ℱn,Pn) and ℋn a sub-σ-algebra of ℱn. Suppose that
(2.7)αn-E(αn∣Hn)⟶P0,
and the sequences (αn), (βn) and (αnβn) are u.i. Then,
(2.8)E(αnβn∣Hn)-αnE(βn∣Hn)⟶P0.Lemma 2.6.
Letn∈ℕΞn be an ℝk-valued random variable given on a probability space (Ωn,ℱn,Pn), and ℋn a sub-σ-algebra of ℱn. Suppose that
(2.9)lim⁡N→∞lim⁡¯n→∞P{|Ξn|>N}=0,
and the relation
(2.10)E(F(Ξn)∣Hn)-F(Ξn)⟶P0
holds for all F from some class of complex-valued bounded continuous functions on ℝk which separates points of the latter. Then, it holds for all F∈Cb.Proof.
Let𝔄 denote the class of all complex-valued bounded continuous functions on ℝk 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 ℝk. The other two conditions of Lemma 2.2 are satisfied trivially. Thus, that lemma asserts that for any F∈Cb,N>0 and ɛ>0, there exists a function G∈𝔄 such that ∥G∥∞≤∥F∥∞ and max|x|≤N|F(x)-G(x)|<ɛ. Then,
(2.11)|F(Ξn)-G(Ξn)|I{|Ξn|≤N}<ɛ,|F(Ξn)-G(Ξn)|I{|Ξn|>N}≤2‖F‖∞I{|Ξn|>N}.
By the choice of G(2.12)E(G(Ξn)∣Hn)-G(Ξn)⟶P0,
whence by the dominated convergence theorem
(2.13)E|E(G(Ξn)∣Hn)-G(Ξn)|⟶0.
Writing the identity
(2.14)E(F(Ξn)∣Hn)-F(Ξn)=E((F(Ξn)-G(Ξn))I{|Ξn|≤N}∣Hn)+E((F(Ξn)-G(Ξn))I{|Ξn|>N}∣Hn)+E(G(Ξn)∣Hn)-G(Ξn)+(G(Ξn)-F(Ξn))I{|Ξn|≤N}+(G(Ξn)-F(Ξn))I{|Ξn|>N},
we get from (2.11)–(2.13)
(2.15)lim⁡¯n→∞E|E(F(Ξn)∣Hn)-F(Ξn)|≤2ɛ+4‖F‖∞lim⁡¯n→∞P{|Ξn|>N},
which together with (2.9) and due to arbitrariness of ɛ proves (2.10).Corollary 2.7.
Let for eachnζn1,…,ζnp 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
(2.16)lim⁡N→∞lim⁡¯n→∞P{|ζnj|>N}=0,(2.17)E(g(ζnj)∣Hn)-g(ζnj)⟶P0asn⟶∞
hold for all j∈{1,…,p} and g from some class 𝔉 of complex-valued bounded continuous functions on ℝd which separates points of the latter. Then,
(2.18)E(F(ζn1,…,ζnp)∣Hn)-F(ζn1,…,ζnp)⟶P0,
for all F∈Cb(ℝpd).Proof.
DenoteΞn=(ζn1,…,ζnp). Condition (2.17) implies by Lemma 2.4 that relation (2.10) is valid for all F of the kind F(x1,…,xp)=∏i=1pgi(xi), where gi∈𝔉. 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 eachnKn be an 𝔖-valued random process given on a probability space (Ωn,ℱn,Pn), ℋn a sub-σ-algebra of ℱn, and ζn0 an ℋn-measurable ℝm-valued random variable. Suppose that the relations
(2.19)lim⁡N→∞lim⁡¯n→∞P{‖Kn(t)‖>N}=0,E(f(zKn(t)z⊤)∣Hn)-f(zKn(t)z⊤)⟶P0,
and (2.16) hold for j=0, all t>0 and any bounded uniformly continuous function f on ℝ. Then, for any l∈ℕ,sl>⋯>s1>0 and F∈Cb(ℝm×𝔖l) the relation
(2.20)E(F(ζn0,Kn(s1),…,Kn(sl))∣Hn)-F(ζn0,Kn(s1),…,Kn(sl))⟶P0
is valid.Recall that for anyB∈𝔖(2.21)‖B‖=max⁡x∈Sd-1|x⊤Bx|,
where Sd-1 is the unit sphere in ℝd.Lemma 2.9.
For any symmetric matricesB1 and B2,
(2.22)max⁡x∈Sd-1|x⊤B1xx⊤B2x|≤‖B1‖‖B2‖.Proof.
It suffices to note that the left-hand side of the equality does not exceedmaxx,y∈Sd-1|x⊤B1xy⊤B2y|.LetX,X1,X2… be ℝd-valued random processes with trajectories in the Skorokhod space D (= càdlàg processes on ℝ+). We write Xn→DX 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 Xn→CX. We say that a sequence (Xn) is relatively compact (r.c.)inD (inC) 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 →d. Likewise =d means equality of distributions.DenoteΠ(t,r)={(u,v)∈ℝ2:(v-r)+≤u≤v≤t},(2.23)ΔU(f;t,r)=sup⁡(u,v)∈Π(t,r)|f(v)-f(u)|(f∈D,t>0,r>0).
Proposition VI.3.26 (items (i), (ii)) [2] together with VI.3.9 [2] asserts that a sequence (ξn) of càdlàg random processes is r.c. in C if and only if for all positive t and ɛ(2.24)lim⁡N→∞lim⁡¯n→∞P{sup⁡s≤t|ξn(s)|>N}=0,lim⁡r→0lim⁡¯n→∞P{ΔU(ξn;t,r)>ɛ}=0.
Hence, two consequences are immediate.Corollary 2.10.
Let(ξn) and (Ξn) be sequences of ℝd-valued and ℝm-valued, respectively, càdlàg processes such that (Ξn) is r.c. in C,|ξn(0)|≤|Ξn(0)| and for any v>u≥0(2.25)|ξn(v)-ξn(u)|≤|Ξn(v)-Ξn(u)|.
Then, the sequence (ξn) is also r.c. in C.Corollary 2.11.
Let(ξn) and (ζn) be r.c. in C sequences of càdlàg processes taking values in ℝd and ℝp, respectively. Suppose also that for each nξn and ζn are given on a common probability space. Then the sequence of ℝd+p-valued processes (ξn,ζn) is also r.c. in C.Lemma 2.12.
Let(ηnl,l,n∈ℕ), (ηl), and (ηn) be sequences of càdlàg random processes such that for any positive t and ɛ(2.26)lim⁡l→∞lim⁡¯n→∞P{sup⁡s≤t|ηnl(s)-ηn(s)|>ɛ}=0,
for each l(2.27)ηnl⟶Dηlasn⟶∞,
the sequence (ηl) is r.c. in D. Then, there exists a random process η such that ηl→Dη.Proof.
Letρ be a bounded metric in D metrizing Skorokhod's 𝒥-convergence (see, e.g., [2, VI.1.26]). Then, condition (2.26) with arbitrary t and ɛ implies that
(2.28)lim⁡l→∞lim⁡¯n→∞Eρ(ηnl,ηn)=0.
Hence, by the triangle inequality, we have
(2.29)lim⁡m→∞k→∞lim⁡¯n→∞Eρ(ηnm,ηnk)=0.
LetF be a uniformly continuous with respect to ρ bounded functional on D. Denote A=supx∈D|F(x)|, ϑ(r)=supx,y∈D:ρ(x,y)<r|F(x)-F(y)|. Then, ϑ(0+)=0 and for any r>0(2.30)E|F(ηnm)-F(ηnk)|≤AP{ρ(ηnm,ηnk)>r}+ϑ(r),
which together with (2.29) yields
(2.31)lim⁡m→∞k→∞lim⁡¯n→∞|EF(ηnm)-EF(ηnk)|=0.
By condition (2.27),
(2.32)lim⁡n→∞|EF(ηnm)-EF(ηnk)|=|EF(ηm)-EF(ηk)|,
which jointly with (2.31) proves fundamentality and, therefore, convergence of the sequence (EF(ηl),l∈ℕ). Now, the desired conclusion emerges from relative compactness of (ηl) in D.Corollary 2.13.
Let the conditions of Lemma2.12 be fulfilled. Then, ηn→Dη, where η is the existing by Lemma 2.12 random process such that ηl→Dη.Proof.
Repeating the derivation of (2.31) from (2.29), we derive from (2.28) the relation
(2.33)lim⁡l→∞lim⁡¯n→∞|EF(ηnl)-EF(ηn)|=0.
It remains to write |EF(ηn)-EF(η)|≤|EF(ηn)-EF(ηnl)|+|EF(ηnl)-EF(ηl)|+|EF(ηl)-EF(η)|.Corollary 2.14.
