Burkholder-Davis-Gundy inequalities in UMD Banach spaces

In this paper we prove Burkholder-Davis-Gundy inequalities for a general martingale $M$ with values in a UMD Banach space $X$. Assuming that $M_0=0$, we show that the following two-sided inequality holds for all $1\leq p<\infty$: \begin{align}\label{eq:main}\tag{{$\star$}} \mathbb E \sup_{0\leq s\leq t} \|M_s\|^p \eqsim_{p, X} \mathbb E \gamma([\![M]\!]_t)^p ,\;\;\; t\geq 0. \end{align} Here $ \gamma([\![M]\!]_t) $ is the $L^2$-norm of the unique Gaussian measure on $X$ having $[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle, \langle M,y^*\rangle]_t$ as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of \eqref{eq:main} was proved for UMD Banach functions spaces $X$. We show that for continuous martingales, \eqref{eq:main} holds for all $0<p<\infty$, and that for purely discontinuous martingales the right-hand side of \eqref{eq:main} can be expressed more explicitly in terms of the jumps of $M$. For martingales with independent increments, \eqref{eq:main} is shown to hold more generally in reflexive Banach spaces $X$ with finite cotype. In the converse direction, we show that the validity of \eqref{eq:main} for arbitrary martingales implies the UMD property for $X$. As an application we prove various It\^o isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide It\^o isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures.


Introduction
In the celebrated paper [10], Burkholder, Davis, and Gundy proved that if M = (M t ) t≥0 is a real-valued martingale satisfying M 0 = 0, then for all 1 ≤ p < ∞ and t ≥ 0 one has the two-sided inequality where where the limit in probability is taken over partitions π = {0 = t 0 < . . . < t N = t} whose mesh approaches 0. Later, Burkholder [7,8] and Kallenberg and Sztencel [32] extended (1.1) to Hilbert space-valued martingales (see also [41]). They showed that if M is a martingale with values in a Hilbert space H satisfying M 0 = 0, then for all 1 ≤ p < ∞ and t ≥ 0 one has where the quadratic variation [M ] is defined as in (1.2) with absolute values replaced by norms in H. A further result along these lines was obtained recently by Veraar and the author [67], who showed that if M is an L p -bounded martingale, 1 < p < ∞, with M 0 = 0, that takes values in a UMD Banach function space X over a measure space (S, Σ, µ) (see Section 2 and 8 for the definition), then for all t ≥ 0: where the quadratic variation [M (σ)] t is considered pointwise in σ ∈ S. Although this inequality seems to be particularly useful from a practical point of view, it does not give any hint how to work with a general Banach space since not every (UMD) Banach space has a Banach function space structure (e.g. noncommutative L q -spaces).
Therefore the following natural question is rising up. Given a Banach space X. Is there an analogue of (1.3) for a general X-valued local martingale M and how then should the right-hand side of (1.3) look like? In the current article we present the following complete solution to this problem for local martingales M with values in a UMD Banach space X. Here γ(V ), where V : X * × X * → R is a given nonnegative symmetric bilinear form, is the L 2 -norm of an X-valued Gaussian random variable ξ with We call γ(V ) the Gaussian characteristic of V (see Section 3).
Let us explain briefly the main steps of the proof of Theorem 1.1. This discussion will also clarify the meaning of the term on the right-hand side, which is equivalent to the right-hand side of (1.3) if X is a Hilbert space, and of (1.4) (up to a multiplicative constant) if X is a UMD Banach function space.
In Section 2 we start by proving the discrete-time version of Theorem 1.1, which takes the following simple form where (d n ) N n=1 is an X-valued martingale difference sequence and (γ n ) N n=1 is a sequence of independent standard Gaussian random variables defined on a probability space (Ω γ , P γ ). (1.6) follows from a decoupling inequality due to Garling [19] and a martingale transform inequality due to Burkholder [6] (each of which holds if and only if X has the UMD property) together with the equivalence of Rademacher and Gaussian random sums with values in spaces with finite cotype due to Maurey and Pisier (see [42]). Theorem 1.1 is derived from (1.6) by finite-dimensional approximation and discretization. This is a rather intricate procedure and depends on some elementary, but nevertheless important properties of a Gaussian characteristic γ(·). In particular in Section 3 we show that for a finite dimensional Banach space X there exists a proper continuous extension of the Gaussian characteristic to all (not necessarily nonnegative) symmetric bilinear forms V : X * × X * → R, with the bound V (x * , x * ).
Next, in Section 5, under the assumptions of Theorem 1.1 we show that M has a well-defined covariation bilinear form, i.e. for each t ≥ 0 and for almost all ω ∈ Ω there exists a symmetric bilinear form [[M ]] t (ω) : X * × X * → R such that for all x * , y * ∈ X * one has Existence of such a covariance bilinear form in the nonhilbertian setting used to be an open problem since 1970's (see e.g. Meyer [45, p. 448] and Métivier [43, p. 156]; see also [1,25,61,66]). In Section 5 we show that such a covariation exists in the UMD case. Moreover, in Proposition 5.5 we show that the process [[M ]] has an increasing adapted càdlàg version.
Next we prove that the bilinear form [[M ]] t (ω) has a finite Gaussian characteristic γ([[M ]] t ) for almost all ω ∈ Ω. After these preparations we prove Theorem 1.1. We also show that the UMD property is necessary for the conclusion of the theorem to hold true (see Subsection 7.3).
In Section 6 we develop three ramifications of our main result: • if M is continuous, the conclusion of Theorem 1.1 holds for all 0 < p < ∞.
• if M is purely discontinuous, the theorem can be reformulated in terms of the jumps of M . • if M has independent increments, the UMD assumption on X can be weakened to reflexivity and finite cotype.
The first two cases are particularly important in view of the fact that any UMD space-valued local martingale has a unique Meyer-Yoeurp decomposition into a sum of a continuous local martingale and a purely discontinuous local martingale (see [69,70]).
A reasonable part of the paper, namely Section 7, is devoted to applications of Theorem 1.1 and results related to Theorem 1.1. Let us outline some of them. In Subsection 7.1 we develop a theory of vector-valued stochastic integration. Our starting point is a result of van Neerven, Veraar, and Weis [48]. They proved that if W H is a cylindrical Brownian motion in a Hilbert space H and Φ : R + × Ω → L(H, X) is an elementary predictable process, then for all 0 < p < ∞ and t ≥ 0 one has the two-sided inequality Here Φ γ(L 2 ([0,t];H),X) is the γ-radonifying norm of Φ as an operator from a Hilbert space L 2 ([0, t]; H) into X (see (2.1) for the definition); this norm coincides with the Hilbert-Schmidt norm given X is a Hilbert space. This result was extended to continuous local martingales in [64,66]. .
Here q M is the quadratic variation derivative of M and γ(L 2 (0, t; [ M ]), X) is a suitable space of γ-radonifying operator associated with M (see Subsection 7.1 for details). This represents a significant improvement of (1.7). In Subsection 7.2 we apply our results to vector-valued stochastic integrals with respect to a compensated Poisson random measure N . We show that if N is a Poisson random measure on R + × J for some measurable space (J, J ), ν is its compensator, N := N −ν is the corresponding compensated Poisson random measure, then for any UMD Banach space X, for any elementary predictable F : J ×R + ×Ω → X, and for any 1 ≤ p < ∞ one has that (1.9) E sup We also show that (1.9) holds if one considers a general quasi-left continuous random measure µ instead of N . In Subsection 7.4 we prove the following martingale domination inequality: for all local martingales M and N with values in a UMD Banach space X such that This extends weak differential subordination L p -estimates obtained in [69,71] (which used to be known to hold only for 1 < p < ∞, see [52,69,71]). Finally, in Section 8, we prove that for any UMD Banach function space X over a measure space (S, Σ, µ), that any X-valued local martingale M has a pointwise local martingale version M (σ), σ ∈ S, such that if 1 ≤ p < ∞, then for µ-almost t p for all t ≥ 0, which extends (1.4) to the case p = 1 and general local martingales. In conclusion we wish to notice that it remains open whether one can find a predictable right-hand side in (1.5): so far such a predictable right-hand side was explored only in the real-valued case and in the case X = L q (S), 1 < q < ∞, see Burkholder-Novikov-Rosenthal inequalities in the forthcoming paper [16].

