On stochastic calculus with respect to q-Brownian motion Aurélien Deya, René Schott
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Aurélien Deya, René Schott. On stochastic calculus with respect to q-Brownian motion. Jour- nal of Functional Analysis, Elsevier, 2018, 274 (4), pp.1047-1075. 10.1016/j.jfa.2017.08.019. hal- 01416950v3
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AURÉLIEN DEYA AND RENÉ SCHOTT
Abstract. Following the approach and the terminology introduced in [A. Deya and R. Schott, On the rough paths approach to non-commutative stochastic calculus, J. Funct. Anal., 2013], we construct a product Lévy area above the q-Brownian motion (for q ∈ [0, 1)) and use this object to study differential equations driven by the process. We also provide a detailled comparison between the resulting “rough” integral and the stochastic “Itô” integral exhibited by Donati-Martin in [C. Donati-Martin, Stochastic integration with respect to q Brownian motion, Probab. Theory Related Fields, 2003].
1. Introduction: the q-Brownian motion The q-Gaussian processes (for q [0, 1)) stand for one of the most stan- dard families of non-commutative random∈ variables in the literature. Their consideration can be traced back to a paper by Frisch and Bourret in the early 1970s [8]: the dynamics is therein suggested as a model to quantify some possible non-commutativity phenomenon between the creation and annihilator operators on the Fock space, the limit case q = 1 morally cor- responding to the classical probability framework. The mathematical con- struction and basic stochastic properties of the q-Gaussian processes were then investigated in the 1990s, in a series of pathbreaking papers by Bożejko, Kümmerer and Speicher [2, 3, 4]. For the sake of clarity, let us briefly recall the framework of this analysis and introduce a few notations that will be used in the sequel (we refer the reader to the comprehensive survey [11] for more details on the subsequent definitions and assertions). First, recall that the processes under consider- ation consist of paths with values in a non-commutative probability space, that is a von Neumann algebra equipped with a weakly continuous, pos- itive and faithful trace ϕ . TheA sole existence of such a trace ϕ on (to be compared with the “expectation” in this setting) is known to giveA the algebra a specific structure, with “Lp”-norms p 1/p X p := ϕ( X ) ( X := √XX ) k kL (ϕ) | | | | ∗ closely related to the operator norm . : k k X p X , X = lim X p , for all X . (1) L (ϕ) p L (ϕ) k k ≤ k k k k →∞ k k ∈A Date: December 1, 2020. 2010 Mathematics Subject Classification. 46L53,60H05,60F17. Key words and phrases. non-commutative stochastic calculus; q-Brownian motion; rough paths theory. 1 2 AURÉLIENDEYAANDRENÉSCHOTT
Now recall that non-commutative probability theory is built upon the fol- lowing fundamental spectral result: any element X in the subset of self- adjoint operators in can be associated with a law that sharesA the∗ same moments. To be moreA specific, there exists a unique compactly supported probability measure µ on R such that for any real polynomial P ,
P (x)dµ(x)= ϕ(P (X)) . (2) ZR Based on this property, elements in are usually referred to as (non- commutative) random variables, and inA∗ the same vein, the law of a given (i) family X i I of random variables in ( , ϕ) is defined as the set of all of its joint{ moments} ∈ A