Let(ηnl), (ηl), and (ηn) be sequences of càdlàg random processes such that for any t∈ℝ+ and ɛ>0 (2.26) holds; for each l∈ℕ relation (2.27) is valid; the sequence (ηl) is r.c. in C. Then, there exists a random process η such that ηl→Cη and ηn→Cη.Below,𝒰 is the symbol of the locally uniform (i.e., uniform in every interval) convergence.Lemma 2.15.
LetX,X1,X2… be càdlàg random processes such that Xn→CX. Then, F(Xn)→dF(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 càdlàg random processes X′,X1′,X2′… such that X′=dX (so that X′ is continuous), Xn′=dXn and ρ(Xn′,X′)→0 a.s. By the choice of ρ, the last relation is tantamount to Xn′→𝒥X′ a.s. Hence, and from continuity of X′, we get by Proposition VI.1.17 [2] Xn′→𝒰X′a.s. and, therefore, by the choice of F,F(Xn′)→F(X′) a.s. It remains to note that F(Xn′)=dF(Xn),F(X′)=dF(X).


### 2.1. Forestopping of Random Processes

Let𝔽 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(0-)=S(0) and denote Sτ(t)=S(t∧τ),(2.34)Sτ(t)=S(t)I[0,τ[(t)+S(τ-)I[τ,∞[(t),ℱτ(t)=ℱ(t)∩ℱ(τ-), 𝔽τ=(ℱτ(t),t∈ℝ+). Obviously,(2.35)τ(Sτ)=Sτ,(2.36)[Sτ]=[S]τ,
provided [S] exists. In case τ is 𝔽-predictable, the operation S↦Sτ was called in [7] the forestopping. The following three statements were proved in [7].Lemma 2.16.
Let a random processU and a stopping time τ be 𝔽-predictable. Then, the process Uτ is 𝔽τ-predictable.Theorem 2.17.
LetX be an 𝔽-martingale and τ an 𝔽-predictable stopping time. Then, Xτ is a 𝔽τ-martingale. If X is uniformly integrable, then so is Xτ.Lemma 2.18.
LetV be an ℝd-valued right-continuous 𝔽-predictable random process and A a closed set in ℝd. Then, the stopping time inf{t:V(t)∈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.
LetU 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
(2.37)σ≤inf⁡⁡{t:tr⁡〈U〉(t)≥N}.
Then, Esups∈ℝ+|Uσ(s)|2≤4N.Proof.
Predictability ofσ implies by Theorem 2.1.13 [3] that there exists a sequence (σn) of stopping times such that
(2.38){σ>0}⊂{σn<σ},(2.39)σn↗σa.s.
By the choice ofU there exists a sequence (τk) of stopping times such that
(2.40)τk↗∞a.s,(2.41)Uk≡Uτk∈M2(F).
Then,
(2.42)sup⁡s≤σn|U(s)|=lim⁡k→∞sup⁡s≤σn∧τk|U(s)|.
Herein, obviously,
(2.43)sup⁡s≤σn∧τk|U(s)|=sup⁡s≤σn|Uk(s)|.
From (2.41) we have by Doob's inequality
(2.44)Esup⁡s≤σn|Uk(s)|2≤4E|Uk(σn)|2.
Noting that: (1) for any x∈ℝd|x|2=trxx⊤, (2) EUk(σn)Uk(σn)⊤=E〈Uk〉(σn), we may rewrite the last inequality in the form
(2.45)Esup⁡s≤σn|Uk(s)|2≤4Etr⁡〈Uk〉(σn).
Writing
(2.46)〈Uk〉(σn)=〈Uτk〉(σn)=〈U〉τk(σn)=〈U〉(τk∧σn),
we get from (2.37) and (2.38) tr〈Uk〉(σn)<N, which together with (2.45) results in Esups≤σn|Uk(s)|2<4N. Then, from (2.42) and (2.43), we have by Fatou's theorem
(2.47)Esup⁡s≤σn|U(s)|2≤4N.
The assumption U(0)=0 yields
(2.48)Esup⁡s<σ|U(s)|2=Esup⁡s<σ|U(s)|2I{σ>0}.
Relations (2.38) and (2.39) imply that
(2.49)sup⁡s<σ|U(s)|2I{σ>0}=lim⁡n→∞sup⁡s≤σn|U(s)|2I{σ>0},
which together with (2.47) yields by Fatou's theorem Esups<σ|U(s)|2I{σ>0}≤4N. It remains to note that sups∈ℝ+|Uσ(s)|=sups<σ|U(s)|I{σ>0} in view of (2.34).Lemma 2.20.
LetU be a locally square integrable martingale with respect to 𝔽 such that
(2.50)E|U(0)|2<∞,
and for any t(2.51)Emax⁡s≤t|ΔU(s)|2<∞.
Let, further, N be a positive number and σ a predictable time satisfying condition (2.37). Then, Uσ∈ℳ¯2(𝔽).Proof.
In view of (2.50) it suffices to show that (U-U(0))σ∈ℳ¯2(𝔽). In other words, we may consider that U(0)=0. Then condition (2.51) and the evident inequality
(2.52)sup⁡s≤t|Uσ(s)|≤sup⁡s≤t|Uσ(s)|+max⁡s≤t|ΔU(s)|
imply by Lemma 2.19 that for any t(2.53)Esup⁡s≤t|Uσ(s)|2<∞.
It remains to prove that for all t2>t1≥0,
(2.54)E(Uσ(t2)∣F(t1))=Uσ(t1).
Taking a sequence(τn) with properties (2.40) and (2.41), we write
(2.55)E(U(t2∧σ∧τk)∣F(t1))=Uσ(t1∧σ∧τk).
To deduce (2.54) from this inequality and (2.40), it suffices to note that
(2.56)|U(t∧σ∧τk)|≤sup⁡s≤t|Uσ(s)|,
so that (2.53) provides uniform integrability of the sequence (U(t2∧σ∧τk),k∈ℕ).Corollary 2.21.
Under the conditions of Lemma2.20  Uσ∈ℳ2(𝔽σ).Theorem 2.22.
LetU 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, 〈Uσ〉=〈U〉σ.Proof.
DenoteX=(U⊗2-〈U〉)σ≡(Uσ)⊗2-〈Uσ〉,Y=U⊗2σ-〈U〉σ≡(U⊗2-〈U〉)σ. It suffices to show that Y is a 𝔽σ-martingale. To deduce this fact from Theorem 2.17, we note that, firstly, X∈ℳ¯(𝔽) by construction and Lemma 2.20, and, secondly, Y=Xσ by construction of both processes and because of (2.35).


## 2.1. Forestopping of Random Processes

Let𝔽 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(0-)=S(0) and denote Sτ(t)=S(t∧τ),(2.34)Sτ(t)=S(t)I[0,τ[(t)+S(τ-)I[τ,∞[(t),ℱτ(t)=ℱ(t)∩ℱ(τ-), 𝔽τ=(ℱτ(t),t∈ℝ+). Obviously,(2.35)τ(Sτ)=Sτ,(2.36)[Sτ]=[S]τ,
provided [S] exists. In case τ is 𝔽-predictable, the operation S↦Sτ was called in [7] the forestopping. The following three statements were proved in [7].Lemma 2.16.
Let a random processU and a stopping time τ be 𝔽-predictable. Then, the process Uτ is 𝔽τ-predictable.Theorem 2.17.
LetX be an 𝔽-martingale and τ an 𝔽-predictable stopping time. Then, Xτ is a 𝔽τ-martingale. If X is uniformly integrable, then so is Xτ.Lemma 2.18.
LetV be an ℝd-valued right-continuous 𝔽-predictable random process and A a closed set in ℝd. Then, the stopping time inf{t:V(t)∈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.
LetU 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
(2.37)σ≤inf⁡⁡{t:tr⁡〈U〉(t)≥N}.
Then, Esups∈ℝ+|Uσ(s)|2≤4N.Proof.
Predictability ofσ implies by Theorem 2.1.13 [3] that there exists a sequence (σn) of stopping times such that
(2.38){σ>0}⊂{σn<σ},(2.39)σn↗σa.s.
By the choice ofU there exists a sequence (τk) of stopping times such that
(2.40)τk↗∞a.s,(2.41)Uk≡Uτk∈M2(F).
Then,
(2.42)sup⁡s≤σn|U(s)|=lim⁡k→∞sup⁡s≤σn∧τk|U(s)|.
Herein, obviously,
(2.43)sup⁡s≤σn∧τk|U(s)|=sup⁡s≤σn|Uk(s)|.
From (2.41) we have by Doob's inequality
(2.44)Esup⁡s≤σn|Uk(s)|2≤4E|Uk(σn)|2.