Burkholder-Davis-Gundy inequalities: the discrete time case
Let us show discrete Burkholder-Davis-Gundy inequalities. First we will provide the reader with the definitions of UMD Banach spaces and γ-radonifying operators. A Banach space X is called a UMD space if for some (equivalently, for all) p ∈ (1, ∞) there exists a constant β > 0 such that for every n ≥ 1, every martingale difference sequence (d j ) n j=1 in L p (Ω; X), and every The least admissible constant β is denoted by β p,X and is called the UMD constant. It is well known (see [27,Chapter 4]) that β p,X ≥ p * − 1 and that β p,H = p * − 1 for a Hilbert space H. We refer the reader to [9,22,27,28,38,53,57] for details. Let H be a separable Hilbert space, X be a Banach space, T ∈ L(H, X). Then T is called γ-radonifying if where (h n ) n≥1 is an orthonormal basis of H, and (γ n ) n≥1 is a sequence of standard Gaussian random variables (otherwise we set T γ(H,X) := ∞). Note that T γ(H,X) does not depend on the choice of (h n ) n≥1 (see [28,Section 9.2] and [47] for details). Often we will call T γ(H,X) the γ-norm of T . γ-norms are exceptionally important in analysis as they are easily computable and enjoy a number of useful properties such as the ideal property, γ-multiplier theorems, Fubini-type theorems, etc., see [28,47]. Now we are able state and prove discrete UMD-valued Burkholder-Davis-Gundy inequalities.
Theorem 2.1. Let X be a UMD Banach space, (d n ) n≥1 be an X-valued martingale difference sequence. Then for any 1 ≤ p < ∞ For the proof we will need Rademacher random variables.
Proof of Theorem 2.1. Without loss of generality we may assume that there exists N ≥ 1 such that d n = 0 for all n > N . Let (r n ) n≥1 be a sequence of independent Rademacher random variables, (γ n ) n≥1 be a sequence of independent standard Gaussian random variables. Then [6, (8.22)], (ii) holds by [ .3), then the final constant will depend only on p and β 2,X (or β q,X for any fixed 1 < q < ∞).
Remark 2.4. If we collect all the constants in (2.3) then one can see that those constants behave well as p → 1, i.e. for any 1 < r < ∞ there exist positive C r,X and c r,X such that for any 1 ≤ p ≤ r Remark 2.5. Fix 1 < p < ∞ and a UMD Banach space X. By Doob's maximal inequality (4.1) and Theorem 2.1 we have that Let us find the constants in the equivalence Since X is UMD, it has a finite cotype q (see [28, where c p,X depends on p, the cotype of X, and the Gaussian cotype constant of X (see [ where (i) and (iii) follow from good-λ inequalities [6, (8.22) (2.5) will depend only on φ and β 2,X (or β q,X for any fixed 1 < q < ∞).
In the following theorem we show that X having the UMD property is necessary for Theorem 2.1 to hold. Theorem 2.7. Let X be a Banach space and 1 ≤ p < ∞ be such that (2.2) holds for any martingale difference sequence (d n ) n≥1 . Then X is UMD.
Proof. Note that for any set (x n ) N n=1 of elements of X and for any Now we turn to the continuous-time case. It turns out that in this case the righthand side of (2.2) transforms to a so-called Gaussian characteristic of a certain bilinear form generated by a quadratic variation of the corresponding martingale. Therefore before proving our main result (Theorem 5.1) we will need to outline some basic properties of a Gaussian characteristic (see Section 3). We will also need some preliminaries concerning continuous-time Banach space-valued martingales (see Section 4).

Gaussian characteristics
The current section is devoted to the definition and some basic properties of one of the main objects of the paper -a Gaussian characteristic of a bilinear form. Many of the statements here might seem to be obvious for the reader. Nevertheless we need to show them before reaching our main Theorem 5.1.

Basic definitions.
Let us first recall some basic facts on Gaussian measures. Let X be a Banach space. An X-valued random variable ξ is called Gaussian if ξ, x * has a Gaussian distribution for all x * ∈ X * . Gaussian random variables enjoy a number of useful properties (see [2,35]). We will need the following Gaussian covariance domination inequality (see [ Lemma 3.1. Let X be a Banach space, ξ, η be centered X-valued Gaussian random variables. Assume that E η, x * 2 ≤ E ξ, x * 2 for all x * ∈ X * . Then Eφ(η) ≤ Eφ(ξ) for any convex symmetric continuous function φ : X → R + .
Let X be a Banach space. We denote the linear space of all continuous R-valued bilinear forms on X × X by X * ⊗ X * . Note that this linear space can be endowed with the following natural norm: where the latter expression is finite due to bilinearity and continuity of V . A bilinear Let X be a Banach space, ξ be a centered X-valued Gaussian random variable. Then ξ has a covariance bilinear form V : Notice that a covariance bilinear form is always continuous, symmetric, and nonnegative. It is worth noticing that one usually considers a covariance operator Q : X * → X * * defined by Qx * , y * = E ξ, x * ξ, y * , x * , y * ∈ X.
But since there exists a simple one-to-one correspondence between bilinear forms and L(X * , X * * ), we will work with covariance bilinear forms instead. We refer the reader to [2,12,24,62] for details.
Let V : X * × X * → R be a symmetric continuous nonnegative bilinear form. Then V is said to have a finite Gaussian characteristic γ(V ) if there exists a centered X-valued Gaussian random variable ξ such that V is the covariance bilinear form of ξ. Then we set γ(V ) := (E ξ 2 ) 1 2 (this value is finite due to the Fernique theorem, see [2,Theorem 2.8.5]). Otherwise we set γ(V ) = ∞. Note that then for all x * , y * ∈ X * one has the following control of continuity of V : Remark 3.2. Note that for any V with γ(V ) < ∞ the distribution of the corresponding centered X-valued Gaussian random variable ξ is uniquely determined (see [2,Chapter 2]).

