Stochastic integration II: the Itô integral

13 Stochastic integration II: the Itô integral We have seen in Lecture 6 how to integrate functions Φ : (, ) L (H, E) with resect to an H-cylindrical Brownian motion W H. In this lecture we address the roblem of extending the theory of stochastic integration to rocesses Φ : (, ) Ω L (H, E). As it turns out, very satisfactory results can be obtained in the setting of UMD Banach saces E. he reason for this is that in these saces we can rove a decouling theorem for certain martingale difference sequence which, in the context of stochastic integrals, enables us to relace W H by an indeendent coy W H. he stochastic integral of Φ with resect to W H can be defined ath by ath using the results of Lecture 6, and the decouling inequality allows us to translate integrability criteria for this integral to the integral with resect to W H. 13.1 Decouling We begin with an abstract decouling result for a suitable class of martingale difference sequences. Let 1 < < be fixed and suose that (ξ n ) N n=1 is a sequence of centred integrable random variables in L (Ω). We assume that a filtration (F n ) N n=1 is given such that the following conditions are satisfied for n = 1,..., N: (1) ξ n is F n -measurable for all 1 n N; (2) ξ n is indeendent of F m for all 1 m < n N. Note that E(ξ n F m ) = Eξ n = for 1 m < n N, so (ξ n ) N n=1 is a martingale difference sequence with resect to (F n ) N n=1. On the roduct sace (Ω Ω, F F, P P) we define, with a slight abuse of notation, ξ n (ω, ω) := ξ n (ω), ξn (ω, ω) := ξ n ( ω). (13.1) he sequences (ξ n ) N n=1 and ( ξ n ) N n=1 are indeendent and identically distributed. he oint here is that we identify each ξ n with a random variable on

182 13 Stochastic integration II: the Itô integral Ω Ω which deends only on the first coordinate and introduce an indeendent coy ξ n which deends only on the second coordinate. Clearly, (ξ n ) N n=1 and ( ξ n ) N n=1 are martingale difference sequences on Ω Ω with resect to the filtrations (F n ) N n=1 and ( F n ) N n=1 defined by F n := F n {, Ω}, Fn := {, Ω} F n, (13.2) where again there is a slight abuse of notation in the first definition. Let (v n ) N n=1 be a redictable sequence of E-valued random variables on Ω. Recall that this means that v n is F n 1 -measurable for n = 1,...,N, with the understanding that F = {, Ω} (so that v 1 is constant almost surely). We identify (v n ) N n=1 with a redictable sequence (v n ) N n=1 on Ω Ω in the same way as above by utting v n (ω, ω) := v n (ω). heorem 13.1 (Decouling). If, in addition to the above assumtions, E is a UMD-sace, then N N ξ n v n,e ξ n v n n=1 with constants deending on and E only. n=1 Proof. he roof uses a trick similar to that of heorem 12.15. For n = 1,...,N define d 2n 1 := 1 2 (ξ n + ξ n )v n and d 2n := 1 2 (ξ n ξ n )v n. We claim that (d j ) 2N j=1 is a martingale difference sequence with resect to the filtration (D j ) 2N j=1, where In view of D 2n 1 = σ(f n 1, F n 1, ξ n + ξ n ), D 2n = σ(f n, F n ). N ξ n v n = 2N d j n=1 j=1 and N ξ n v n = 2N n=1 j=1 ( 1) j+1 d j, the result then follows from the definition of the UMD -roerty. It remains to rove the claim. We begin by observing that (d n ) 2N (D n ) 2N n=1-adated. Moreover, E(d 2n D 2n 1 ) (i) = 1 2 v ne(ξ n ξ n F n 1, F n 1, ξ n + ξ n ) (ii) = 1 2 v ne(ξ n ξ n ξ n + ξ n ) (iii) =. n=1 is Here (i) follows from the F n 1 -measurability of v n, (ii) from Proosition 11.7 and the indeendence of σ(ξ n, ξ n ) and σ(f n 1, F n 1 ) (which follows from the