Noting that: (1) for any x∈ℝd|x|2=trxx⊤, (2) EUk(σn)Uk(σn)⊤=E〈Uk〉(σn), we may rewrite the last inequality in the form
(2.45)Esup⁡s≤σn|Uk(s)|2≤4Etr⁡〈Uk〉(σn).
Writing
(2.46)〈Uk〉(σn)=〈Uτk〉(σn)=〈U〉τk(σn)=〈U〉(τk∧σn),
we get from (2.37) and (2.38) tr〈Uk〉(σn)<N, which together with (2.45) results in Esups≤σn|Uk(s)|2<4N. Then, from (2.42) and (2.43), we have by Fatou's theorem
(2.47)Esup⁡s≤σn|U(s)|2≤4N.
The assumption U(0)=0 yields
(2.48)Esup⁡s<σ|U(s)|2=Esup⁡s<σ|U(s)|2I{σ>0}.
Relations (2.38) and (2.39) imply that
(2.49)sup⁡s<σ|U(s)|2I{σ>0}=lim⁡n→∞sup⁡s≤σn|U(s)|2I{σ>0},
which together with (2.47) yields by Fatou's theorem Esups<σ|U(s)|2I{σ>0}≤4N. It remains to note that sups∈ℝ+|Uσ(s)|=sups<σ|U(s)|I{σ>0} in view of (2.34).Lemma 2.20.
LetU be a locally square integrable martingale with respect to 𝔽 such that
(2.50)E|U(0)|2<∞,
and for any t(2.51)Emax⁡s≤t|ΔU(s)|2<∞.
Let, further, N be a positive number and σ a predictable time satisfying condition (2.37). Then, Uσ∈ℳ¯2(𝔽).Proof.
In view of (2.50) it suffices to show that (U-U(0))σ∈ℳ¯2(𝔽). In other words, we may consider that U(0)=0. Then condition (2.51) and the evident inequality
(2.52)sup⁡s≤t|Uσ(s)|≤sup⁡s≤t|Uσ(s)|+max⁡s≤t|ΔU(s)|
imply by Lemma 2.19 that for any t(2.53)Esup⁡s≤t|Uσ(s)|2<∞.
It remains to prove that for all t2>t1≥0,
(2.54)E(Uσ(t2)∣F(t1))=Uσ(t1).
Taking a sequence(τn) with properties (2.40) and (2.41), we write
(2.55)E(U(t2∧σ∧τk)∣F(t1))=Uσ(t1∧σ∧τk).
To deduce (2.54) from this inequality and (2.40), it suffices to note that
(2.56)|U(t∧σ∧τk)|≤sup⁡s≤t|Uσ(s)|,
so that (2.53) provides uniform integrability of the sequence (U(t2∧σ∧τk),k∈ℕ).Corollary 2.21.
Under the conditions of Lemma2.20  Uσ∈ℳ2(𝔽σ).Theorem 2.22.
LetU 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, 〈Uσ〉=〈U〉σ.Proof.
DenoteX=(U⊗2-〈U〉)σ≡(Uσ)⊗2-〈Uσ〉,Y=U⊗2σ-〈U〉σ≡(U⊗2-〈U〉)σ. It suffices to show that Y is a 𝔽σ-martingale. To deduce this fact from Theorem 2.17, we note that, firstly, X∈ℳ¯(𝔽) by construction and Lemma 2.20, and, secondly, Y=Xσ 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
(3.1)Ml⟶dM,
and for any t the sequence (|Ml(t)|) is uniformly integrable. Then, M is a martingale.Lemma 3.2.
Let(Ml) be a sequence of martingales such that (3.1) holds and
(3.2)sup⁡l∈N,t∈R+Etr⁡〈Ml〉(t)<∞.
Then, suptE|M(t)|2<∞.Proof.
By condition (3.2) and the definition of quadratic characteristic, there exists a constant C such that E|Ml(t)|2≤C for all t 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) E|M(t)|2≤C.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.
LetY be a local martingale and K be an 𝔖-valued random process. Suppose that they are given on a common probability space and (Y,K)=d(Y,[Y]). Then for any tK(t)=[Y](t) a.s.Proof.
By assumption,(3.3)∑i=1n(Y(ti)-Y(ti-1))⊗2-K(t)=d∑i=1n(Y(ti)-Y(ti-1))⊗2-[Y](t),
for all n,t and t0<t1<⋯tn. Hence, recalling the definition of quadratic variation, we get [Y](t)-K(t)=d0.We shall identify indistinguishable processes, writing simplyξ=η if P{∀t∈ℝ+,ξ(t)=η(t)}=1. Theorem 2.3.5 [3] asserts that(3.4)[Y]=〈Y〉,
for a continuous local martingale Y. Hence, and from Lemma 3.4, we haveCorollary 3.5.
LetY be a continuous local martingale and K an 𝔖-valued random process. Suppose that they are given on a common probability space and (Y,K)=d(Y,〈Y〉). Then, K=〈Y〉.Proof.
Lemma3.4 and formula (3.4) yield P{∀t∈ℚ+,K(t)=〈Y〉(t)}=1. Continuity of both processes enables us to substitute ℚ+ by ℝ+.Lemma 3.6.
LetU be a locally square integrable martingale. Then, ∥〈U〉∥ is an increasing process.Proof.
For anyx∈ℝd, the process x⊤U is a numeral locally square integrable martingale and, therefore, the process 〈x⊤U〉 increases. It remains to note that 〈x⊤U〉=x⊤〈U〉x and to recall formula (2.21).Lemma 3.7.
LetZ1 and Z2 be locally square integrable martingales with respect to a common filtration. Then,
(3.5)‖〈Z1,Z2〉+〈Z2,Z1〉‖≤2‖〈Z1〉‖‖〈Z2〉‖.Proof.
Ford=1 (then 〈Z2,Z1〉=〈Z1,Z2〉), this is the Kunita-Watanabe inequality [3, page 118]. In the general case, we take an arbitrary vector x∈Sd-1 and write
(3.6)x⊤(〈Z1,Z2〉+〈Z2,Z1〉)x=2〈x⊤Z1,x⊤Z2〉≤2〈x⊤Z1〉〈x⊤Z2〉=2x⊤〈Z1〉xx⊤〈Z2〉x,
hereupon the required conclusion ensues from (2.21) and Lemma 2.9.Lemma 3.8.
LetU1 and U2 be locally square integrable martingales with respect to some common filtration. Then, for any t>0(3.7)sup⁡s≤t‖〈U1〉(s)-〈U2〉(s)‖≤‖〈U1-U2〉(t)‖+2‖〈U1-U2〉(t)‖‖〈U2〉(t)‖.Proof.
Writing the identities(3.8)〈U1〉=〈U1-U2+U2〉=〈U1-U2〉+〈U1-U2,U2〉+〈U2,U1-U2〉+〈U2〉,
we deduce from Lemma 3.7 that
(3.9)‖〈U1〉(s)-〈U2〉(s)‖≤‖〈U1-U2〉(s)‖+2‖〈U2-U1〉(s)‖‖〈U2〉(s)‖.
It remains to note that the right-hand side increases in s by Lemma 3.6.For a functionf∈D we denote Δf(t)=f(t)-f(t-).Let us introduce the conditions:(RC)
The sequence(tr〈Yn〉) is r.c. in C.(UI1)
The sequence(|Yn(t)-Yn(0)|2) is u.i.(UI2)
For anyz∈ℝd* the sequence (tr〈zYn〉(t)) is u.i.(UI3)
The sequence(sups≤t|Yn(s)-Yn(0)|2) is u.i.Lemma 3.9.
Let(Yn) be a sequence of local square integrable martingales satisfying the conditions: (RC),
(3.10)lim⁡L→∞lim⁡¯n→∞P{|Yn(0)|>L}=0,
and, for each t>0, the condition
(3.11)max⁡s≤t|ΔYn(s)|⟶P0.
Then, (Yn) is r.c. in C.Proof.
It follows from (RC) and (3.10) by Rebolledo's theorem [2, VI.4.13] that (Yn) 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 Lemma3.9 with Corollary 2.11, we getCorollary 3.10.
Under the assumptions of Lemma3.9, the sequence of compound processes (Yn,〈Yn〉) is r.c. in C.Some statements below deal with random processes on[0,t], not on ℝ+. In this case, the time variable is denoted by s and C means C[0,t] instead of C(ℝ+).Lemma 3.11.
Let(Yn) be a r.c. in C and satisfying condition (UI1) sequence of martingales on [0,t]. Then, for any z∈ℝd*, the sequence ([zYn](t)) is u.i.Proof.