Remark 3.3.
Note that if X is finite dimensional, then γ(V ) < ∞ for any nonnegative symmetric bilinear form V . Indeed, in this case X is isomorphic to a finite dimensional Hilbert space H, so there exists an eigenbasis (h n ) d n=1 making V diagonal, and then the corresponding Gaussian random variable will be equal to ξ := d n=1 V (h n , h n )γ n h n , where (γ n ) d n=1 are independent standard Gaussian. 3.2. Basic properties of γ(·). Later we will need the following technical lemmas.
Lemma 3.5. Let X be a separable Banach space, H be a separable Hilbert space, T ∈ L(H, X), V : X * × X * → R be a symmetric continuous nonnegative bilinear form such that V (x * , y * ) = T * x * , T * y * for all x * , y * ∈ X * . Then γ(V ) = T γ(H,X) . Remark 3.6. Fix a Hilbert space H and a Banach space X. Note that even though by the lemma above there exists a natural embedding of γ-radonifying operators from L(H, X) to the space of symmetric nonnegative bilinear forms on X * × X * , this embedding is neither injective nor linear. This also explains why we need to use bilinear forms with finite Gaussian characteristics instead of γradonifying operators: in the proof of our main result -Theorem 5.1 -we will need various statements (like triangular inequalities and convergence theorems) for bilinear forms, not operators. Now we will prove some statements about approximation of nonnegative symmetric bilinear forms by finite dimensional ones in γ(·).
Proof. First of all notice that γ(V 0 ) is well-defined since Y is finite dimensional, hence reflexive, and thus has a predual space coinciding with its dual. Without loss of generality assume that V γ < ∞. Let ξ V be a centered X-valued Gaussian random variable with V as the covariance bilinear form. Define ξ V0 := P * ξ V (note that Y * ֒→ X due to the Hahn-Banach theorem). Then for all , so V 0 is the covariance bilinear form of ξ V0 and since P * = P = 1 Proposition 3.8. Let X be a separable reflexive Banach space, V : X * ×X * → R be a symmetric continuous nonnegative bilinear form.
Proof. First of all notice that V m 's are well-defined since each of the Y m is finite dimensional, hence reflexive, and thus has a predual space coinciding with its dual (which we will call X m and which can even be embedded into X due to the Hahn-Banach theorem). Let P m : Y m ֒→ X * be the inclusion operator (thus is particular P m ≤ 1). Let a Hilbert space H and an operator T ∈ L(H, X) be as constructed in Lemma 3.4. Let (h n ) n≥1 be an orthonormal basis of H, and (γ n ) n≥1 be a sequence of standard Gaussian random variables. For each N ≥ 1 define a centered Gaussian random variable ξ N := N n=1 γ n T h n . Then for each m ≥ 1 the centered Gaussian random variable ∞ n=1 γ n P * m T h n is well-defined (since P * m T has a finite rank, and every finite rank operator has a finite γ-norm, see [28, Section 9.2]), and for any x * ∈ Y m we have that so V m is the covariance bilinear form of ∞ n=1 γ n P * m T h n , and The latter expression converges to γ(V ) by Lemma 3.5 and due to the fact that P * m x → x monotonically for each x ∈ X as m → ∞. The next lemma provides the Gaussian characteristic with the triangular inequality. Lemma 3.9. Let X be a reflexive Banach space, V, W : X * × X * be symmetric continuous nonnegative bilinear forms. Then Let ξ V and ξ W be X-valued centered Gaussian random variables corresponding to V and W respectively. Without loss of generality we can set ξ V and ξ W independent. Let ξ V +W = ξ V + ξ W . Then ξ V +W is an X-valued centered Gaussian random variable (see [2]) and for any x * ∈ X * due to the independence of ξ V and ξ W So ξ V +W has V + W as the covariation bilinear form, and therefore Now we discuss such important properties of γ(·) as monotonicity and monotone continuity.
Proof. Without loss of generality assume that γ(V 1 ) < ∞. Note that by Lemma 3.10 the sequence (γ(V n )) n≥1 is monotone and bounded by γ( By Lemma 3.4 we may assume that there exists a separable Hilbert space H and a sequence of operators (T n ) n≥1 from H to X such that V n (x * , x * ) = T * n x * 2 for all x * ∈ X * (note that we are working with one Hilbert space since all the separable Hilbert spaces are isometrically isomorphic). Let T ∈ L(H, X) be the zero operator. Then T * n x * → T * x * = 0 as n → ∞ for all x * ∈ X * , and hence by [28, Theorem 9.4.2], Lemma 3.5, and the fact that The following lemma follows for Lemma 3.9 and 3.11.
3.3. γ(·) and γ(·) 2 are not norms. Notice that γ(·) is not a norm. Indeed, it is easy to see that γ(αV ) = √ αγ(V ) for any α ≥ 0 and any nonnegative symmetric bilinear form V : if we fix any X-valued Gaussian random variable ξ having V as its covariance bilinear form, then √ αξ has αγ(V ) as its covariance bilinear form. It is a natural question whether γ(·) 2 satisfies the triangle inequality and hence has the norm properties. It is easy to check the triangle inequality if X is Hilbert: indeed, for any V and W where ξ V , ξ W , and ξ V +W are as in the latter proof. It turns out that if such a triangular inequality holds for some Banach space X, then this Banach space must have a Gaussian type 2 (see [28,Subsection 7.1.d]). Indeed, let X be such that for all nonnegative symmetric bilinear forms V and W on X * × X * Fix (x i ) n i=1 ⊂ X and a sequence of independent standard Gaussian random variables

Then by (3.4) and the induction argument
where ( * ) follows from the fact that while ( * * ) follows analogously by exploiting the fact that ξ i x i is a centered Gaussian random variable with the covariance bilinear form V i . Therefore by [28,Definition 7.1.17], X has a Gaussian type 2 with the corresponding Gaussian type constant τ γ 2,X = 1. In the following proposition we show that this condition yields that X is Hilbert, and thus we conclude that γ(·) 2 defines a norm if and only if X is a Hilbert space.
Proposition 3.13. Let X be a Banach space such that its Gaussian type 2 constant equals 1. Then X is Hilbert.
Proof. Due to the parallelogram identity it is sufficient to show that every two dimensional space of X is Hilbert; consequently, without loss of generality we can assume that X is two dimensional. We need to show that the unit ball of X is an ellipse as any ellipse corresponds to an inner product (see e.g. [13]). Let B ∈ X ≃ R 2 be the unit ball of X. Then by [58,Theorem 1] there exists an ellipse E ∈ X containing B such that ∂B and ∂E intersect in at least two pairs of points. Let us denote these pairs by (x 1 , −x 1 ) and (x 2 , −x 2 ). Notice that both x 1 and x 2 are nonzero and are not collinear. Let |||·||| be the norm associated to E. Then Note that X endowed with |||·||| is a Hilbert space by [13], thus it has an inner product ·, · E . Let γ 1 and γ 2 be independent standard Gaussian random variables. Then we have that where ( * ) holds by (3.5), and ( * * ) holds since τ 1 2,X = 1 (see [28,Definition 7.1.17]). Therefore we have that every inequality in the estimate above is actually an equality, and hence E|||γ 1 s., and as x 1 and x 2 are not collinear and X is two dimensional, γ 1 x 1 + γ 2 x 2 has a nonzero distribution density on the whole X, so we have that |||x||| = x for a.e. x ∈ X (and by continuity for any x ∈ X), and the desired follows.
Assume that X has a Gaussian cotype 2 constant equals 1. Then the same proof will yield that X is Hilbert, but now one needs to find an ellipse E inside B such that ∂B and ∂E intersect in at least two pairs of points. In order to find such an ellipse it is sufficient to find an ellipse E ′ ∈ X * containing the unit ball B ′ ⊂ X * such that ∂B ′ and ∂E ′ intersect in at least two pairs of points, and then set B to be the unit ball of a space Y * , where Y is a Hilbert space having E ′ as its unit ball. Then (3.6) will hold true but with ≥ instead of ≤.