13.2 Stochastic integration 183 indeendence of ξ n and F n 1 ), and (iii) uses that ξ n and ξ n are indeendent and identically distributed (Exercise 11.2). Similarly, E(d 2n 1 D 2n 2 ) = 1 2 v ne(ξ n + ξ n F n 1, F n 1 ) = 1 2 v ne(ξ n + ξ n ) = since ξ n + ξ n is indeendent of σ(f n 1, F n 1 ) and ξ n, ξ n are centred. 13.2 Stochastic integration Let (Ω, F, P) be a robability sace. A function Φ : (, ) Ω L (H, E) is said to be a finite rank adated ste rocess with resect to a given filtration F = (F t ) t [,] if it is of the form Φ(t, ω) = M m=1 n=1 N 1 (tn 1,t n)(t)1 Amn (ω) k h j x jmn, (13.3) where t < < t N, for each n = 1,...,N the sets A 1n,..., A Mn are disjoint and belong to F tn 1, the vectors h 1,...,h k H are orthonormal, and the vectors x jmn belong to E. In what follows we assume that W H is an H-cylindrical Brownian motion on (Ω, F, P), adated to F in the sense that the random variables W H (t)h are F t -measurable and the increments W H (t)h W H (s)h are indeendent of F s for t > s. It follows from Exercise 11.4 that the filtration F WH generated by W H has these roerties. he stochastic integral with resect to W H of a finite rank adated ste rocess Φ of the form (13.3) is defined as Φ(t)dW H (t) := M m=1 n=1 j=1 N k 1 Amn (W H (t n )h j W H (t n 1 )h j )x jmn. j=1 We leave it to the reader to check that this definition does not deend on the articular reresentation of Φ in (13.3). Note that Φ(t)dW H(t) belongs to L (Ω, F ; E) for all 1 <, and satisfies he latter follows by linearity from E Φ(t)dW H (t) =. E ( 1 Amn (W H (t n )h j W H (t n 1 )h j ) ) = E ( E(1 Amn (W H (t n )h j W H (t n 1 )h j ) F tn 1 ) ) = E(1 Amn E(W H (t n )h j W H (t n 1 )h j ) F tn 1 ) =.

184 13 Stochastic integration II: the Itô integral For each ω Ω the trajectory t Φ ω (t) := Φ(t, ω) is a finite rank ste function and therewith defines an element R Φω of γ(l 2 (, ; H), E). his results in a simle random variable R Φ : Ω γ(l 2 (, ; H), E). In order to extend the above stochastic integral to a more general class of L (H, E)-valued rocesses we shall roceed as in Lecture 6 by estimating the L (Ω; E)-norm of the stochastic integral in terms of R Φ. Due to the resence of the random variables 1 Amn, however, the Gaussian comutation of heorem 6.14 breaks down. In the roof of the next theorem we circumvent this roblem by relacing W H by an indeendent coy W H and use the decouling estimate of heorem 13.1. heorem 13.2 (Itô isomorhism). Let E be a UMD sace and fix 1 < <. For all finite rank adated ste rocesses Φ : (, ) Ω L (H, E) we have Φ(t)dW H (t),e E R Φ γ(l 2 (,;H),E), with constants deending only on and E. Proof. As in (13.1) we identify W H with an H-cylindrical Brownian motion on the roduct Ω Ω and define an indeendent coy on W H on Ω Ω by utting W H (t)h(ω, ω) := W H (t)h(ω), WH (t)h(ω, ω) := W H (t)h( ω). If Φ : (, ) Ω L (H, E) is a finite rank adated ste rocess of the form (13.3), we define the decouled stochastic integral Φ(t)d W H (t) := N n=1 m=1 M k 1 Amn ( WH (t n )h j W ) H (t n 1 )h j xjmn. j=1 he lan of the roof is to aly heorem 13.1 to the real-valued sequence (ξ jn ) 1 j k and the E-valued sequence (v jn ) 1 j k, 1 n N 1 n N ξ jn := W H (t n )h j W H (t n 1 )h j, v jn := With these notations, M 1 Amn x jmn. m=1 Φ(t)dW H (t) = N k ξ jn v jn, n=1 j=1 Φ(t)d W H (t) = N k ξ jn v jn. n=1 j=1