The obvious equality[V-V(0)]=[V] allows us to consider that Yn(0)=0. Then, condition (UI1) together with Doob's inequality yields
(3.12)sup⁡nEsup⁡s≤t|Yn(s)|2<∞,
whence
(3.13)sup⁡nEmax⁡s≤t|ΔYn(s)|<∞.
By assumption, for any infinite setJ0⊂ℕ, there exist an infinite subset J⊂J0 and a random process Y such that
(3.14)Yn⟶CYasn⟶∞,n∈J.
Condition (UI1) implies that Y is a square integrable martingale and for any z∈ℝd*,
(3.15)E(zYn(t))2⟶E(zY(t))2asn⟶∞,n∈J.
From (3.14) and (3.13), we have by Corollary VI.6.7 [2]
(3.16)[zYn]⟶C[zY]asn⟶∞,n∈J.
Hence, and from (3.15), recalling that for any ℝ-valued M∈ℳ¯2 one has E(M-M(0))2=E[M][3, Theorem 2.2.4], we get
(3.17)E[zYn](t)⟶E[zY](t)asn⟶∞,n∈J.
Comparing this relation with (3.16), we conclude that the sequence ([zYn](t),n∈J) is uniformly integrable. Hence, in view of arbitrariness of J0, we deduce by Lemma 2.3 uniform integrability of ([zYn](t),n∈ℕ).Lemma 3.12.
Let(Yn) be a sequence of martingales on [0,t] satisfying condition (UI1) and (UI2). Suppose that there exists an ℝd×𝔖+-valued random process (Y,K) such that
(3.18)(Yn,〈Yn〉)⟶C(Y,K).
Then, firstly,
(3.19)[Yn]-〈Yn〉⟶CO,
where O is the null matrix, Y is a continuous martingale, and, secondly,
(3.20)(Y,K)=d(Y,〈Y〉).Proof.
For the same reason as in the proof of Lemma3.11, we may consider that Yn(0)=0. Then, as was shown above, condition (UI1) implies (3.13). Combining the latter with
(3.21)Yn⟶CY,
(a part of (3.18)), we get by Corollary VI.6.7 [2] that
(3.22)(Yn,[Yn])⟶C(Y,[Y]).
From (3.18) and (3.22), we get by Corollary 2.11 that for any infinite set J0⊂ℕ, there exist an infinite subset J⊂J0 and an 𝔖+×𝔖+-valued random process (QJ,RJ) such that
(3.23)([Yn],〈Yn〉)⟶(QJ,RJ)asn⟶∞,n∈J.
(Of course QJ=dK,RJ=d[Y].)
DenoteZn=[Yn]-〈Yn〉. This is a martingale by Lemma 10.4 in [4]. Relation (3.23) implies that
(3.24)Zn⟶CQJ-RJasn⟶∞,n∈J.
For any z∈ℝd*, the sequence (zZn(t)z⊤) is, by Lemma 3.11 and condition (UI2), u.i. So, relation (3.24) implies by Lemma 3.1 that z(QJ-RJ)z⊤ is a martingale. Also, it implies its continuity. Relation (3.23) shows that the processes zQJz⊤ and zRJz⊤ increase and start from zero. So, z(QJ-RJ)z⊤ starts from zero and has finite variation in [0,t]. These four properties of QJ-RJ imply together that QJ(s)-RJ(s)=O for all s∈[0,t]. Thus, any subsequence (Zn,n∈J0) contains, in turn, a subsequence (Zn,n∈J) such that Zn→Oasn→∞,n∈J. This proves (3.19).
From (3.22) and (3.19), we have (Yn,〈Yn〉)→C(Y,[Y]). And this is, in view of (3.4), tantamount to
(3.25)(Yn,〈Yn〉)⟶C(Y,〈Y〉).
Comparing this relation with (3.18), we arrive at (3.20).Remark 3.13.
The second conclusion of Lemma3.12 implies by Corollary 3.5 thatK=〈Y〉.Corollary 3.14.
Let a sequence(Yn) of martingales on [0,t] satisfy conditions (RC), (3.11), (UI1), and (UI2). Then, relation (3.19) holds.Proof.
By Corollary3.10 for any infinite set J0⊂ℕ, there exist an infinite set J⊂J0 and an ℝd×𝔖+-valued random process (Y,K) such that relation (3.18) holds as n→∞,n∈J. Then, by Lemma 3.12, so does (as n→∞,n∈J) (3.19). Due to arbitrariness of J0 this relation holds when n ranges over ℕ, too.Corollary 3.15.
Let(Yn) be a sequence of martingales on ℝ+ satisfying conditions (RC) and for all t>0, conditions (3.11), (UI1), (UI2). Then, relation (3.19) holds.Lemma 3.16.
Let a sequence(Yn) of martingales on ℝ+ satisfy conditions (RC) and, for any t>0, (3.11) and (UI3). Then, relation (3.19) holds.Proof.
DenoteσnN=inf{s:tr〈Yn〉(s)≥N},YnN=YnσnN. Obviously,
(3.26){σnN<t}⊂{tr⁡〈Yn〉(t)≥N}.
By Corollary 2.21  YnN is a square integrable martingale with respect to 𝔽nσnN. By Theorem 2.22,
(3.27)〈YnN〉=〈Yn〉σnN.
Condition (RC) implies relative compactness of the sequence (〈YnN〉,n∈ℕ). By construction |YnN(t)-YnN(0)|≤sups≤t|Yn(t)-Yn(0)|. So, condition (UI3) implies that for any t and N the sequence (|YnN(t)-YnN(0)|2,n∈ℕ) is u.i. Thus, Corollary 3.15 asserts that for any N(3.28)[YnN]-〈YnN〉⟶COasn⟶∞.
Equalities (3.27) and (2.36) yield the relation
(3.29){sup⁡s≤t(‖[YnN](s)-[Yn](s)‖+‖〈YnN〉(s)-〈Yn〉(s)‖)>0}⊂{σnN<t},
which together with (3.26), (3.28) and (RC) entails (3.19).Corollary3.15 and Lemma 3.16 are only the steps towards the final result about asymptotic proximity of quadratic variations and quadratic characteristics—Corollary 5.3.Corollary 3.17.
Let a sequence(Yn) of martingales on ℝ+ satisfy conditions (RC) and for any t>0, (3.11) and (UI3). Suppose also that there exists an ℝd×𝔖+-valued random process (Y,K) such that relation (3.18) is valid. Then, Y is a continuous martingale, and (3.20) holds.Proof.
Lemma3.16 asserts (3.19). The implications ((3.21) and (UI1) ⇒ (3.22)); ((3.22) and (3.19) ⇒ (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 eachnMn be an ℝ-valued square integrable martingale on [0,t] with respect to a flow (𝒢n(s),s∈[0,t]) and ℋn a sub-σ-algebra of 𝒢n(0). Suppose that conditions (UI1) and (RC) are fulfilled for Yn=Mn,
(4.1)Mn(0)=0,(4.2)max⁡s≤t|ΔMn(s)|⟶P0
and there exists a nonrandom number N such that for all n(4.3)〈Mn〉(t)≤N.
Then,
(4.4)E(eiMn(t)+〈Mn〉(t)/2∣Hn)⟶P1.Proof.
Conditions (RC)(Yn=Mn), (4.1), and (4.2) entail, by Lemma 3.9, relative compactness of (Mn) in C.
DenoteTn=〈Mn〉/2, ξn=eiMn+Tn, Xn=([Mn]-〈Mn〉)/2, γn=ξn-∘Xn,
(4.5)ηn=∑s≤tξn(s-)(eΔTn(s)+iΔMn(s)-1-ΔTn(s)-iΔMn(s)+12(ΔMn(s))2).
Condition (RC) (Yn=Mn) implies that
(4.6)max⁡s≤tΔTn(s)⟶P0.
In view of (4.1) ξn(0)=1. Then, by Itô's formula
(4.7)ξn(t)=1+iξn-⋅Mn(t)+ξn-∘Tn(t)-12ξn-∘〈Mnc〉(t)+∑s≤tξn(s-)(eiΔMn(s)+ΔTn(s)-1-iΔMn(s)-ΔTn(s)).
Hence, recalling that 〈Mnc〉(t)=[Mn](t)-∑s≤t(ΔMn(s))2, we get
(4.8)ξn(t)=1+iξn-⋅Mn(t)-ξn-∘Xn(t)+ηn.
By the definition ofξn and by condition (4.3),
(4.9)sup⁡s≤t|ξn(s)|≤eN/2.