3.4.
Finite dimensional case. Even though a Gaussian characteristic is welldefined only for some nonnegative symmetric forms, it can be extended in a proper continuous way to all the symmetric forms given X is finite dimensional. Let X be a finite dimensional Banach space. Notice that in this case γ(V ) < ∞ for any nonnegative symmetric bilinear form V (see Remark 3.3). Let us define γ(V ) for a general symmetric V ∈ X * * ⊗ X * * = X ⊗ X in the following way: Notice that γ(V ) is well-defined and finite for any symmetric V . Indeed, by a well known linear algebra fact (see e.g. [60, Theorem 6.6 and 6.10]) any symmetric bilinear form V has an eigenbasis ( λ n a n b n .
Therefore it is sufficient to define 1 λn<0 (−λ n )a n b n and then γ(V ) ≤ γ(V + ) + γ(V − ) < ∞ due to the fact that V + and V − are nonnegative and by Remark 3.3. (In fact, one can check that γ(V ) = γ(V + ) + γ(V − ), but we will not need this later, so we leave this fact without a proof). Now we will develop some basic and elementary (but nonetheless important) properties of such a general γ(·). Lemma 3.15. Let V : X * × X * → R be a nonnegative symmetric bilinear form. Then γ(V ) defined by (3.7) coincides with γ(V ) defined in Subsection 3.1. In other words, these definitions agree given V is nonnegative.
Lemma 3.16. Let V, W : X * × X * → R be symmetric bilinear forms. Then Then since V = U + W by (3.7) and Lemma 3.9 and by sending ε → 0 we conclude the desired.
Proof. The first part follows directly from (3.7). For the second part we have that due to (3.7) it is enough to justify γ(αV ) = √ αγ(V ) only for nonnegative V , which was done in Subsection 3.3.
Proposition 3.18. The function γ(·) defined by (3.7) is continuous on the linear space of all symmetric bilinear forms endowed with · defined by (3.1). Moreover, Proof. Due to Lemma 3.16 and 3.17 it is sufficient to show that γ(·) is bounded on the unit ball with respect to the norm · in order to prove the first part of the proposition. Let us show this boundedness. Let U be a fixed symmetric nonnegative element of X ⊗ X such that U + V is nonnegative and such that U (x * , x * ) ≥ V (x * , x * ) for any symmetric V with V ≤ 1 (since X is finite dimensional, one can take U (x * ) := c|||x * ||| 2 for some Euclidean norm |||·||| on X * and some big enough constant c > 0). Fix a symmetric V : which does not depend on V . Let us show the second part. Due to the latter consideration there exists a constant C X depending only on X such that γ(V ) ≤ C X if V ≤ 1. Therefore by Lemma 3.17 we have that for a general symmetric V Later we will also need the following elementary lemma.
defines a norm on the space of all symmetric bilinear forms on X * × X * . In particular we have that V X |||V ||| for any symmetric bilinear form V : X * × X * → R.
We will demonstrate here the proof for the convenience of the reader.
Proof. First notice that |||·||| clearly satisfies the triangular inequality. Let us show that there exists a set ( . This means that for these i and j It remains to notice that the latter sum has the form (3.8) for a proper choice of (x * i ) n i=1 independent of V . In order to show the last part of the lemma we need to notice that the space of symmetric bilinear forms is finite dimensional if X is so, so all the norms on the linear space of symmetric bilinear forms are equivalent, and therefore V X |||V ||| for any symmetric bilinear form V : X * × X * → R.

Preliminaries
We continue with some preliminaries concerning continuous-time martingales.
4.1. Banach space-valued martingales. Let (Ω, F , P) be a probability space with a filtration F = (F t ) t≥0 which satisfies the usual conditions. Then F is rightcontinuous (see [29,30] for details).
Let X be a Banach space. An adapted process M : R + × Ω → X is called a martingale if M t ∈ L 1 (Ω; X) and E(M t |F s ) = M s for all 0 ≤ s ≤ t (we refer the reader to [27] for the details on vector-valued integration and vector-valued conditional expectation). It is well known that in the real-valued case any martingale is càdlàg (i.e. has a version which is right-continuous and that has limits from the left-hand side). The same holds for a general X-valued martingale M as well (see [63,71]), so one can define Since · : X → R + is a convex function, and M is a martingale, M is a submartingale by Jensen's inequality, and hence by Doob's inequality (see e.g. [33, Theorem 1.3.8(i)]) we have that for all 1 < p ≤ ∞

Quadratic variation.
Let H be a Hilbert space, M : R + × Ω → H be a local martingale. We define a quadratic variation of M in the following way: where the limit in probability is taken over partitions 0 = t 0 < . . . < t N = t. Note that [M ] exists and is nondecreasing a.s. The reader can find more on quadratic variations in [43,44,66] for the vector-valued setting, and in [30,44,54] for the real-valued setting.
As it was shown in [45, Proposition 1] (see also [56,Theorem 2.13] and [66,Example 3.19] for the continuous case) that for any H-valued martingale M there exists an adapted process q M : R + × Ω → L(H) which we will call a quadratic variation derivative, such that the trace of q M does not exceed 1 on R + × Ω, q M is self-adjoint nonnegative on R + × Ω, and for any h, g ∈ H a.s.

Burkholder-Davis-Gundy inequalities: the continuous-time case
The following theorem is the main theorem of the paper.
with the mesh not exceeding 1/N . For each ω ∈ Ω and N ≥ 1 define a bilinear form V N : X * × X * → R as follows: is a sequence of independent Gaussian standard random variables, and the latter equality holds due to the fact that for any fixed ω ∈ Ω the random variable Indeed, if this is the case, then by (5.3) and by Lemma 3.16 where the letter holds by the dominated convergence theorem as any martingale has a càdlàg version (see Subsection 4.1). Let us show this convergence. Note that by Proposition 3.18 and Lemma 3.19 a.s.
(Ω), which follows from the fact that for any x * i from Lemma 3.19, i = 1, . . . , n, we have that Step 2: infinite dimensional case. First assume that M is an L p -bounded martingale. Without loss of generality we can assume X to be separable. Since X is UMD, X is reflexive, so X * is separable as well. Let Y 1 ⊂ Y 2 ⊂ . . . ⊂ Y n ⊂ . . . be a family of finite dimensional subspaces of X * such that ∪ n Y n = X * . For each n ≥ 1 let P n : Y n → X * be the inclusion operator. Then P * n ≤ 1 and P * n M is a well-defined Y * n -valued L p -bounded martingale.
Let Ω 0 ⊂ Ω be a subset of measure 1 such that (5.5) holds for all m ≥ n ≥ 1. Fix ω ∈ Ω 0 . Then by (5.5) we can define a bilinear form (not necessarily continuous!) for all x * , y * ∈ Y n and n ≥ 1.
Let us show that V is continuous (and hence has a continuous extension to X * × X * ) and γ(V ) < ∞ a.s. on Ω 0 . Notice that by Lemma 3.7 the sequence (γ([[P * n M ]] t )) n≥1 is increasing a.s. on Ω 0 . Moreover, by the monotone convergence theorem and (5.4) (γ([[P * n M ]] t )) n≥1 has a limit a.s. on Ω 0 . Let Ω 1 ⊂ Ω 0 be a subset of full measure such that (γ([[P * n M ]] t )) n≥1 has a limit on Ω 1 . Then by (3.2) V is continuous on Ω 1 and hence has a continuous extension to X * × X * (which we will denote by V as well for simplicity). Then by Proposition 3.8 n M ]] t ) monotonically on Ω 1 and hence by monotone convergence theorem ans the fact that P * n x → x as n → ∞ monotonically for all