13.2 Stochastic integration 185 We consider the filtration (F jn ) 1 j k, where F jn is the σ-algebra generated by all ξ j n with (j, n ) (j, n); the airs are ordered lexicograhically 1 n N according to the rule (j, n ) (j, n) n < n or [n = n & j j]. With resect to this filtration, the sequence (ξ jn ) 1 j k is centred and 1 n N has the roerties (1) and (2) stated at the beginning of Section 13.1 and (v jn ) N 1 j k 1 n N is redictable. Let us denote by E 1 and E 2 the exectations with resect to the first and second coordinate of Ω Ω. Alying successively heorem 13.1, the Kahane- Khintchine inequality, and heorem 6.14 (ointwise with resect to Ω 1 ), we obtain E 1 E 2 Φ(t)dW H (t),e E 1 E 2 Φ(t)d W H (t),e E 1 ( E 2 Φ(t)d W H (t) 2 ) 2,E E 1 R Φ γ(l 2 (,;H),E). Definition 13.3. A random variable R L (Ω; γ(l 2 (, ; H), E)) is called adated if it belongs to the closure in L (Ω; γ(l 2 (, ; H), E)) of the finite rank adated ste rocesses. he closed subsace in L (Ω; γ(l 2 (, ; H), E)) of all adated elements with be denoted by L F (Ω; γ(l2 (, ; H), E)). heorem 13.2 shows that the stochastic integral extends uniquely to an isomorhic embedding J WH : L F (Ω; γ(l2 (, ; H), E)) L (Ω; E). Definition 13.4. Let E be a Banach sace and fix 1 < <. A rocess Φ : (, ) Ω L (H, E) is said to be L -stochastically integrable with resect to the H-cylindrical Brownian motion W H if there exists a sequence of finite rank adated ste rocesses Φ n : (, ) Ω L (H, E) such that: (1) for all h H we have lim n Φ n h = Φh in measure; (2) there exists a random variable X L (Ω; E) such that lim n X in L (Ω; E). he L -stochastic integral of Φ is then defined as ΦdW H := lim Φ n dw H. n Φ n dw H =

186 13 Stochastic integration II: the Itô integral he remarks (a) and (b) following Definition 6.15 extend to the resent situation, but (c) is no longer automatic since stochastic integrals of ste rocesses are no longer Gaussian. his is the reason why the adjective L - has been built into the definition. heorem 13.5. Let 1 < <. If Φ : (, ) Ω L (H, E) is L - stochastically integrable with resect to W H, then the stochastic integral rocess ( t ΦdW ) H t [,] is an E-valued L -martingale which has a continuous version satisfying the maximal inequality ( E su t [,] t ΦdW H ) ( ) E ΦdW H. 1 Proof. Choose a sequence (Φ n ) n 1 of finite rank adated ste rocesses such that the conditions of Definition 13.4 are satisfied and ut X n (t) := t Φ n dw H. Clearly, there exists a continuous version X n of each X n, and by the Pettis measurability theorem we have X n L (Ω; C([, ]; E)). o see that this theorem can be alied in the resent situation, first note that there exists a searable closed subsace E of E such that each X n has trajectories in C([, ]; E ). he sace C([, ]; E ) is searable, and the linear san of the functionals δ t x is norming in its dual; moreover, X n, δ t x = t Φ nx dw H almost surely and the right hand side is measurable as a function on Ω. By Doob s maximal inequality (we use that the stochastic integral rocess is a martingale; see Exercise 3), for every choice of t 1 < < t N we have E ( su X n (t j ) X m (t j ) ) ( ) E Xn () X m (). 1 j=1,...,n Hence, by ath continuity and Fatou s lemma, E ( su X n (t) X m (t) ) ( ) E Xn () X m (). t [,] 1 his inequality shows that the sequence ( X n ) n 1 is a Cauchy sequence in L (Ω; C([, ]; E)). Since for all t [, ] we have lim n X n (t) = X(t) in L (Ω; E), the limit X = lim n Xn defines a continuous version of X. he final inequality follows from Doob s maximal inequality in the same way as above (relace X n X m by X). As in Lecture 6, in the secial case E = R we may identify L (H, R) with H and heorem 13.2 reduces to the statement that the L -stochastic integral of an adated ste rocess φ : (, ) Ω H satisfies φdw H E φ L 2 (,;H) (13.4)