Consequently,
(4.10)E(ξn-⋅Mn(t)∣Hn)=0,
and E|ξn-·Mn(t)|2=E(|ξn-|2∘〈Mn〉(t)). The right-hand side of the last equality being less than eNN, the sequence (ξn-·Mn(t)) is u.i., and so is ([Mn](t)) by Lemma 3.11 whose conditions (those not postulated) we have verified. This together with (4.9) and (4.3) implies uniform integrability of (ξn-∘Xn(t)). Now, (4.8) and inequality (4.9) show that (ηn) has this property, too.
By construction and Lemma 10.4 in [4], Xn is a martingale. Then, it follows from (4.9) that E(ξn-∘Xn(t)∣ℋn)=0, which together with (4.10) and (4.8) yields
(4.11)E(ξn(t)∣Hn)=1+E(ηn∣Hn).
So, it suffices to show that
(4.12)ηn⟶P0.
Obviously, for any reala and b(4.13)ea+bi-a=(ea-1-a)ebi+a(ebi-1)+ebi,|ea-1-a|≤|a|2e|a|,|ebi-1|≤|b|,|ebi-1-bi+b22|≤|b|3.
Hence, from (4.5), (4.9), we get
(4.14)|ηn|≤eN(eeN∑s≤t(ΔTn(s))2+max⁡s≤t|ΔMn(s)|(∑s≤tΔTn(s)+∑s≤t(ΔMn(s))2)).
Now, (4.12) ensues from (4.6), (4.3), (4.2), and stochastic boundedness of the sequence ([Mn](t)).Lemma 4.2.
Let for eachn,Mn be an ℝ-valued starting from zero locally square integrable martingale with respect to some flow (𝒢n(t),t∈ℝ+) and ℋn a sub-σ-algebra of 𝒢n(0). Suppose that condition (RC) is fulfilled for Yn=Mn;
(4.15)Emax⁡s≤t(ΔMn(s))2⟶0,
for any t>0; there exists a nonrandom function q:ℝ+→ℝ+ such that
(4.16)〈Mn〉(t)≤q(t),
for all n and t. Then, for any t, relation (4.4) holds.Proof.
Let us denote, only in this proof,τNn=inf{t:|Mn(t)|≥N}, MNn(t)=Mn(t∧τNn), TNn(t)=Tn(t∧τNn) (so that MNn∈ℓℳ2 and 〈MNn〉=2TNn),ξNn=eiMNn+TNn. The evident inequality
(4.17)sup⁡s≤t|MNn(s)|≤N+max⁡s≤t|ΔMn(s)|,
and condition (4.15) show us that for any positive t and N, the sequence (MNn(t)2,n∈ℕ) is u.i.
By assumption, there exists a sequence(σk) of stopping times such that σk↗∞ a.s. and for each k,Mσk∈ℳ2. Then, for any t>s≥0, N>0 and n, k∈ℕ,
(4.18)E(MNn(t∧σk)∣Gn(s))=E(Mσk(t∧τNn)∣Gn(s))=Mσk(s∧τNn)=MNn(s∧σk).
Writing |MNn(t∧σk)|≤sups≤t|MNn(s)|, we deduce from (4.17) and (4.15) uniform integrability of the sequence (MNn(t∧σk),k∈ℕ). So, letting k→∞ in (4.18), we get
(4.19)E(MNn(t)∣Gn(s))=MNn(s),
that is, MNn is a martingale. It is square integrable because of (4.17) and (4.15). Thus, for any N and t the sequence (MNn,n∈ℕ) satisfies all the conditions of Lemma 4.1 which, therefore, asserts that
(4.20)E(ξNn(t)∣Hn)⟶P1asn⟶∞.
Here, in |ξn(t)|∨|ξNn(t)|≤eq(t) because of (4.16), so
(4.21)|ξn(t)-ξNn(t)|≤2eq(t)I{τNn≤t}.
Obviously, {τNn≤t}⊂{sups≤t|Mn(s)|≥N}. From (4.16), we have by the Lenglart-Rebolledo inequality
(4.22)lim⁡N→∞lim⁡¯n→∞P{sup⁡s≤t|Mn(s)|≥N}=0.
The last three relations imply that
(4.23)lim⁡N→∞lim⁡¯n→∞E|ξn(t)-ξNn(t)|=0,
which together with (4.20) yields (4.4).Lemma 4.3.
Let for eachn,Mn,Mn1,Mn2…  be ℝ-valued starting from zero locally square integrable martingales with respect to a flow (𝒢n(t),t∈ℝ+) and ℋn a sub-σ-algebra of 𝒢n(0). Suppose that for all m∈ℕ, t>0, ɛ>0,
(4.24)lim⁡n→∞Emax⁡u≤t(ΔMnm(u))2=0,(4.25)lim⁡l→∞lim⁡¯n→∞P{〈Mnl-Mn〉(t)>ɛ}=0,
there exists a nonrandom function q:ℝ+→ℝ+ such that
(4.26)〈Mn〉(t)∨〈Mnm〉(t)≤q(t),
for all m,n, and t; for each m, the sequence (〈Mnm〉,n∈ℕ) is r.c. in C. Then, for any t relation (4.4) holds.Proof.
DenoteTnm=〈Mnm〉/2. By Lemma 4.2  E(eiMnm(t)+Tnm(t)∣ℋn)→P1 as n→∞. So, in view of (4.26), it suffices to prove that for any ɛ>0,
(4.27)lim⁡l→∞lim⁡¯n→∞E|eiMnl(t)+Tnl(t)-eiMn(t)+Tn(t)|=0.
Conditions (4.25) and (4.26) imply by Lemma 3.8 that for all positive t and ɛ(4.28)lim⁡l→∞lim⁡¯n→∞P{|Tnl(t)-Tn(t)|>ɛ}=0.
Furthermore, (4.25) together with the Lenglart-Rebolledo inequality and the assumed equalities Mn(0)=0=Mnl(0) yields
(4.29)lim⁡l→∞lim⁡¯n→∞P{|Mnl(t)-Mn(t)|>ɛ}=0,
which jointly with the previous relation and condition (4.25) entails (4.27).Lemma 4.4.
Let for eachn,Mn be an ℝ-valued starting from zero locally square integrable martingale with respect to a flow (𝒢n(t),t∈ℝ+) and ℋn a sub-σ-algebra of 𝒢n(0). Suppose that conditions (RC) (Yn=Mn), (4.24) (for all m and t) and (4.25) (for all t and ɛ) are fulfilled; for all m∈ℕ and u2>u1≥0,
(4.30)〈Mnm〉(u2)-〈Mnm〉(u1)≤〈Mn〉(u2)-〈Mn〉(u1);
for any t>0 and bounded uniformly continuous function f:ℝ+→ℝ,
(4.31)E(f(〈Mn〉(t))∣Hn)-f(〈Mn〉(t))⟶P0.
Then, for any t,
(4.32)E(eiMn(t)∣Hn)-e-〈Mn〉(t)/2⟶P0.Proof.
(1∘) Let us fix t and denote αn=eTn(t), βn=eiMn(t) so that ξn(t)=αnβn. If there exists a nonrandom constant N such that 〈Mn〉(s)≤N for all n and s, then all the conditions of Lemma 4.3 are fulfilled, and therefore,
(4.33)E(αnβn∣Hn)⟶P1.
Also, under this assumption αn=gN(Tn(t)), where gN(x)=ex[N],
(4.34)x[N]=NxN∨|x|.
So, substituting f(x)=gN(2x) to (4.31), we obtain (2.7), whence by Lemma 2.5, relation (2.8) follows. Juxtaposing it with (4.33), we get αnE(βn∣ℋn)-1→P0. Dividing both sides of this relation by αn(≥1), we arrive at (4.32).
(2∘) Let us waive the extra assumption.
Denoteσkn=inf{s:〈Mn〉(s)≥k},
(4.35)Tkn(s)=Tn(s)I[0,σkn[(s)+Tn(σkn-)I[σkn,∞[(s),(4.36)Tknm(s)=Tnm(s)I[0,σkn[(s)+Tnm(σkn-)I[σkn,∞[(s),
and likewise with M instead of T. Lemma 2.18 asserts predictability of σkn. By construction and condition (4.33), σkn≤inf{s:〈Mnm〉(s)≥k}. Thus, Theorem 2.22 asserts that Mkn and Mknm are square integrable martingales and 〈Mkn〉=2Tkn,〈Mknm〉=2Tknm. Consequently, for any t2>t1>0(4.37)〈Mknm〉(t2)-〈Mknm〉(t1)≤〈Mkn〉(t2)-〈Mkn〉(t1).
In view of (4.35) and (4.34),
(4.38)(Tn(t))[k]-Tkn(t)=(kTn(σkn)k∨Tn(σkn)-Tn(σkn-))I[σkn,∞[(t),
whence
(4.39)|(Tn(t))[k]-Tkn(t)|≤ΔTn(σkn∧t)≤max⁡s≤tΔTn(s).