(i1) (ir) ϕ X X , i1,...,ir I , r N . · · · ∈ ∈ With this stochastic approach in mind and using the terminology of [2], the definition of a q-Gaussian family can be introduced along the following combinatorial description: Definition 1.1. 1. Let r be an even integer. A pairing of 1,...,r is any partition of 1,...,r into r/2 disjoint subsets, each of cardinality{ }2. We { } denote by 2( 1,...,r ) or 2(r) the set of all pairings of 1,...,r . 2. WhenP π{ (}1,...,rP ), a crossing in π is any{ set of the} form ∈ P2 { } x1,y1 , x2,y2 with xi,yi π and x1 The existence of such a non-commutative process (in some non-commutative space ( , ϕ)) has been established by Bożejko and Speicher in [3]. In the same spiritA as above, the q-Bm distribution can be regarded as a straight- forward extension of two well-known processes: When q 1, one recovers the classical Brownian-motion dynamics, with •independent,→ stationary and normally-distributed increments. The 0-Brownian motion coincides with the celebrated free Brownian mo- tion,• whose freely-independent increments are known to be closely related to the asymptotic behaviour of large random matrices, following Voiculescu’s breakthrough results [14]. Thus, we have here at our disposal a family of processes which, as far as distribution is concerned, provides a natural “smooth” interpolation between two of the most central objects in probability theory: the standard and the free Brownian motions. It is then natural to wonder whether the classical stochastic properties satisfied by each of these two processes can be “lifted” on the level of this interpolation, or in other words if the properties known for q = 0 and q 1 can be extended to every q [0, 1). Of course, any such extension potentially→ offers an additional piece∈ of evidence in favor of this interpolation model, as a privileged link between the free and the commutative worlds. Some first results in this direction, focusing on the stationarity property and the marginal-distribution issue, can be found in [2]: Proposition 1.4. For any fixed q [0, 1), let Xt t 0 be a q-Brownian motion in some non-commutative probability∈ space{ }( ≥, ϕ). Then for all 0 s 1 ∞ 2 cos θ µ (dx)= 1 q sin θ (1 qn) 1 qne2iθ 2 , where x = with θ [0,π] . q π − − | − | √1 q ∈ p nY=1 − A next natural step is to examine the possible extension, to all q [0, 1), of the stochastic integration results associated with the free/classical∈ Brownian motion. Let us here recall that the foundations of stochastic calculus with respect to the free Brownian motion (that is, for q = 0) have been laid in a remarkable paper by Biane and Speicher [1]. Among other results, the latter study involves the construction of a free Itô integral, as well as an analysis of the free Wiener chaoses generated by the multiple integrals of the free Brownian motion. These lines of investigation have been followed by Donati-Martin in [7] to handle the general q-Bm case, with the construction of a q-Itô integral and a study of the q-Wiener chaos. Let us also mention the results of [5] related 4 AURÉLIENDEYAANDRENÉSCHOTT to the extension of the fourth-moment phenomenon that prevails in Wiener chaoses. In this paper, we intend to go further with the analysis related to the q-Brownian motion. To be more specific, we propose, in the continuation of [6], to adapt some of the main rough-path principles to this setting. The aim here is to derive a very robust integration theory allowing, in particu- lar, to consider the