13.2 Stochastic integration 187 he constants deend only on since the UMD -constant of Hilbert saces only deend on. From this it is not hard to see (Exercise 2) that a strongly adated measurable rocess φ : (, ) Ω H is L -stochastically integrable with resect to W H if and only if φ L (Ω; L 2 (, ; H)), and the isomorhism (13.4) extends to this situation. Definition 13.6. A rocess Φ : (, ) Ω L (H, E) is called H-strongly measurable if for each h H the rocess Φh : (, ) Ω E is strongly measurable. Such a rocess Φ is called adated if for each h H the rocess Φh is adated. We are now in a osition to state the main result of this section, which extends heorem 6.17 to L (H, E)-valued rocesses. heorem 13.7. Let E be a UMD sace and fix 1 < <. For an H-strongly measurable adated rocess Φ : (, ) Ω L (H, E) the following assertions are equivalent: (1) Φ is L -stochastically integrable with resect to W H ; (2) Φ x L (Ω; L 2 (, ; H)) for all x E, and there exists a random variable X L (Ω; E) such that for all x E, X, x = Φ x dw H (t) in L (Ω). (3) Φ x L (Ω; L 2 (, ; H)) for all x E, and there exists a random variable R L (Ω; γ(l 2 (, ; H), E)) such that for all f L 2 (, ; H) and x E, Rf, x = Φ(t)f(t), x dt in L (Ω). If these equivalent conditions are satisfied, the random variables X and R are uniquely determined, we have X = ΦdW H in L (Ω; E), and ΦdW H,E E R γ(l 2 (,;H),E). Moreover, R L F (Ω; γ(l2 (, ; H), E)), that is, R is adated. Proof. We sketch the main stes and refer to the Notes for more information. (1) (2): his is roved in the same way as in heorem 6.17. Note that the stochastic integrals Φ x dw H are well-defined by the above remarks. (2) (3): For the secial case where F is the filtration generated by W H, a roof will be outlined below. (1) (3): his is an immediate consequence of heorem 13.2: if (Φ n ) n=1 is an aroximating sequence for Φ, then the oerators (R Φn ) n=1 form a Cauchy sequence in L F (Ω; γ(l2 (, ; H), E)) and its limit has the desired roerties.