Here, in condition (4.6) is fulfilled because of (RC).
Letf be a bounded uniformly continuous function. Then,
(4.40)E(f((Tn(t))[k])∣Hn)-f((Tn(t))[k])⟶P0,
by condition (4.30);
(4.41)f(Tkn(t))-f((Tn(t))[k])⟶P0,
on the strength of (4.39), (4.6) and uniform continuity of f. From the second relation, we get, since f is bounded,
(4.42)E(f(Tkn(t))-f((Tn(t))[k])∣Hn)⟶P0.
These three relations together yield
(4.43)E(f(Tkn(t))∣Hn)-f(Tkn(t))⟶P0.
Thus, the sequences (Mkn,n∈ℕ) and (Mknm,n∈ℕ) satisfy all the conditions of the lemma plus the above extra assumption. Then, according to item  (1∘)E(eiMkn(t)∣ℋn)-e-〈Mkn〉(t)/2→P0 as n→∞. Hence, and from (RC), relation (4.32) emerges by the same argument as (4.4) was derived from (4.20) and (4.22).Theorem 4.5.
Let for eachn∈ℕXn,Xn1,Xn2… be locally square integrable martingales with respect to a flow 𝔽n. Suppose that the sequence (tr〈Xn〉) is r.c. in C and for all m∈ℕ,t>s>0,ɛ>0,t2>t1≥0,z∈ℝd* and bounded uniformly continuous functions f:ℝ+→ℝ(4.44)lim⁡n→∞Emax⁡u≤t|ΔXnm(u)|2=0,(4.45)lim⁡l→∞lim⁡¯n→∞P{tr⁡〈Xnl-Xn〉(t)>ɛ}=0,(4.46)〈zXnm〉(t2)-〈zXnm〉(t1)≤〈zXn〉(t2)-〈zXn〉(t1),(4.47)E(f(〈zXn〉(t))∣Fn(s))-f(〈zXn〉(t))⟶P0.
Then, (1) for any p,l∈ℕ, tp>⋯>t1>t0≥s>0, z1,…,zp∈ℝd*, and sl>⋯>s1>0(4.48)E(exp⁡{i∑j=1pzj(Xn(tj)-Xn(tj-1))}∣Fn(s))-exp⁡{-12∑j=1pzj(〈Xn〉(tj)-〈Xn〉(tj-1))zj⊤}⟶P0(2) under the extra assumption that the sequence (Xn(0)) is stochastically bounded,
(4.49)E(exp⁡{i∑j=1pzj(Xn(tj)-Xn(tj-1))}F(Xn(0),〈Xn〉(s1),…,〈Xn〉(sl))∣Fn(s))-exp⁡{-12∑j=1pzj(〈Xn〉(tj)-〈Xn〉(tj-1))zj⊤}F(Xn(0),〈Xn〉(s1),…,〈Xn〉(sl))⟶P0,
for all p,l∈ℕ, tp>⋯>t1>t0≥s>0, z1,…,zp∈ℝd*,sl>⋯>s1>0, and F∈Cb(ℝd×𝔖l).Proof.
The relative compactness condition implies that for anyt, the sequence (∥〈Xn〉∥(t)) is stochastically bounded. Then, it follows from (4.47) by Corollary 2.8 that for any t>t′≥s>0, ɛ>0,z∈ℝd* and φ∈Cb(ℝ+2)(4.50)E|E(φ(〈zXn〉(t),〈zXn〉(t′))∣Fn(s))-φ(〈zXn〉(t),〈zXn〉(t′))|⟶0.
If, moreover, the sequence (Xn(0)) is stochastically bounded, then the same corollary asserts that, in the notation of formula (4.49),
(4.51)E(F(Xn(0),〈Xn〉(s1),…,〈Xn〉(sl))∣Fn(s))-F(Xn(0),〈Xn〉(s1),…,〈Xn〉(sl))⟶P0.
Let us fixj and denote Mn(u)=zjXn(tj-1+u)-zjXn(tj-1) (likewise with a superscript), 𝒢n(u)=ℱn(tj-1+u). Then,
(4.52)〈Mn〉(u)=〈zjXn〉(tj-1+u)-〈zjXn〉(tj-1),(4.53)Mn(tj-tj-1)=zjXn(tj)-zjXn(tj-1),〈Mn〉(tj-tj-1)=〈zjXn〉(tj)-〈zjXn〉(tj-1),max⁡u≤t|ΔMnm(u)|=max⁡u≤tj-1+t|zjΔXnm(u)|,〈Mnl-Mn〉(u)=〈zj(Xnl-Xn)〉(tj-1+u)-〈zj(Xnl-Xn)〉(tj-1).
So we have the implications: (4.44) ⇒ (4.24); (4.45) ⇒ (4.25). Setting in (4.50) t=tj-1+u, t′=tj-1, z=zj, φ(x,y)=f(x-y) (f∈Cb(ℝ+)), and taking to account (4.52), we get (4.31) with ℋn=𝒢n(0). Equality (4.52) shows that the sequence (〈Mn〉) is r.c. in C, since (〈Xn〉) has this property. The similar equality for Mnm and condition (4.46) imply (4.30). Thus, Lemma 4.4 asserts that for any t relation (4.32) with ℋn=𝒢n(0) holds. Putting t=tj-tj-1, we convert it to
(4.54)E(eizj(Xn(tj)-Xn(tj-1))∣Fn(tj-1))-e-zj(〈Xn〉(tj)-〈Xn〉(tj-1))zj⊤/2⟶P0.
Denote the left-hand side of this relation by ϰn. Inequality |ϰn|≤2 allows to rewrite it in the form E|ϰn|→0. Consequently, for any s∈[0,tj-1](4.55)E(eizj(Xn(tj)-Xn(tj-1))∣Fn(s))-E(e-zj(〈Xn〉(tj)-〈Xn〉(tj-1))zj⊤/2∣Fn(s))⟶P0.
Hence, and from (4.50) (φ(x,y)=f(x-y)), we have for j=1,…,p(4.56)E(eizj(Xn(tj)-Xn(tj-1))∣Fn(s))-e-zj(〈Xn〉(tj)-〈Xn〉(tj-1))zj⊤/2⟶P0.
Now, (4.48) emerges from Lemma 2.4.
Relation (4.49) follows from (4.48) and (4.51) by Lemma 2.5.Remark 4.6.
Relation (4.49) implies that every partial limit (with respect to the weak convergence in law) of a sequence(Xn)is a process with conditionally independent increments.The following result can facilitate the verification of condition (4.47).Lemma 4.7.
Let for eachn∈ℕQn be an 𝔖-valued or ℝk-valued random process adapted to a flow 𝔽n on a probability space (Ωn,ℱn,Pn). Suppose that there exists a sequence (Λn) of scalar random processes such that, for any n∈ℕ and u>0,Λn(u) is an ℱn(0)-measurable positive random variable; for all t>s>0,
(4.57)Qn(t)-Λn(t)Λn(s)Qn(s)⟶P0.
Then, for all for t>s>0 and bounded uniformly continuous functions g on 𝔖 (or on ℝk),
(4.58)E(g(Qn(t))∣Fn(s))-g(Qn(t))⟶P0.Proof.
Denoteλn(t,s)=Λn(t)/Λn(s). Condition (4.57) implies that
(4.59)P{|Qn(t)-λn(t,s)Qn(s)|>ɛ∣Hn}⟶P0,
for any t>s>0, ɛ>0 and sequence (ℋn) whose nth member is a sub-σ-algebra of ℱn. Hence, and from the evident inequality
(4.60)E(|g(Qn(t))-g(λn(t,s)Qn(s))|∣Hn)≤2∥g∥∞P{‖Qn(t)-λn(t,s)Qn(s)‖>ɛ∣Hn}+sup⁡‖A-B‖≤ɛ|g(A)-g(B)|,
we get by the choice of g(4.61)E(g(Qn(t))-g(λn(t,s)Qn(s))∣Hn)⟶P0.
Setting here at first ℋn=ℱn(s) and then ℋn=ℱn(t), subtracting the second relation from the first and recalling that the random variable λn(t,s)Qn(s) is, by the assumptions about Qn and Λn, ℱn(s)-measurable, we arrive at (4.58).Example 4.8.
Letℱn(t)=ℱ(nt) and Qn(t)=n-1R(nt), where R is an 𝔽-adapted random process (so that Qn is 𝔽n-adapted). Writing
(4.62)Qn(t)-tsQn(s)=t(R(nt)nt-R(nt)ns),
we see that condition (4.57) will be fulfilled with Λn(t)=t if we demand that t-1R(t) tend in probability to some limit as t→∞.


## 5. The Convergence Theorems

Theorem 5.1.