It remains to show that
s. for any x * ∈ X * . If x * ∈ Y , then the desired follows from the construction of V . Fix Remark 5.3. Because of Lemma 3.5 the reader may suggest that if X is a UMD Banach space, then for any X-valued local martingale M , for any t ≥ 0, and for a.a. ω ∈ Ω there exist a natural choice of a Hilbert space H(ω) and a natural choice of an operator T (ω) ∈ L(H(ω), X) such that for all x * , y * ∈ X * a.s. Remark 5.4. As in Remark 2.6, by a limiting argument shown in the proof of Theorem 5.1 one can prove that for any UMD Banach space X, for any martingale M : R + × Ω → X, and for any convex increasing function φ : R + → R + with φ(0) = 0 and with φ(2λ) ≤ cφ(λ) for some fixed c > 0 for any λ > 0 one has that To this end, one first needs to prove the finite-dimensional case by using the proof of [10, Theorem 5.1] and the fact that for any convex increasing ψ : R + → R + with ψ(0) = 0 and with ψ(2λ) ≤ cφ(λ) one has that ψ • φ satisfies the same properties (perhaps with a different constant c), and then apply the same extending argument.
Let X be a UMD Banach space, M : R + × Ω → X be a martingale. Then by Theorem 5.1 there exists a process [[M ]] : R + × Ω → X ⊗ X such that for any x * , y * ∈ X * and a.e. (t, ω) ∈ R + × Ω In our final proposition we show that this process is adapted and has a càdlàg version (i.e. a version which is right-continuous with left limits). Let F : R + × Ω → X ⊗ X be a bilinear form-valued process such that for any n ≥ 1, which existence can be shown analogously proof of Theorem 5.1. First note that F is adapted by the definition. Let us show that F is increasing càdlàg on Ω 0 . Fix ω ∈ Ω 0 . Then F t (x * , x * ) ≥ F s (x * , x * ) for any t ≥ s ≥ 0 and any x * ∈ Y := ∪ n Y n ⊂ X * , and thus we have the same for any x * ∈ X * by continuity of F t and F s and the fact that Y is dense in X * . Now let us show that F is right-continuous. By (5.7) and the fact that [[P n M ]] is càdlàg we have that Finally, F has left-hand limits. Indeed, fix t > 0 and let F t− be a bilinear form defined by Then  The process γ(F ) is finite a.s. by the fact that γ(F T ) < ∞ a.s., increasing a.s. by the fact that F is increasing a.s. and by Lemma 3.10, and adapted and càdlàg as F is adapted and càdlàg and by the fact that the map V → γ(V ) is continuous by(3.2).

Ramifications of Theorem 5.1
Let us outline some ramifications of Theorem 5.1.
6.1. Continuous and purely discontinuous martingales. In the following theorems we will consider separately the cases of continuous and purely discontinuous martingales. Recall that an X-valued martingale is called purely discontinuous if [ M, x * ] is a.s. a pure jump process for any x * ∈ X * (see [29,30,69,71] for details).
First we show that if M is continuous, then Theorem 5.1 holds for the whole range 0 < p < ∞. Theorem 6.1. Let X be a UMD Banach space, M : R + × Ω → X be a continuous local martingale. Then we have that for any 0 < p < ∞ For the proof we will need the following technical lemma, which extends Proposition 5.5 in the case of a continuous martingale.  Proof. The proof follows from the classical argument due do Lenglart (see [37]), but we will recall this argument for the convenience of the reader. We will show only one direction, the other direction follows analogously. Fix ε, δ > 0. For each n ≥ 1 define a stopping time τ n in the following way: τ n := inf{t ≥ 0 : M n t > ε}. Then by (5.1) and Chebyshev's inequality and the latter vanishes for any fixed δ > 0 as ε → 0 and n → ∞.
Remark 6.4. Note that Theorem 6.3 does not hold for general martingales even in the real-valued case, see [30,Exercise 26.5].
Proof. It is sufficient to notice that for any x * ∈ X * a.s.
and apply Theorem 5.1 and Lemma 3.5.
Remark 6.6. Note that martingales in Theorem 6.1 and 6.5 cover all the martingales if X is UMD. More specifically, if X has the UMD property, then any X-valued local martingale M has a unique decomposition M = M c + M d into a sum of a continuous local martingale M c and a purely discontinuous local martingale M d (such a decomposition is called the Meyer-Yoeurp decomposition, and it characterizes the UMD property, see [69,70]).

Martingales with independent increments.
Here we show that both Theorem 2.1 and 5.1 hold in much more general Banach spaces given the corresponding martingale has independent increments. Proposition 6.7. Let X be a Banach space, (d n ) n≥1 be an X-valued martingale difference sequence with independent increments. Then for any Moreover, if X has a finite cotype, then Proof. Let (r n ) n≥1 be a sequence of independent Rademacher random variables, (γ n ) n≥1 be a sequence of independent standard Gaussian random variables. Then where (i) follows from (4.1), (ii) follows from [36,Lemma 6.3], (iii) holds by [28,Proposition 6.3.2], and finally (iv) follows from [28,Proposition 6.3.1].
If X has a finite cotype, then one has p,X instead of p in (iii) (see [28,Corollary 7.2.10]), and the second part of the proposition follows.
Based on Proposition 6.7 and the proof of Theorem 5.1 one can show the following assertion. Notice that we presume the reflexivity of X since it was assumed in the whole Section 3. Proof. The proof coincides with the proof of Theorem 5.1, but one needs to use Proposition 6.7 instead of Theorem 2.1. 6.3. One-sided estimates. In practice one often needs only the upper bound of (2.2). It turns out that existence of such estimates for a fixed Banach space X is equivalent to the fact that X has the UMD − property. Definition 6.9. A Banach space X is called a UMD − space if for some (equivalently, for all) p ∈ (1, ∞) there exists a constant β > 0 such that for every n ≥ 1, every martingale difference sequence (d j ) n j=1 in L p (Ω; X), and every sequence (r j ) n j=1 of independent Rademachers we have The least admissible constant β is denoted by β − p,X and is called the UMD − constant. By the definition of the UMD property and a triangular inequality one can show that UMD implies UMD − . Moreover, UMD − is a strictly bigger family of Banach spaces and includes nonreflexive Banach spaces such as L 1 . The reader can find more information on UMD − spaces in [11,19,20,21,27,65].
The following theorem presents the desired equivalence.
Theorem 6.10. Let X be a Banach space, 1 ≤ p < ∞. Then X has the UMD − property if and only if one has that for any X-valued martingale difference sequence Proof. Assume that X has the UMD − property. Let (d n ) m n=1 be an X-valued martingale difference sequence. Then we have that for a sequence (r n ) n≥1 of independent Rademacher random variables and for a sequence (γ n ) n≥1 of independent standard Gaussian random variables where (i) follows from [6, (8.22)], (ii) holds by [ Remark 6.11. Unfortunately, it remains open whether one can prove the upper bound of (5.1) given X has the UMD − property. The problem is in the approximation argument employed in the proof of (5.1): we can not use an increasing sequence (Y n ) n≥1 of finite dimensional subspaces of X * since we can not guarantee that β − p,Y * n does not blow up as n → ∞ (recall that β p,Y * n ≤ β p,X by the duality argument, see [27,Proposition 4.2.17]). Nonetheless, such an upper bound can be shown for X = L 1 by an ad hoc argument (by using an increasing sequence of projections onto finite-dimensional L 1 -spaces).