188 13 Stochastic integration II: the Itô integral (3) (1): First one shows that R is adated (see Exercise 1). Knowing this, the roof can be finished in the same way as the corresonding imlication of heorem 6.17. Unfortunately we are not able to give a fully self-contained roof of the imlication (2) (3). In the sequel we shall not need this imlication; we only use the equivalence (1) (3) which is the most useful art of the theorem. In site of this we want to sketch a roof of (2) (3) under the simlifying assumtion that the filtration is the one generated by W H. In this situation we can aly a version of the so-called martingale reresentation theorem for H- cylindrical Brownian motions W H. In most textbook roofs, the integrator is a Brownian motion (or a more general martingale); the extension to cylindrical Brownian motions is obtained from it by an aroximation argument as in the roof of the martingale convergence theorem (heorem 11.21). Recall that the filtration F WH has been defined in Exercise 11.4. Lemma 13.8. Let 1 < < and ξ L (Ω, F WH ). here exists unique φ L F (Ω; W L2 (, ; H)) such that H ξ = Eξ + φdw H. he roof of this lemma is beyond the scoe of these lectures. Roughly seaking it roceeds like this. First, we may assume that Eξ =. By aroximation we may further assume that H is finite-dimensional. From W H (t)h = 1 (,t) h dw H (t) we see that every X in the linear san of the random variables W H (t)h can be reresented by a stochastic integral. Since the stochastic integral defines an isomorhic embedding, it remains to show that this san is dense in the closed subsace of L (Ω, F WH ; H) consisting of all mean elements. he next result extends the lemma to UMD saces. Recall that J WH : L F (Ω; γ(l2 (, ; H), E)) L (Ω; E) is the isomorhic embedding of heorem 13.2. heorem 13.9. Let E be a UMD sace, let 1 < <, and let X L (Ω, F WH ; E). here exists a unique R L F (Ω; W γ(l2 (, ; H), E)) such H that X = EX + J WH (R). Proof. Choose a sequence of simle F WH -measurable random variables X n such that lim n X n = X in L (Ω; E). Let us write X n = M n m=1 1 A mn x mn.

13.3 Stochastic integrability of L -martingales 189 By Lemma 13.8, there exist unique rocesses φ mn L (Ω; F W L2 (, ; H)) H such that 1 Amn = E1 Amn + φ mn dw H. Put Φ n (t)h := M m=1 [φ mn, h]x mn. he rocess Φ n : (, ) Ω L (H, E) is L -stochastically integrable with resect to W H and X n = EX n + Φ n dw H. Let R n L F (Ω; γ(l2 (, ; H), E) be defined by R n (ω)f := M n m=1 φ mn (ω) x mn, f L 2 (, ; H). Since lim n X n = X in L (Ω; E), the isomorhism of heorem 13.2 imlies that the sequence (R n ) n=1 is Cauchy in L (Ω; γ(l 2 (, ; H), E)). he limit R has the desired roerties. Uniqueness follows from the injectivity of J WH. As a corollary we observe that the stochastic integral defines an isomorhism of Banach saces J WH : L F W H (Ω; γ(l2 (, ; H), E)) L WH (Ω, F ; E), where L WH (Ω, F ; E) is the closed subsace of L (Ω, F WH ; E) consisting of all elements with mean. Proof (Proof of heorem 13.7 (2) (3) for the filtration F WH ). By the Pettis measurability theorem, the random variable X belongs to L WH (Ω, F ; E). he element R rovided by heorem 13.9 has the desired roerties. 13.3 Stochastic integrability of L -martingales We return to the setting where W H is an H-cylindrical Brownian motion, adated to a filtration F. he main result of this section states that if E is a UMD sace and M is a γ(h, E)-valued L -martingale with resect to F, then M is L -stochastically integrable with resect to W H. he roof has three ingredients: the characterisation of L -stochastic integrability (the equivalence (1) (3) of heorem 13.7)), the γ-multilier theorem (heorem 9.14), and the vector-valued Stein inequality (heorem 12.15). heorem 13.1. Let E be a UMD sace and fix 1 < <. Let W H be an H-cylindrical Brownian motion, adated to a filtration F, and let M :