Let(Yn) be a sequence of local square integrable martingales satisfying conditions (RC), (3.10), and, for each t, the condition
(5.1)Emax⁡s≤t|ΔYn(s)|2⟶0.
Then, for any infinite set J0⊂ℕ there exist an infinite set J⊂J0 and a continuous local martingale YJ such that
(5.2)(Yn,〈Yn〉)⟶C(YJ,〈YJ〉)asn⟶∞,n∈J.Proof.
(1∘) Denote τnl=inf{t:|Yn(t)-Yn(0)|≥l}, Ynl(t)=Yn(t∧τnl),Kn=〈Yn〉, Knl=〈Ynl〉 (so that Knl(t)=Kn(t∧τnl)),
(5.3)ηn=(Yn,Kn),ηnl=(Ynl,Knl),
regarding ηn and ηnl as ℝd+d2-valued processes.
Conditions (RC), (3.10) and (5.1) imply by Corollary 3.10 that the sequence (ηn) is r.c. in C. Then, by Corollaries 2.10 and 2.11, for any l∈ℕ the sequence of compound processes (ηn1,…,ηnl,Kn) is r.c. in C, too. Hence, using the diagonal method, we deduce that for any infinite set J0⊂ℕ, there exist an infinite set J⊂J0 and random processes Y1,K1,Y2,K2… such that for all l∈ℕ(5.4)(ηn1,…,ηnl,Kn)⟶C(η1,…,ηl,K)asn⟶∞,n∈J,
where
(5.5)ηi=(Yi,Ki).
The distribution of the right-hand side of (5.4) may depend on J, so the minute notation would be something like (ηJ,1,…,ηJ,l,KJ). We suppress, “for technical reasons”, the superscript J, keeping, however, it in mind.
(2∘) By the definition of Ynl,
(5.6)sup⁡s≤t|Ynl(s)-Ynl(0)|≤l+max⁡s≤t|ΔYn(s)|,
which together with (5.1) shows that for any l and t, the sequence (sups≤t|Ynl(s)-Ynl(0)|2,n∈ℕ) is uniformly integrable. Then, it follows from (5.3)–(5.5) by Corollary 3.17 and Remark 3.13 that Yl is a continuous martingale and
(5.7)Kl=〈Yl〉.
(3∘) Writing
(5.8){sup⁡s≤t|ηnl(s)-ηn(s)|>0}⊂{τnl<t}⊂{sup⁡s≤t|Yn(s)-Yn(0)|≥l},
and recalling that (Yn) is r.c. in C, we arrive at (2.26).
(4∘) Note that the processes η1,η2… are given, in view of (5.4), on a common probability space. Let us show that
(5.9)lim⁡l→∞sup⁡i>lEϱ(ηi,ηl)=0,
where ϱ is the metric in D defined by
(5.10)ϱ(f,q)=∑m=1∞2-m(1∧sup⁡s≤m|f(s)-g(s)|).
From (5.4), we have by Lemma 2.15(5.11)sup⁡s≤m|ηni(s)-ηnl(s)|⟶dsup⁡s≤m|ηi(s)-ηl(s)|asn⟶∞,n∈J,
for all natural m,i and l. Then, Alexandrov's theorem asserts that for any ɛ>0,
(5.12)P{sup⁡s≤m|ηi(s)-ηl(s)|>ɛ}≤lim⁡̲n→∞,n∈JP{sup⁡s≤m|ηni(s)-ηnl(s)|>ɛ},
which together with the definitions of ηnk, lim̲ and lim¯ yields, for i>l,
(5.13)P{sup⁡s≤m|ηi(s)-ηl(s)|>ɛ}≤lim⁡¯n→∞,n∈JP{sup⁡s≤m|Yn(s)|>l}.
Hence, and from the evident inequality
(5.14)E(1∧γ)≤ɛ+P{γ>ε},
where γ is an arbitrary nonnegative random variable, we get for i>l,
(5.15)E(1∧sup⁡s≤m|ηi(s)-ηl(s)|)≤ɛ+lim⁡¯n→∞,n∈JP{sup⁡s≤m|Yn(s)|≥l}.
By the Lenglart-Rebolledo inequality,(5.16)P{sup⁡s≤m|Yn(s)|≥l}≤al2+P{tr⁡Kn(m)≥a},
for any a>0. Relation (5.4) implies, by Alexandrov's theorem, that
(5.17)lim⁡¯n→∞,n∈JP{tr⁡Kn(m)≥a}≤P{tr⁡K(m)≥a},
which together with (5.10)–(5.16) yields
(5.18)sup⁡i>lEϱ(ηi,ηl)≤ɛ+al2+∑m=1∞2-mP{tr⁡K(m)≥a}.
Hence, letting l→∞, then a→∞ finally ɛ→0, we obtain (5.9).
(5∘) Obviously, ϱ metrizes the 𝒰-convergence and the metric space (C,ϱ) is complete. Relation (5.9) means that the sequence (ηl) of C-valued random elements is fundamental in probability. Then, by the Riesz theorem, each of its subsequences contains a subsequence converging w.p.1. The limits of every two convergent subsequences coincide w.p.1 because of (5.9). So, there exists a C-valued random element (= continuous random process) η such that
(5.19)lim⁡l→∞Eϱ(ηl,η)=0.
And this is a fortified form of the relation
(5.20)ηl⟶Cη.
In particular, the sequence (ηl) is r.c. in C (which can be proved directly, but such proof does not guarantee that partial limits are given on the same probability space that the prelimit processes are).
(6∘) Relation (5.4) together with the conclusions of items (3∘) and (5∘) shows that all the conditions of Corollary 2.14 (with the range of n restricted to J) are fulfilled (and even overfulfilled: relation (5.20) proved above without recourse to Corollary 2.14 contains both an assumption and a conclusion of the latter). So, Corollary 2.14 asserts, in addition to (5.20), that
(5.21)ηn⟶Cηasn⟶∞,n∈J.
This pair of relations can be rewritten, in view of (5.3) and (5.5), in the form
(5.22)(Yl,Kl)⟶C(Y,K),(5.23)(Yn,Kn)⟶C(Y,K)asn⟶∞,n∈J,
where (Y,K) is a synonym of η. We wish to stress again that, firstly, all the processes in (5.22) are given on a common probability space and, secondly, they depend on the choice of J.
(7∘) Let us show that Y is a local martingale.
Denoteσm=inf{t:trK(t)≥m}, and Mm(t)=Y(t∧σm), Mml(t)=Yl(t∧σm). Equalities (5.19), (5.10), and (5.5) yield
(5.24)lim⁡l→∞Eϱ(Mml,Mm)=0,
whence
(5.25)Mml⟶dMmasl⟶∞.
On the strength of (5.7),
(5.26)〈Mml〉(t)=Kl(t∧σm).
By the construction of the processes Ynl and Knl for any s∈ℝ+ and n∈ℕ, the sequence (trKnl(s),l∈ℕ) increases. Then, due to (5.4) so does (trKl(s),l∈ℕ). Hence, we have with account of (5.19), (5.10), and (5.5)
(5.27)tr⁡Kl(s)≤tr⁡K(s),
for all s and l. Comparing this with (5.26), we see that
(5.28)Etr⁡〈Mml〉(t)≤Etr⁡K(t∧σm).
But trK is a continuous increasing process, so trK(σm)=m,trK(t∧σm)≤m. Now, it follows from (5.25) and (5.28) by Corollary 3.3 that Mm is a uniformly integrable martingale. Thus, the sequence (σm) localizes Y.
(8∘) Relation Yl→CY (a part of (5.22)) where the prelimit processes are, according to item (2∘), continuous martingales implies by Corollary VI.6.7 [2] that
(5.29)(Yl,[Yl])⟶C(Y,[Y]).
Comparing this with (5.22), we get with account of (3.4) (Y,K)=d(Y,〈Y〉), hereupon Corollary 3.5 asserts that K=〈Y〉.Corollary 5.2.
Let(Yn) be a sequence of local square integrable martingales satisfying conditions (RC), (3.21), and, for all t>0, (5.1). Then, Y is a continuous local martingale and relation (3.25) holds.Proof.
LetJ0 be an arbitrary infinite set of natural numbers. Then, Theorem 5.1 whose condition (3.10) is covered by (3.21) asserts existence of an infinite set J⊂J0 and a continuous local martingale YJ such that (5.2) holds. By assumption, the distribution of YJ and, consequently, of (YJ,〈YJ〉) does not depend on J, which allows to delete the superscript in (5.2). Hence, taking to account arbitrariness of J0, we conclude that (5.2) holds for J=ℕ.Corollary 5.3.