study of differential equations driven by the q-Bm, i.e. sophisticated dynamics of the form dY = f(Y ) dX g(Y ) , (6) t t · t · t for smooth functions f, g. In fact, thanks to the general (non-commutative) rough-path results proved in [6] (and which we will recall in Section 2), the objective essentially reduces to the exhibition of a so-called product Lévy area above the q-Bm, that is a kind of iterated integral of the process involving the product structure of . Let us briefly recall that the definition of such an object (which appears Aas quite natural in this algebra setting) has been introduced in [6] as a way to overcome the possible non-existence problems arising from the study of more general Lévy areas, in the classical Lyons’ sense [10] (see [13] for a description of the non-existence issue in the free case). At this point, we would like to draw the reader’s attention to the fact that the construction in [7] of a q-Itô integral as an element of L2(ϕ) would not be not sufficient for our purpose. Indeed, the rough-path techniques are based on Taylor-expansion procedures, which, for obvious stability reasons, forces us to consider an algebra norm in the computations. As a result, any satisfying notion of product Lévy area requires some control with respect to the operator norm, that is in L∞(ϕ) (along (1)), and not only with respect to the L2(ϕ)-norm (see Section 2 and especially Definition 2.2 for more details on the topology involved in this control). In the particular case of the free Bm (q = 0), the Burkholder-Gundy inequality established by Biane and Speicher in [1, Theorem 3.2.1] immedi- ately gives rise to operator-norm controls on the free Itô integral, which we could readily exploit in [6] to deal with rough paths in the free situation. Unfortunately, and at least for the time being, no similar operator-norm control has been shown for the q-Itô integral when q (0, 1). With our rough-path objectives in mind, we will be able to overcome∈ this difficulty though, by resorting to a straightforward L∞(ϕ)-construction of a product Lévy area - the latter object being actually much more specific than a gen- eral Itô integral. This is the purpose of the forthcoming Section 3, which leads to the main result of the paper. Injecting this construction into the general rough-path theory will immediately answer our original issue, that is the derivation of a robust stochastic calculus for the q-Bm. It is then possible to compare, a posteriori, the resulting rough integral with more familiar q-Itô or q-Stratonovich integrals, through a standard ON STOCHASTIC CALCULUS WITH RESPECT TO q-BROWNIAN MOTION 5 L2(ϕ)-analysis and the involvement of the so-called second-quantization op- erator. This comparison will be the topic of Section 4. Let us however insist, one more time, on the fact that this sole L2(ϕ)-analysis would not have been sufficient for the rough-path theory (and the powerful rough-path results) to be applied in this situation. Our construction of a product Lévy area will only rely on the consideration of the law of the q-Bm, that is on the process as given by Definition 1.3. In other words, no reference will be made to any particular representation of the process as a map with values in some specific algebra (just as classical probability theory builds upon the law of the Brownian motion and not upon its representation). The only reference to some particular representation of the q-Brownian