19 13 Stochastic integration II: the Itô integral [, ] Ω γ(h, E) be an L -martingale with resect to F. hen M is L - stochastically integrable with resect to W H and we have ( M(t)dW H (t) ) 1 with a constant deending only on and E. (,E E M() ) 1 γ(h,e), Proof. First we rove the result under the additional assumtion that M() L (Ω; γ(h, E)). By the L -contractivity of the conditional exectation we then have M L ((, ) Ω; γ(h, E)). In articular, for all x E we have M x L (Ω; L 2 (, ; H)), and even M x L F (Ω; L2 (, ; H)) since M is adated. Let us write B := L (Ω; E) for brevity. Define the bounded function N : [, ] L (B) by N(t)ξ := E(ξ F t ), ξ B, t [, ]. Since E is a UMD sace, by heorem 12.15 the family {N(t) : t [, ]} is R-bounded on B, and therefore γ-bounded, with γ-bound deending only on and E. By heorem 11.21, for every ξ B the function t N(t)ξ has left limits at every oint [, ]. In articular, these functions are strongly measurable. By the γ-fubini isomorhism (heorem 5.22), for each t [, ] we may identify the random variable M(t) L (Ω; γ(h, E)) with a unique oerator M(t) γ(h, B) by the formula ( M(t)h)(ω) = M(t, ω)h. Define a constant function G : [, ] γ(h, B) by G(t) := M(), t [, ]. his function reresents the element R G γ(l 2 (, ; H), B) satisfying R G γ(l2 (,;H),B) = ( M() γ(h,b) E M() ) 1 γ(h,e), where we used the result of Exercise 5.3. By the martingale roerty, for all t [, ] we have M(t) = N(t) M() in B. Now we aly heorem 9.14 to conclude that M reresents an element R γ(l 2 (, ; H), B) satisfying R γ(l2 (,;H),B),E R G γ(l2 (,;H),B). Using heorem 5.22 once more, we can identify R with an element X L (Ω; γ(l 2 (, ; H), E)) by the formula X(ω)f = (Rf)(ω). Below we check that X x = M x in L (Ω; L 2 (, ; H)) for all x E. Assuming this for the moment, it follows from heorem 13.7(3) (1) that M is L -stochastically integrable and satisfies

( M(t)dW H (t) ) 1 (,E E X ) 1 γ(l 2 (,;H),E) 13.4 Exercises 191 R γ(l2 (,;H),B),E (E M() γ(h,e) ) 1. o rove that X x = M x for all x E, let f L 2 (, ; H) and x E be arbitrary and note that for all A F, E( Mf, x 1 A ) = M(t, ω)f(t), x 1 A (ω)dt dp(ω) = Ω = E Ω M(t, ω)f(t), x 1 A (ω)dp(ω)dt M(t)f(t), 1 A x dt = E Rf, 1 A x = E( Xf, x 1 A ). o conclude the roof we remove the assumtion M() L (Ω; E). Choose a sequence of F -measurable simle random variables M n () converging to M() in L (Ω; E), and define M n (t) := E(M n () F t ). Since M n () L (Ω; E), we may aly what we roved above to the martingales M n. We obtain that each M n is L -stochastically integrable with resect to W H and ( M n (t)dw H (t) ) 1 C ( E M n () ) 1 γ(h,e), with a constant C indeendent of n. Similarly, by the above alied to the martingales M n M m, we find that the stochastic integrals M n(t)dw H (t) are Cauchy in L (Ω; E). By the Itô isomorhism, this means that the corresonding elements R n L F (Ω; γ(l2 (, ; H), E)) are Cauchy and therefore converge to a limit R L F (Ω; γ(l2 (, ; H), E)). Clearly, R x = lim n Rn x = lim n Mn x = M x in L (Ω; L 2 (, ; H)), and the conclusion of the theorem now follows via heorem 13.7. 13.4 Exercises In the exercises 1-3 we fix 1 < <. 1. In this exercise we comare the two notions of adatedness given in Definitions 13.3 and 13.6. a) Show that R L (Ω; γ(l 2 (, ; H), E)) is adated if and only if the random variables R(1 (,t) f) : Ω E have strongly F t -measurable versions for all t (, ) and f L 2 (, ; H). Hint: For the if art, aroximate with simle random variables and use that the finite rank ste functions are dense in γ(l 2 (, ; H), E). o secure adatedness, build in a small shift before aroximating.