Applications and miscellanea
Here we provide further applications of Theorem 5.1. 7.1. Itô isomorphism: general martingales. Let H be a Hilbert space, X be a Banach space. For each x ∈ X and h ∈ H we denote the linear operator g → g, h x, g ∈ H, by h⊗x. The process Φ : R + ×Ω → L(H, X) is called elementary predictable with respect to the filtration F = (F t ) t≥0 if it is of the form where 0 = t 0 < . . . < t K < ∞, for each k = 1, . . . , K the sets B 1k , . . . , B Mk are in F t k−1 and the vectors h 1 , . . . , h N are in H. Let M : R + × Ω → H be a local martingale. Then we define the stochastic integral Φ · M : R + × Ω → X of Φ with respect to M as follows: Notice that for any t ≥ 0 the stochastic integral Φ · M obtains a covariation bilinear form [[Φ · M ]] t which is a.s. continuous on X * × X * and which has the following form due to (4.3) and (7.2) so one can extend the definition of a stochastic integral to all predictable Φ : by extending the stochastic integral operator from a dense subspace of all elementary predictable processes satisfying (7.4). We refer the reader to [30,44,45] for details.  T HS(H1,H2) is the Hilbert-Schmidt norm of an operator T acting form a Hilbert space H 1 to a Hilbert space H 2 , and L 2 (R + ; A) for a given increasing A : R + → R is a Hilbert space of all functions f : Now we present the Itô isomorphism for vector-valued stochastic integrals with respect to general martingales, which extends [48,64,66]. is the γ-norm (see (2.1)).
Proof. Fix t ≥ 0. Then the theorem holds by Theorem 5.1, Lemma 3.5, and the fact that by (7.3) for any fixed x * ∈ X * a.s.
Theorem 7.3 allows us to provide the following general stochastic integration result. Recall that a predictable process Φ : R + × Ω → L(H, X) is called strongly predictable if there exists a sequence (Φ n ) n≥1 of elementary predictable L(H, X)valued processes such that Φ is a pointwise limit of (Φ n ) n≥1 . Corollary 7.4. Let H be a Hilbert space, X be a UMD Banach space, M : R + × Ω → H be a local martingale, Φ : R + × Ω → L(H, X) be strongly predictable such that E Φq 1/2 M γ(L 2 (R+,[ M];H),X) < ∞. Then there exists a martingale Φ · M which coincides with the stochastic integral given Φ is elementary predictable such that where the latter integral is defined as in Remark 7.1. Moreover, then we have that for any 1 ≤ p < ∞ .
For the proof we will need the following technical lemma.
. . ⊂ X * be finite dimensional subspaces such that ∪ n Y n = X * . Let P n : Y n ֒→ X * , n ≥ 1, and P n,m : Y n ֒→ Y m , m ≥ n ≥ 1, be the inclusion operators. For each n ≥ 1 let x n ∈ Y * n be such that P * n,m x m = x n for all m ≥ n ≥ 1. Assume also that sup n x n < ∞. Then there exists x ∈ X such that P * n x = x n for all n ≥ 1 and x = lim n→∞ x n monotonically.
Proof. Set C = sup n x n . First notice that (x n ) n≥1 defines a bounded linear functional on Y = ∪ n Y n . Indeed, fix y ∈ Y n for some fixed n ≥ 1 (then automatically y ∈ Y m for any m ≥ n). Define ℓ(y) = x n , y . Then this definition of ℓ agrees for different n's since for any m ≥ n we have that x m , y n = x m , P n,m y n = P * n,m x m , y n = x n , y n . Moreover, this linear functional is bounded since | x n , y n | ≤ x n y n ≤ C y n . So, it can be continuously extended to the whole space X * . Since X is reflexive, there exists x ∈ X such that ℓ(x * ) = x * , x for any x * ∈ X * . Then for any fixed n ≥ 1 and for any y ∈ Y n we have that x n , y = ℓ(y) = x, y = x, P n y = P * n x, y , so P * n x = x n . The latter follows from the fact that P * n x → x monotonically as n → ∞ for any x ∈ X.
Proof of Corollary 7.4. We will first consider the finite dimensional case and then deduce the infinite dimensional case.
Finite dimensional case. Since X is finite dimensional, it is isomorphic to a finite dimensional Euclidean space, and so the γ-norm is equivalent to the Hilbert-Schmidt norm (see e.g. [28, Proposition 9.1.9]). Then Φ is stochastically integrable with respect to M due to Remark 7.2, so (7.5) clearly holds and we have that for any x * ∈ X * a.s. Infinite dimensional case. Let now X be general. Since Φ is strongly predictable, it takes values in a separable subspace of X, so we may assume that X is separable. Since X is UMD, it is reflexive, so X * is separable as well, and there exists a sequence Y 1 ⊂ Y 2 ⊂ . . . ⊂ Y n ⊂ . . . ⊂ X * of finite dimensional subsets of X * such that ∪ n Y n = X * . For each m ≥ n ≥ 1 define inclusion operators P n : Y n ֒→ X * and P n,m : Y n ֒→ Y m . Notice that by the ideal property [28, Theorem 9.1.10] E P * n Φq 1/2 M γ(L 2 (R+,[ M];H),Y * n ) < ∞ for any n ≥ 1, so since Y * n is finite dimensional, the stochastic integral (P * n Φ) · M is well-defined by the case above and , where the equivalence is independent of n since Y n ⊂ X * for all n ≥ 1 and due to [27,Proposition 4.2.17] and Theorem 5.1. Denote the stochastic integral (P * n Φ) · M by Z n . Note that Z n is Y * n -valued, and since P * n,m P * m Φ = P * n Φ for all m ≥ n ≥ 1, P * m,n Z m t = Z n t a.s. for any t ≥ 0. Therefore by Lemma 7.5 there exists a process Z : R + × Ω → X such that P * n Z = Z n for all n ≥ 1. Let us show that Z is integrable. Fix t ≥ 1. Notice that by Lemma 7.5 the limit Z t = lim n→∞ P * n Z t = lim n→∞ Z n t is monotone, so by the monotone convergence theorem, (7.7), and the ideal property [28, Theorem 9.1.10] Now let us show that Z is a martingale. Since Z is integrable, due to [27,Section 2.6] it is sufficient to show that E( Z t , x * |F s ) = Z s , x * for all 0 ≤ s ≤ t for all x * from some dense subspace Y of X * . Set Y = ∪ n Y n and x * ∈ Y n for some n ≥ 1.
x * |F s ) = Z n s , x * = Z s , x * , so Z is a martingale. Finally, let us show (7.6). First notice that for any n ≥ 1 and x * ∈ Y n ⊂ X * a.s.
the same holds for a general x * ∈ X * by a density argument. Then (7.6) follows from Theorem 5.1 and Lemma 3.5. Remark 7.6. As the reader can judge, the basic assumptions on Φ in Corollary 7.4 can be weakened by a stopping time argument. Namely, one can assume that Φq ; H), X)) (i.e. there exists an increasing sequence (τ n ) n≥1 of stopping times such that τ n → ∞ a.s. as n → ∞ and Φq ; H), X)) for all n ≥ 1). Notice that such an assumption is a natural generalization of classical assumptions for stochastic integration in the real-valued case (see e.g. [30, p. 526]).
In the theory of stochastic integration one might be interested in one-sided estimates. In the following proposition we show that such type of estimates is possible if X satisfies the UMD − property (see Subsection 6.3). Then there exists an L p -bounded martingale Φ · M as a strong L p -limit of (Φ n · M ) n≥1 , and we have that for any 1 ≤ p < ∞ .
Proof. Inequality (7.8) for Φ = Φ n follows from Theorem 6.10, while the proposition together with (7.8) for a general Φ follows from a simple limiting argument.
7.2. Itô isomorphism: Poisson and general random measures. Let (J, J ) be a measurable space, N be a Poisson random measure on J × R + , N be the corresponding compensated Poisson random measure (see e.g. [15,23,30,34,59] for details). Then by Theorem 6.5 for any UMD Banach space X, for any 1 ≤ p < ∞, and for any elementary predictable F : J × R + × Ω → X we have that The same holds for a general quasi-left continuous random measure (see [17,26,29,31,40,50] for the definition and the details): if µ is a general quasi-left continuous random measure on J × R + , ν is its compensator, andμ := µ − ν, then for any 1 ≤ p < ∞ The disadvantage of right-hand sides of (7.9) and (7.10) is that both of them are not predictable and do not depend continuously on time a.s. on Ω (therefore they seem not to be useful from the SPDE's point of view since one may not produce a fixed point argument). For example, if X = L q for some 1 < q < ∞, then such predictable a.s. continuous in time right-hand sides do exist (see [15,17]). 7.3. Necessity of the UMD property. As it follows from Remark 5.2, Theorem 5.1 holds only in the UMD setting. The natural question is whether there exists an appropriate right-hand side of (5.1) in terms of ([ M, x * , M, y * ]) x * ,y * ∈X * for some non-UMD Banach space X and some 1 ≤ p < ∞. Here we show that this is impossible.