motion (namely its standard representation on the q-Fock space) will occur in Section 4, as a way to compare our rough objects with the constructions of [7], based on the Fock space. Besides, we have chosen in this study to focus on the case where q [0, 1) and introduce the q-Brownian motion as a natural interpolation between∈ the free and the standard Brownian motions. We are aware that the definition of a q-Bm can also be extended to every q ( 1, 0), that is up to the “an- ticommutative” situation q 1. In fact,∈ we− must here specify that the positivity assumption on q will→ be− used in an essential way for the construc- tion of the product Lévy area (see for instance (24)), and at this point, we do not know if such an object could also be exhibited in the case q < 0. As we already sketched it in the above description of our results, the study is organized as follows. In Section 2, we will recall the general non- commutative rough-path results obtained in [6] and at the core of the present analysis. Section 3 is devoted to the construction of the main object involved in the rough-path procedure, that is a product Lévy area above the q-Bm. Finally, Section 4 focuses on the L2(ϕ)-comparison of the rough construc- tions with more standard Itô/Stratonovich definitions. Acknowledgements. We are very grateful to two anonymous referees for their attentive reading and insightful suggestions. 2. General rough-path results in C∗-algebras Our strategy to develop a robust L∞(ϕ)-stochastic calculus for the q-Bm is based on the non-commutative rough-path considerations of [6, Section 4]. Therefore, before we can turn to the q-Bm situation, it is necessary for us to recall the main results of the theoretical analysis carried out in [6]. This requires first a few brief preliminaries on functional calculus in a C∗- algebra (along the framework of [1]), as well as precisions on the topologies involved in the study. Special emphasis will be put on the cornerstone of the rough-path machinery, the product Lévy area, around which the whole integration procedure can be naturally expanded. Note that the considerations of this section apply to a general C∗-algebra , that we fix from now on. In particular, no additional trace operator will A 6 AURÉLIENDEYAANDRENÉSCHOTT be required here. As before, we denote by . the operator norm on , and will stand for the set of self-adjoint operatorsk k in . We also fixA an arbitraryA∗ time horizon T > 0 for the whole section. A 2.1. Tensor product. Let be the algebraic tensor product generated by , and just as in [1], denoteA⊗A by ♯ the natural product interaction between andA , that is the linear extension of the formula A A⊗A (U U )♯X = X♯(U U ) := U XU , for all U , U , X . 1 ⊗ 2 1 ⊗ 2 1 2 1 2 ∈A In a similar way, set, for all U , U , U , X , 1 2 3 ∈A X♯(U U U ) := (U XU ) U , (U U U )♯X := U (U XU ) . 1 ⊗ 2 ⊗ 3 1 2 ⊗ 3 1 ⊗ 2 ⊗ 3 1 ⊗ 2 3 Our developments will actually involve the projective tensor product ˆ of , that is the completion of with respect to the norm A⊗A A A⊗A U = U ˆ := inf Ui Vi , k k k kA⊗A Xi k kk k where the infimum is taken over all possible representation U = U V of i i ⊗ i U. It is readily checked that for all U and X , one hasP U♯X U X , and so the above ♯-product∈ continuously A⊗A extends∈A to ˆk . Thesek≤ kconsiderationskk k can, of course, be generalized to the n-th projectiveA⊗A tensor ˆ n product ⊗ , n 1, and we will still denote by . the projective tensor A