192 13 Stochastic integration II: the Itô integral Now suose that Φ : (, ) L (H, E) is H-strongly measurable and assume that the conditions of heorem 13.7 (3) be satisfied; let R L (Ω; γ(l 2 (, ; H), E)) be as in (3). b) Show that if Φ is adated, then R is adated. 2. In the discussion after Definition 13.4 it was observed that a strongly measurable adated rocess φ : (, ) Ω H is L -stochastically integrable with resect to W H if and only if φ L (Ω; L 2 (, ; H)). Prove this. Hint: If φ L (Ω; L 2 (, ; H)), then by the revious exercise φ is adated as an element of L (Ω; L 2 (, ; H)). 3. Let Φ : (, ) Ω L (H, E) be L -stochastically integrable with resect to W H. Show that the stochastic integral rocess ( t ΦdW ) H t [,] is a martingale. Hint: Aroximate with finite rank adated ste rocesses. If E is a UMD sace with tye 2 and Φ : (, ) Ω γ(h, E) is an adated and strongly measurable rocess such that E Φ(t) 2 γ(h,e) dt <, then Φ is stochastically integrable with resect to H-cylindrical Brownian motions W H ; this follows from heorem 13.7 and Exercise 5.4. In the next two exercises we show that the UMD assumtion can essentially be droed from this statement. 4. A Banach sace E has martingale tye [1, 2] if there exists a constant M (E) such that for any L -martingale sequence (d n ) N n=1 with values in E we have N N d n (M (E)) E d n. n=1 n=1 a) Show that every martingale tye sace has tye. b) Show that every UMD sace with tye has martingale tye. In both cases, give relations between the constants. c) Deduce that L -saces, 1 < <, have martingale tye min{, }, where 1 + 1 = 1. 5. Let E be a martingale tye 2 sace. a) Show that if W H is an H-cylindrical Brownian motion and Φ : (, ) Ω L (H, E) is an adated finite rank ste rocess, then ΦdW H 2 (M2 (E)) 2 E Φ(t) 2 γ(h,e) dt.

13.4 Exercises 193 b) Conclude that if Φ : (, ) Ω γ(h, E) is an adated strongly measurable rocess satisfying E Φ(t) 2 dt <, then Φ is stochastically integrable with resect to W H, with the same estimate as before. Notes. A systematic treatment of decouling inequalities is resented in the monograh of Gine and de la Peña [31]. he roof of the decouling inequality (heorem 13.1) is based on an idea of Montgomery-Smith [78]. he idea to use decouling inequalities for obtaining bounds on stochastic integrals in UMD saces was first used by Garling [4], who only considered ste rocesses and used the resulting estimates to investigate certain geometric roerties of UMD saces. Using a more delicate decouling result together with Burkholder s characterisation of UMD saces through ζ-convexity (heorem 12.17), McConnell [75] roved that a UMD-valued rocess is stochastically integrable if almost surely its trajectories are stochastically integrable with resect to an indeendent coy of the Brownian motion. In view of heorem 6.17 this result can be viewed as an almost sure version of the imlication (3) (1) of heorem 13.7. Our aroach to vector-valued stochastic integration in UMD saces via γ-radonifying norms is taken from [82], where heorem 13.7 was roved. In that aer, McConnell s result is recovered using a stoing time argument. he equivalence of norms in (13.4) is a secial case of an inequality of Burkholder, Davis, Gundy which, in the more general situation where the integrator is a continuous-time martingale M, relates the norms of stochastic integrals to the norms of the quadratic variation rocess of M. For more details we refer to Karatzas and Shreve [59], Revuz and Yor [94], or Kallenberg [55]. An alternative roof of the imlication (2) (3) of heorem 13.7, which is based on finite-dimensional aroximations, covariance domination, and the theorem of Hoffmann-Jorgensen and Kwaień (heorem 5.9) is given in [82]. A detailed roof of the imlication (3) (1) is contained in [81]. A systematic theory of stochastic integration in martingale tye 2 sace has been develoed by Neidhardt [85], Dettweiler [33], and Brzeźniak [13]. he first two authors assumed that E be 2-uniformly smooth, a roerty which was subsequently shown to be equivalent to the martingale tye 2 roerty by Pisier [91]. For an overview, see Brzeźniak [15].