Let a sequence(Yn) of locally square integrable martingales satisfy conditions (RC) and, for all t>0, (5.1). Then, relation (3.19) holds.Proof.
It was shown in items(1∘) and (2∘) of the proof of Theorem 5.1 that for each l, the sequence (Ynl,n∈ℕ) satisfies all the conditions of Lemma 3.16 which, therefore, asserts that [Ynl]-〈Ynl〉→CO as n→∞. Hence, by the same argument as in item (3∘), relation (3.19) follows.Theorem 5.4.
Let for eachn∈ℕXn,Xn1,Xn2… be locally square integrable martingales with respect to a common filtration. Suppose that for all m∈ℕ, t>0 and ɛ>0 conditions (4.44) and (4.45) are fulfilled, and
(5.30)lim⁡L→∞sup⁡llim⁡¯n→∞P{|Xnl(0)|>L}=0,(5.31)lim⁡L→∞sup⁡llim⁡¯n→∞P{tr⁡〈Xnl〉(t)>L}=0,(5.32)lim⁡r→0sup⁡llim⁡¯n→∞P{sup⁡(t1,t2)∈Π(t,r)(tr⁡〈Xnl〉(t2)-tr⁡〈Xnl〉(t1))>ɛ}=0,(5.33)lim⁡l→∞lim⁡¯n→∞P{|Xnl(0)-Xn(0)|>ɛ}=0.
Then, for any infinite set J0⊂ℕ, there exist an infinite set J⊂J0 and a continuous local martingale X such that
(5.34)(Xn,〈Xn〉)⟶C(X,〈X〉)asn⟶∞,n∈J.Note that relation (5.34) is, up to notation, a duplicate of (5.2). So, the superscript J on the right-hand side is tacitly implied (but suppressed because the conditions of the theorem contain another superscript).Proof.
Conditions (5.31) and (5.32) imply that for each m, the sequence (〈Xnm〉,n∈ℕ) is r.c. in C. Then, it follows from (4.44) and (5.30) by Lemma 3.9 that the sequence (Xnm,n∈ℕ) is r.c. in C. So, there exist an infinite set Jm⊂Jm-1 and a random process Xm such that
(5.35)Xnm⟶CXmasn⟶∞,n∈Jm.
Consequently, if we denote by J the set whose mth member is that of Jm, then for each m,
(5.36)Xnm⟶CXmasn⟶∞,n∈J.
And this together with (4.44) and relative compactness of (〈Xnm〉,n∈ℕ) implies by Corollary 5.2 that Xm is a continuous local martingale and
(5.37)ηnm≡(Xnm,〈Xnm〉)⟶Cηm≡(Xm,〈Xm〉)asn⟶∞,n∈J.
Then, it follows from (5.30)–(5.32) that
(5.38)lim⁡L→∞sup⁡lP{|Xl(0)|>L}=0,lim⁡L→∞sup⁡lP{tr⁡〈Xl〉(t)>L}=0,lim⁡r→0sup⁡lP{sup⁡(t1,t2)∈Π(t,r)(tr⁡〈Xl〉(t2)-tr⁡〈Xl〉(t1))>ɛ}=0,
and therefore, the sequences (〈Xl〉), (Xl) and (ηl) are r.c. in C.
Conditions  (5.33) and (4.45) imply by the Lenglart-Rebolledo inequality that for all positive t and ɛ(5.39)lim⁡l→∞lim⁡¯n→∞P{sup⁡s≤t|Xnl(s)-Xn(s)|>ɛ}=0.
Conditions(5.31) and (4.45) imply by Lemma 3.8 that
(5.40)lim⁡l→∞lim⁡¯n→∞P{sup⁡s≤t‖〈Xnl〉(s)-〈Xn〉(s)‖>ɛ}=0,
which together with the previous relation yields (2.26). Then, Corollary 2.14 asserts existence of a random process η≡(X,H) such that
(5.41)(Xl,〈Xl〉)⟶C(X,H),(5.42)(Xn,〈Xn〉)⟶C(X,H)asn⟶∞,n∈J.
The ensuing relation Xl→CX, continuity (due to (5.37)) of all Xl and relative compactness of (〈Xl〉) imply by Corollary 5.2 that X is a continuous local martingale and (Xl,〈Xl〉)→C(X,〈X〉). Comparing this with (5.41), we get (X,H)=d(X,〈X〉), which converts (5.42) to (5.34).Repeating the deduction of Corollary5.2 from Theorem 5.1, we get from Theorem 5.4 the following conclusion.Corollary 5.5.
Let for eachn∈ℕXn,Xn1,Xn2… be locally square integrable martingales with respect to a common filtration. Suppose that, they have the same initial value; conditions (4.44), (4.45), (5.31), and (5.32) are fulfilled for all m and t; there exists a random process X such that Xn→CX. Then, X is a continuous local martingale and (Xn,〈Xn〉)→C(X,〈X〉).Theorem 5.6.
Let for eachn∈ℕXn,Xn1,Xn2… be locally square integrable martingales with respect to a flow 𝔽n. Suppose that, conditions (4.44)–(4.47) and (5.30)–(5.33) are fulfilled for all m∈ℕ,t>s≥0,>0,t2>t1≥0,z∈ℝd*, and bounded uniformly continuous functions f:ℝ+→ℝ; there exist an ℝd-valued random variable X∘ and an 𝔖+-valued random process H such that
(5.43)(Xn(0),〈Xn〉)⟶C(X∘,H).
Then, (1) for any p,l∈ℕ,tp>⋯>t0≥0,sl>⋯>s1>0,z1,…zp∈ℝd* and bounded continuous function F:ℝd×𝔖+l→ℝ(5.44)E(exp⁡{i∑j=1pzj(Xn(tj)-Xn(tj-1))}F(Xn(0),〈Xn〉(s1),…,〈Xn〉(sl)))⟶E(exp⁡{-12∑j=1pzj(H(tj)-H(tj-1))zj⊤}F(X∘,H(s1),…,H(sl))), (2) there exists a continuous local martingale X with initial value X∘ and quadratic characteristic H such that (Xn,〈Xn〉)→C(X,H) and
(5.45)E(exp⁡{i∑j=1pzj(X(tj)-X(tj-1))}F(X(0),〈X〉(s1),…,〈X〉(sl)))=E(exp⁡{-12∑j=1pzj(H(tj)-H(tj-1))zj⊤}F(X∘,H(s1),…,H(sl))),
for any p,l∈ℕ,tp>⋯>t0≥0,sl>⋯>s1>0,z1,…zp∈ℝd* and bounded continuous function F:ℝd×𝔖+l→ℝ.Proof.
Since the assumptions of this theorem contain those of Theorem5.4, the conclusion of the latter is valid. It implies, in particular, that the sequence (〈Xn〉) is r.c. in C. So, firstly, the assumptions of Theorem 4.5 are also fulfilled (and therefore the conclusions are valid), and, secondly, the relation
(5.46)lim⁡t→0lim⁡¯n→∞P{|Xn(t)-Xn(0)|>ɛ}=0
holds.
Ift0>0, then Theorem 4.5 asserts relation (4.49) which together with (5.43) yields, by the dominated convergence theorem, (5.44). Relation (5.46) and continuity (due to (5.43)) of H enable us to let t0→0 in (5.44), thus waiving the interim assumption t0>0.
Combining (4.49) with the conclusion of Theorem 5.4 and with the dominated convergence theorem, we see that for any infinite set J0⊂ℕ, there exist an infinite set J⊂J0 and a continuous local martingale X such that for all p,l∈ℕ,tp>⋯>t0>0,sl>⋯>s1>0,z1,…zp∈ℝd* and bounded continuous function F:ℝd×𝔖+l→ℝ(5.47)E(exp⁡{i∑j=1pzj(Xn(tj)-Xn(tj-1))}F(Xn(0),〈Xn〉(s1),…,〈Xn〉(sl)))⟶E(exp⁡{-12∑j=1pzj(〈X〉(tj)-〈X〉(tj-1))zj⊤}F(X(0),〈X〉(s1),…,〈X〉(sl))),
as n→∞,n∈J. The comparison of (5.43) and (5.34) shows that the right-hand side of (5.47) equals
(5.48)E(exp⁡{-12∑j=1pzj(H(tj)-H(tj-1))zj⊤}F(X∘,H(s1),…,H(sl))),
and, therefore, does not depend on the choice of J0 and J. So, (5.47) holds as n ranges over ℕ, too. This together with (5.44) proves the second statement under the extra assumption t0>0 which can be waived exactly as above.Corollary 5.7.
Let the conditions of Theorem5.6 be fulfilled. Then, X has conditionally with respect to 𝒢≡σ(X∘,〈X〉(·)) independent increments.



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*Source: 580292-2011-11-22.xml*