ˆ n ≥ k k norm on ⊗ . Along theA same terminology as in [1], we will call any process with values in ˆ , resp. ˆ ˆ , a biprocess, resp. a triprocess. A⊗A A⊗A⊗A 2.2. Functional calculus in a C∗-algebra. Following again the presenta- tion of [1], let us introduce the class of functions f defined for every integer k 0 by ≥ ıξx i Fk := f : R C : f(x)= e µf (dξ) with ξ µf (dξ) < for every i 0,...,k , { → ZR ZR | | ∞ ∈ { }} (7) k i and set, if f F , f := R ξ µ (dξ). Then, with all f F and ∈ k k kk i=0 | | f ∈ 0 X , we associate the operatorP R f(X) along the formula ∈A∗ ıξX f(X) := e µf (dξ) , ZR where the integral in the right-hand side is uniformly convergent in . This straightforward operator extension of functional calculus happens toA be com- patible with Taylor expansions of f, a central ingredient towards the appli- cation of rough-path techniques. The following notion of tensor derivatives naturally arises in the procedure (see the subsequent Examples 2.5 and 2.6): Definition 2.1. For every f F1, resp. f F2, we define the tensor derivative, resp. second tensor∈ derivative, of f∈ by the formula: for every X , ∈A∗ 1 ıαξX ı(1 α)ξX ∂f(X) := dα ıξ [e e − ] µf (dξ) ˆ , Z0 ZR ⊗ ∈A⊗A 2 2 ıαξX ıβξX ı(1 α β)ξX resp. ∂ f(X) := dα dβ ξ [e e e − − ] µf (dξ) ˆ ˆ . − ZZ α,β 0 ZR ⊗ ⊗ ∈A⊗A⊗A α+β≥ 1 ≤ ON STOCHASTIC CALCULUS WITH RESPECT TO q-BROWNIAN MOTION 7 2.3. Filtration and Hölder topologies. From now on and for the rest of Section 2, we fix a process X : [0, T ] and assume that X is γ-Hölder regular, that is →A∗ Xt Xs sup k − γ k < , 0 s α α hst 2 (I; V ) := h 2(I; V ) : [h; 2 (I; V )] := sup k kα < . C ∈C N C s T ( ⇀) := L = (Lst)0 s λ Lst[U] [L; 2 ( T ( ⇀))] := sup k λ k . (8) N C L A s Remark 2.3. Recall that Definition 2.2 is derived from the theoretical analy- sis performed in [6, Section 4] with equation (6) in mind. At some heuristic level, and following the classical rough-path approach, the notion of product Lévy area must be seen as some abstract version of the iterated integral t X2 st[U]= (U♯δXsu)dXu , (10) Zs noting that definition of this integral is not clear a priori for a non-differentiable process X. As pointed out in [6], the above notion of “Lévy area” is specifi- cally designed to handle the non-commutative algebra dynamics of (6), and it offers a much more efficient approach than general rough-path theory based on “tensor” Lévy areas (the object considered in [10]). In a com- mutative setting (i.e., if were a commutative algebra), the basic process A 1 A st(U) := 2 (U♯δXst) δXst would immediately provide us with such a prod- uct Lévy area. In the general (non-commutative) situation though, this path only satisfies 1 A [U] A [U] A [U]= (U♯δX ) δX + (U♯δX ) δX , st su ut 2 su ut ut su − − so that A may not meet the product-Chen condition (ii), making Definition 2.2 undoubtedly relevant. 2.5. Controlled (bi)processes and integration. A second ingredient in the rough-path machinery (in addition to a “Lévy area”) consists in the identification of a suitable class of integrands for the future rough integral with respect to X. The following definition naturally arises in this setting: Definition 2.4. Given a time interval I [0, T ], we call adapted controlled ⊂ γ process, resp. biprocess, on I any adapted process Y 1 (I; ), resp. biprocess U γ (I; ˆ ), with increments of the form ∈ C A ∈C1 A⊗A (δY ) = YX ♯(δX) + Y ♭ , s Example 2.5. If f, g F2 and Y X∗ (I) with decomposition (11), then U := f(Y ) g(Y ) Q∈ (I) with ∈ Q ⊗ ∈ X X,1 := [∂f(Y ) YX ] g(Y ) , X,2 = f(Y ) [∂g(Y ) YX ] . Us s s ⊗ s Us s ⊗ s s X,1 X,2 Example 2.6. If f F3, then U := ∂f(X) QX ([0, T ]) with s = s = 2 ∈ ∈ U U ∂ f(Xs). We are finally in a position to recall the definition of the rough integral with respect to X, which can be expressed (among other ways) as the limit of “corrected Riemann sums”: ON STOCHASTIC CALCULUS WITH RESPECT TO q-BROWNIAN MOTION 9 Proposition 2.7. [6, Proposition 4.12] Assume that we are given a product Lévy area X2 above X, in the sense of Definition 2.2, as well as a time interval I = [ℓ ,ℓ ] [0, T ]. Then for every U Q (I) with decomposition 1 2 ⊂ ∈ X (12), all s Proposition 2.9. [6, Proposition 4.16] Assume that there exists a product Lévy area X2 above X such that, as n tends to infinity, n γ X2,n X2 2γ [X X; 1 ([0, T ]; )] 0 and [ ; 2 ( T ( ⇀))] 0 . N − C A → N − C L A → (15) Then for all f, g F , it holds that ∈ 3 . . n n n n X γ f(Xu )dXu g(Xu ) →∞ f(Xu)d u g(Xu) in 2 ([0, T ]; ) , (16) Z. −−−→ Z. C A where the integral in the limit is interpreted with Proposition 2.7. Similarly, for all f F , one has ∈ 3 . . n n n X γ ∂f(Xu )♯dXu →∞ ∂f(Xu)♯d u in 2 ([0, T ]; ) , (17) Z. −−−→ Z. C A which immediately yields Itô’s formula: for all s n n m n n n Y = A , dY = fi(Y )dX gi(Y ) . 0 t i=1 t t t ∈A∗ X Theorem 2.10. [6, Theorem 4.17] Under the assumptions of Proposition n n γ 2.9, one has Y →∞ Y in 1 ([0, T ]; ), where Y is the solution of (13) given by Theorem−−−→ 2.8. C A As we pointed it out in the introduction, these convergence results are based on Taylor-expansion procedures and accordingly, the consideration of an algebra norm for the control of U and Lst[U] in the roughness assumption (8) is an essential ingredient. 3. A product Lévy area above the q-Brownian motion We go back here to the q-Bm setting described in Section 1. Namely, we fix q [0, 1) and consider a q-Brownian motion (Xt)t 0 in some non- commutative∈ probability space ( , ϕ). With the developments≥ of the previ- ous section in mind, the route towardsA an efficient operator-norm calculus for X is now clear: we need to exhibit a product Lévy area above X, in the sense of Definition 2.2. Our main result thus reads as follows: n Theorem 3.1. Denote by Xt t 0 the linear interpolation of X along the { } ≥ dyadic partition Dn := tn , i 0 , tn := i . Then there exists a product { i ≥ } i 2n Lévy area X2,S above X, in the sense of Definition 2.2, such that for every T > 0 and every 0 <γ< 1/2, one has n γ X2,n X2,S 2γ X X in 1 ([0, T ]; ) and in 2 ( T ( )) , (19) → C A → C L A→ where X2,n is defined by (14). We call X2,S the Stratonovich product Lévy area above X. ON STOCHASTIC CALCULUS WITH RESPECT TO q-BROWNIAN MOTION 11 Based on this result, the conclusions of Proposition 2.7, Theorem 2.8, Proposition 2.9 and Theorem 2.10 can all be applied to the q-Brownian mo- tion, with limits understood as rough integrals with respect to the “product rough path” XS := (X, X2,S). The Stratonovich terminology is here used as a reference to the classical commutative situation, where the (almost sure) limit of the sequence of approximated Lévy areas would indeed coincide with the Stratonovich iterated integral (see also Corollary 4.10 for another justification of this terminology). Before we turn to the proof of Theorem 3.1, let us recall that the whole difficulty in constructing a stochastic integral with respect to the general q- Bm, in comparison