Assume that for some Banach space X and some 1 ≤ p < ∞ there exists a function G acting on families of stochastic processes parametrized by X * × X * (i.e. each family has the form V = (V x * ,y * ) x * ,y * ∈X * ) taking values in R such that for any X-valued local martingale M starting in zero we have that For every x * , y * ∈ X * define a stochastic process V x * ,y * : n=1 e n , x * · e n , y * , t ≥ 0 (recall that [t] is the integer part of t). Let V := (V x * ,y * ) x * ,y * ∈X * . Then by (7.11) Since N , (d n ) N n=1 , and (ε n ) N n=1 are general, (7.12) implies that X is a UMD Banach space (see the proof of Theorem 2.7). 7.4. Martingale domination. The next theorem shows that under some natural domination assumptions on martingales one gets L p -estimates. Note that the assumptions in Theorem 7.8 are a way more general than the weak differential subordination assumptions (recall that N is weakly differentially is nondecreasing a.s. for any x * ∈ X * , see [52,69,71]), so Theorem 7.8 significantly improves the L p -bounds obtained previously for weakly differentially subordinated martingales in [69,71] and extends the results to the case p = 1 as well.
Proof of Theorem 7.8. First notice that by a triangular inequality Consequently Remark 7.9. It is not known what the sharp constant is in (7.13). Nevertheless, sharp inequalities of such type have been discovered in the scalar case by Osękowski in [51]. It was shown there that if M and N are real-valued L p -bounded martingales such that a.s.
Martingale approximations. The current subsection is devoted to approximation of martingales. Namely, we will extend the following lemma by Weisz (see [68,Theorem 6]) to general UMD Banach space-valued martingales.
Lemma 7.10. Let X be a finite dimensional Banach space, M : R + × Ω → X be a martingale such that E sup t≥0 M t < ∞. Then there exists a sequence (M n ) n≥1 of X-valued uniformly bounded martingales such that E sup t≥0 M t − M n t → 0 as n → ∞.
Here is the main theorem of the current subsection.
Though this theorem easily follows from Doob's maximal inequality (4.1) in the case p > 1, the case p = 1 (which is the most important one for the main application of Theorem 7.11, Theorem 8.2) remains problematic and requires some work.
For the proof of the theorem we will need to find its analogues for purely discontinuous martingales. Let us first recall some definitions.
A random variable τ : Ω → R + is called an optional stopping time (or just a stopping time) if {τ ≤ t} ∈ F t for each t ≥ 0. With an optional stopping time τ we associate a σ-field F τ = {A ∈ F ∞ : A ∩ {τ ≤ t} ∈ F t , t ∈ R + }. Note that M τ is strongly F τ -measurable for any local martingale M . We refer to [30,Chapter 7] for details.
Recall that due to the existence of a càdlàg version of a martingale M : R + ×Ω → X, we can define an X-valued random variables M τ − and ∆M τ for any stopping time τ in the following way: A stopping time τ is called predictable if there exists a sequence of stopping times (τ n ) n≥1 such that τ n < τ a.s. on {τ > 0} for each n ≥ 1 and τ n → τ monotonically a.s. A stopping time τ is called totally inaccessible if P{τ = σ < ∞} = 0 for each predictable stopping time σ. The reader can find more information on quasi-left continuous martingales and martingales with accessible jumps in [17,29,30,69,70].
In order to prove Theorem 7.11 we will need to show similar approximation results for quasi-left continuous purely discontinuous martingales and purely discontinuous martingales with accessible jumps. Both cases will be considered separately. 7.5.1. Quasi-left continuous purely discontinuous martingales. Before stating the corresponding approximation theorem let us show the following proposition.
Proposition 7.13. Let X be a Banach space, M : R + × Ω → X be a purely discontinuous quasi-left continuous martingale. Then there exist sequences of positive numbers (a n ) n≥1 , (b n ) n≥1 , and a sequence of X-valued purely discontinuous quasi-left continuous martingales (M n ) n≥1 such that sup t ∆M n t ≤ a n , #{t ≥ 0 : ∆M n t = 0} ≤ b n a.s. ∀n ≥ 1,  [17,Lemma 5.20]. We will recall the construction here for the convenience of the reader. First of all we refer the reader to [17,29,30,31] for the basic definitions and facts on random measures, which presenting we will omit here for the brevity and simplification of the proof. Let µ M be a random measure defined on (R + × X, B(R + ) ⊗ B(X)) by Let ν M be the corresponding compensator,μ M := µ M − ν M . Due to the proof of [17,Lemma 5.20] there exists an a.s. increasing sequence (τ n ) n≥1 of stopping times such τ n → ∞ a.s. as n → ∞, and such = that there exist positive sequences (a n ) n≥1 , (b n ) n≥1 with (a n ) n≥1 being increasing natural and with #{t ≥ 0 : ∆M τn t ∈ [1/a n , a n ]} ≤ b n . Lemma 7.15. Let X be a Banach space, M : R + × Ω → X be a local martingale with accessible jumps. Then there exists a set (τ n ) n≥0 of predictable stopping times with disjoint graphs (i.e. τ n = τ m a.s. for all m = n) such that a.s. (7.17) {t ≥ 0 : ∆M t = 0} ⊂ {τ 1 , τ 2 , . . . , τ n , . . .}.
Lemma 7.16. Let X be a Banach, M : R + × Ω → X be an L 1 -bounded martingale, τ be a predictable stopping time. Then defines an L 1 -bounded martingale.
Let X be a Banach space, M : R + ×Ω → X be a purely discontinuous martingale with accessible jumps, (τ n ) n≥0 be a set of predictable stopping times with disjoint graphs such that (7.17) holds. Thanks to Lemma 7.16 for each n ≥ 1 we can define a martingale Does (M n ) n≥1 converge to M in strong L p -sense? The following theorem answers this question in the UMD case.
Theorem 7.17. Let X be a UMD Banach space, M : R + × Ω → X be a martingale with accessible jumps, (M n ) n≥1 be as in (7.18).
Proof. The proof is fully analogous to the proof of Theorem 7.14. 7.5.3. Proof of Theorem 7.11. Let us now prove Theorem 7.11. Since X is a UMD Banach space, M has the canonical decomposition, i.e. there exist an X-valued continuous local martingale M c , an X-valued purely discontinuous quasi-left continuous local martingale M q , and an X-valued purely discontinuous local martingale M a with accessible jumps such that M c 0 = M q 0 = 0 and M = M c + M q + M a (see Subsection 7.6 for details). Moreover, by (7.20) and a triangle inequality Case 2: M is purely discontinuous quasi-left continuous. By Theorem 7.14 one can assume that M has uniformly bounded jumps. Then the theorem follows from the fact that any adapted càdlàg process with uniformly bounded jumps is local uniformly bounded and the fact that M t = M T for all t ≥ T .
Case 3: M is purely discontinuous with accessible jumps. By Theorem 7.17 we can assume that there exist predictable stopping times (τ n ) N n=1 with disjoint graphs such that Fix ε > 0. Without loss of generality we may assume that the stopping times (τ n ) N n=1 are bounded a.s. Due to [17,Subsection 5.3] we may additionally assume that (τ n ) N n=1 is a.s. increasing. Then by [17,Subsection 5.3] (or [30,Lemma 26.18] in the real-valued case) the sequence (0, ∆M τ1 , 0, ∆M τ2 , . . . , 0, ∆M τN ) is a martingale difference sequence with respect to the filtration [30,Lemma 25.2] for the definition of F τ − ). As any discrete L p -bounded martingale difference sequence, (0, ∆M τ1 , 0, ∆M τ2 , . . . , 0, ∆M τN ) can be approximated in a strong L p -sense by a uniformly bounded X-valued G-martingale difference sequence (0, d ε The martingale difference sequence (0, d ε 1 , 0, d ε 2 , . . . , 0, d ε N ) can be translated back to a martingale on R + in the same way as it was shown in [17,Subsection 5.3], i.e. one can define a process N ε : R + × Ω → X such that , an X-valued purely discontinuous quasi-left continuous local martingale M q (a compensated Poisson-like part), and an X-valued purely discontinuous local martingale M a with accessible jumps (a discrete-like part) such that M c 0 = M q 0 = 0 and M = M c + M q + M a . We refer the reader to [17,30,69,70] for the details on the canonical decomposition.
As it was shown in [17,69,70], the canonical decomposition is unique, and by [69, Section 3] together with (4.1) we have that for any 1 < p < ∞ and for any i = c, q, a Theorem 7.8 allows us to extend (7.19) to the case p = 1. Indeed, it is known due to [69,70] that for any x * ∈ X * a.s. (M (t n ) − M (t n−1 ))(N (t n ) − N (t n−1 )), where the limit in probability is taken over partitions 0 = t 0 < . . . < t n = t, and it is taken in a weak sense (i.e. (7.22) holds under action of any linear functional x * ∈ X * ). It remains open whether (7.22) holds in a strong sense.

UMD Banach function spaces
Here we are going to extend (1.4) to the case p = 1. Let us first recall some basic definitions on Banach function spaces. For a given measure space (S, Σ, µ), the linear space of all real-valued measurable functions is denoted by L 0 (S). We endow L 0 (S) with the local convergence in measure topology.
for all x, y ∈ L 0 (S) and λ ∈ R, n(λx) = |λ|n(x) and n(x + y) ≤ n(x) + n(y), (iii) if x ∈ L 0 (S), y ∈ L 0 (S), and |x| ≤ |y|, then n(x) ≤ n(y), (iv) there exists ζ ∈ L 0 (S) with ζ > 0 and n(ζ) < ∞, (v) if 0 ≤ x n ↑ x with (x n ) ∞ n=1 a sequence in L 0 (S) and x ∈ L 0 (S), then n(x) = sup n∈N n(x n ). Let X denote the space of all x ∈ L 0 (S) for which x := n(x) < ∞. Then X is called a normed function space associated to n. It is called a Banach function space when (X, · X ) is complete. We will additionally assume the following natural property of X: (vi) X is continuously embedded into L 0 (S) with the local convergence in measure topology.
Notice that the condition (vi) holds automatically if one changes the measure on (S, Σ) in an appropriate way (see [39,Theorem 1.b.14]). We refer the reader to [39,46,57,67,72] for the details on Banach function spaces.
Given a Banach function space X over a measure space S and Banach space E, let X(E) denote the space of all strongly measurable functions f : S → E with f E ∈ X. The space X(E) becomes a Banach space when equipped with the norm f X(E) = σ → f (σ) E X . Let X be a UMD Banach function space over a σ-finite measure space (S, Σ, µ). According to [67] any X-valued L p -bounded martingale M , 1 < p < ∞, has a pointwise martingale version, i.e. there exists a process N : R + × Ω × S → R such that (i) N | [0,t]×Ω×S is B([0, t]) ⊗ F t ⊗ Σ-measurable for all t ≥ 0, (ii) N (·, ·, σ) is a local martingale for a.e. σ ∈ S, (iii) N (ω, t, ·) = M t (ω) for any t ≥ 0 for a.a. ω ∈ Ω.
A process N satisfying (i) and (ii) is called a local martingale field. Moreover, it was shown in [67] that for any 1 < p < ∞ 1/2 ∞ , σ ∈ S, defines an element of X a.s. The goal of the present subsection is to show that (8.1) holds for p = 1.
Theorem 8.2. Let X be a UMD Banach function space over a σ-finite measure space (S, Σ, µ), M : R + × Ω → X be a local martingale. Then there exists a local martingale field N : R + × Ω × S → R such that N (ω, t, ·) = M t (ω) for all t ≥ 0 for a.a. ω ∈ Ω, and for all 1 ≤ p < ∞  Proof. The proof follows from Theorem 2.1 and the equivalence [28, (9.26)] between the γ-norm and the square function.
Remark 8.4. By Remark 2.4 and [28, (9.26)] one has that for any r ∈ (1, ∞) there exist positive C r,X and c r,X such that for any 1 ≤ p ≤ r c r,X E ∞ n=1 |d n | 2 We will also need the following technical lemma proved in [67,Section 4]. Recall that D b ([0, ∞); X) is the Banach space of all bounded X-valued càdlàg functions on R + , which is also known as a Skorohod space.  . Then (MQ 1 (X), · MQ 1 (X) ) is a Banach space. Moreover, if N n → N in MQ 1 , then there exists a subsequence (N n k ) k≥1 such that pointwise a.e. in S, we have N n k → N in L 1 (Ω; D b ([0, ∞))).
Let us finally show (8.2). Since N n → N in MQ 1 (X) and by (8.5) which terminates the proof. i.e. one has the same equivalence with a pointwise supremum in S. The techniques that provide such an improvement were discovered by Rubio de Francia in [57]. Unfortunately, it remains open whether (8.6) holds for p = 1. Surprisingly, (8.6) holds for p = 1 and for X = L 1 (S) by a simple Fubini-type argument, so it might be that (8.6) holds for p = 1 even for other nonreflexive Banach spaces.