with the free (q = 0) or the commutative (q 1) cases, lies in the absence of any satisfying “q-freeness” property for the→ increments of the process when q (0, 1) (as reported by Speicher in [12]). For instance, if s The proof of Theorem 3.1 will also appeal to the two following elementary lemmas. The first one (whose proof follows immediately from (3)) is related to the linear stability of q-Gaussian families: Lemma 3.2. For any fixed q [0, 1), let Y := Y1,...,Yd be a q-Gaussian vector in some non-commutative∈ probability space{ ( , ϕ)}, and consider a real-valued (d m)-matrix Λ. Then Z := ΛY is also aAq-Gaussian vector in ( , ϕ). × A We will also need the following general topology property on the space accommodating any Lévy area: λ Lemma 3.3. The space 2 ( T ( ⇀)), endowed with the norm (8), is com- plete. C L A Proof. Although the arguments are classical, let us provide a few details λ here, since the 2 ( T ( ⇀))-structure is not exactly standard. C L A n λ Consider a Cauchy sequence L in 2 ( T ( ⇀)). For every fixed s [0, T ], the sequence Ln defines a Cauchy sequenceC L inA the space L ([s, T ]; (∈ ˆ , )) s. ∞ L As⊗As A of bounded functions on [s, T ] (with values in ( s ˆ s, )), endowed with the uniform norm. Therefore it converges in theL latterA ⊗A spaceA to some function λ Ls.. The fact that the so-defined family Lst s m ε λ Mε,s,t 0 such that for all m Mε,s,t, Lst Lst ( s ˆ s, ) 2 t s . ≥ ≥ k − kL A ⊗A A ≤ | − | On the other hand, there exists Nε 0 such that for all n,m Nε and all n m ε ≥ λ ≥ s n n m m λ Lst Lst ( s ˆ s, ) Lst Lst ( s ˆ s, )+ Lst Lst ( s ˆ s, ) ε t s , k − kL A ⊗A A ≤ k − kL A ⊗A A k − kL A ⊗A A ≤ | − | n λ and so L L in 2 ( T ( ⇀)), which achieves to prove that the latter space is complete.→ C L A Proof of Theorem 3.1. Throughout the proof, we will denote by A . B any bound of the form A cB, where c is a universal constant independent from the parameters under≤ consideration. The first-order convergence statement in (19) is a straightforward consequence of the 1/2-Hölder regularity of X. In fact, using (5), it can be checked that for all n 0 and s n 1/2 n γ n(1/2 γ) δXst . X1 t s and δ(X X)st . X1 t s 2− − . k k k k| − | k − k k k| − | (20) Let us turn to the second-order convergence and to this end, fix n 0 and s 2,n+1 2,n 2,n+1 2,n 2 2γ n(1/2 γ) X [U] X [U] X [U] + X [U] . X t s 2− − . k st − st k ≤ k st k k st k k 1k | − | Assume from now on that ℓ k + 2 and in this case consider the decompo- sition, for every U ˆ ≥, ∈As⊗As X2,n+1 X2,n st [U] st [U] tn− tn ℓ n+1 n+1 ℓ n n = U♯δXtn u dXu U♯δXtn u dXu Ztn k+1 − Ztn k+1 k+1 k+1 tn t tn k+1 n+1 n+1 n+1 n+1 k+1 n n + U♯δXsu dXu + U♯δXsu dXu U♯δXsu dXu Zs Ztn − Zs ℓ t n n U♯δXsu dXu − Ztn ℓ tn tn ℓ n+1 n+1 ℓ n n + U♯δXstn dXu U♯δXstn dXu . (21) Ztn k+1 − Ztn k+1 k+1 k+1 ON STOCHASTIC CALCULUS WITH RESPECT TO q-BROWNIAN MOTION 13 The “boundary” integrals within the second and third brackets can again be bounded individually using the first estimate in (20) only. For instance, t n+1 n+1 U♯δXsu dXu Ztn ℓ t 1/2 n+1 . X1 U 1 tn+1 t Therefore, we only have to focus on the first bracket in decomposition (21). In fact, noting that tn i+1 n n 1 n n n n n U♯δXt u dXu = U♯δXti ti+1 δXti ti+1 , Ztn i 2 i 14 AURÉLIENDEYAANDRENÉSCHOTT we get tn tn ℓ n+1 n+1 ℓ n n U♯δXtn u dXu U♯δXtn u dXu Ztn k+1 − Ztn k+1 k+1 k+1 ℓ 1 tn+1 tn+1 − 2i+1 n+1 n+1 2i+2 n+1 n+1 = U♯δXtn u dXu + U♯δXtn u dXu Z n+1 k+1 Z n+1 k+1 i=Xk+1 t2i t2i+1 tn i+1 n n U♯δXtn u dXu − Z n k+1 ti ℓ 1 tn+1 tn+1 − 2i+1 n+1 n+1 2i+2 n+1 n+1 = U♯δX n+1 dX + U♯δX n+1 dX +1 u +1 u Z n t2i u Z n t2i+1 u i=Xk+1 t2i t2i+1 tn i+1 n n U♯δXtnu dXu − Z n i ti + U♯δX n n+1 δX n+1 n+1 + U♯δX n n+1 δX n+1 n+1 tk+1 t2i t2i t2i+1 tk+1t2i+1 t2i+1t2i+2 h U♯δX n n+1 δX n+1 n+1 tk+1 t2i t2i t2i+2 − i ℓ 1 − 1 = U♯δX n+1 n+1 δX n+1 n+1 + U♯δX n+1 n+1 δX n+1 n+1 2 t2i t2i+1 t2i t2i+1 t2i+1 t2i+2 t2i+1t2i+2 i=Xk+1 U♯δX n+1 n+1 δX n+1 n+1 t2i t2i+2 t2i t2i+2 − + U♯δX n n+1 δX n+1 n+1 + U♯δX n n+1 δX n+1 n+1 tk+1 t2i t2i+1 t2i+2 tk+1 t2i+1 t2i+1 t2i+2 h − i ℓ 1 − 1 = U♯δXtn+1 tn+1 δXtn+1 tn+1 U♯δXtn+1 tn+1 δXtn+1 tn+1 2 − 2i 2i+1 2i+1 2i+2 − 2i+1 2i+2 2i 2i+1 i=Xk+1 + U♯δX n+1 n+1 δX n+1 n+1 t2i t2i+1 t2i+1t2i+2 h i ℓ 1 1 − = U♯(δX) n+1 n+1 (δX) n+1 n+1 U♯(δX) n+1 n+1 (δX) n+1 n+1 (22). 2 t2i t2i+1 t2i+1 t2i+2 − t2i+1 t2i+2 t2i t2i+1 i=Xk+1 Let us bound the two sums ℓ 1 1,n − S [U] := U♯(δX) n+1 n+1 (δX) n+1 n+1 st t2i t2i+1 t2i+1 t2i+2 i=Xk+1 and ℓ 1 2,n − S [U] := U♯(δX) n+1 n+1 (δX) n+1 n+1 st t2i+1 t2i+2 t2i t2i+1 i=Xk+1 separately. Consider first the case where U = o U V , with j=1 j ⊗ j P Uj := X j X j , Vj := X j X j , s sm s s 1 · · · j mj +1 · · · mj +pj ON STOCHASTIC CALCULUS WITH RESPECT TO q-BROWNIAN MOTION 15 j and sp s for all j, p. Besides, let us set Yi = Yi,n := (δX)tn+1 tn+1 . With ≤ i i+1 these notations, and for every r 1, we have ≥ r ∗ ϕ S1,n[U] 2r = ϕ U Y V Y U Y V Y | st | j1 2i1 j1 2i1+1 j2 2i2 j2 2i2+1 Xi1 Xj1 Xi2 Xj2 = ϕ U Y V Y Y V ∗ Y U ∗ j1 2i1 j1 2i1+1 2i2+1 j2 2i2 j2 · · · i1,...,iX2r j1,...,jX2r Uj2r−1 Y2i2r−1 Vj2r−1 Y2i2r−1+1Y2i2r+1Vj∗2r Y2i2r Uj∗2r(23), where each index i runs over k + 1,...,ℓ 1 and each index j runs overs 1,...,o . At this point, observe{ that for− all} fixed i := (i ,...,i ) and { } 1 2r j := (j1,...,j2r), the family Xsj ,...,Xsj ,Y2i,Y2i+1, i i1,...,i2r , j j1,...,j2r { 1 mj +pj ∈ { } ∈ { }} is a q-Gaussian family (due to Lemma 3.2) and accordingly the associated joint moments obey Formula (3). Besides, we have trivially (n+1) 2 ϕ Y2ia Y2ib+1 = 0 , ϕ Y2ia Y2ib = ϕ Y2ia+1Y2ib+1 = 1 ia=i 2− ϕ X1 ) { b} | | and ϕ Y2iX j )= ϕ Y2i+1X j ) = 0 . sa sa Using these basic observations and going back to (23), it is clear that, when applying Formula (3) to the expectation in (23), we can restrict the sum to the set of pairings π ( 1,...,N ) (N := 2 (m + p )+ . . . + ∈ P2 { r} r j1 j1 (mj2r + pj2r )] + 8r) that decompose - in a unique way - as a combination of 1 three sub-pairings, namely: 1) a pairing π 2( 1,..., 2r ) that connects ∈ P { 2 } the random variables Y2i to each other; 2) a pairing π 2( 1,..., 2r ) { } ∈ P { 3 } that connects the random variables Y2i+1 to each other; 3) a pairing π ( 1,...,N ) (N := 2 (m +p )+{ . . . +(} m +p )]) that connects the∈ P2 { r′ } r′ j1 j1 j2r j2r random variables X j to each other. Moreover, with this decomposition { si } in mind, one has clearly Cr(π) Cr(π1)+Cr(π2) + Cr(π3) . ≥ Consequently, it holds that for all fixed i := (i1,...,i2r) and j := (j1,...,j2r), ϕ Uj1 Y2i1 Vj1 Y2i1+1Y2i2+1Vj∗2 Y2i2 Uj∗2 · · · Uj2r−1 Y2i2r−1 Vj2r−1 Y2i2r−1+1Y2i2r+1Vj∗2r Y2i2r Uj∗2r 1 2 3 qCr(π )+Cr(π )+Cr(π ) ≤ 1 2 π ,π X2( 1,...,2r ) 3 ∈P { ′ } π 2( 1,...,N ) ∈P { r} j j ϕ Y2ia Y2ib 1 ia=ib ϕ Y2ic+1Y2id+1 1 ic=id ϕ ZeZf(24) { } { } a,bY π1 c,dY π2 e,fY π3 { }∈ { }∈ { }∈ 2r(n+1) 2 2r Cr(π1) 2− ϕ X1 ) q 1 ia=ib ≤ | | 1 1 { } π 2X( 1,...,2r ) a,bY π ∈P { } { }∈ Cr(π2) Cr(π3) j j q 1 ic=id q ϕ ZeZf , 2 2 { } 3 ′ 3 π 2(X1,...,2r ) c,dY π π 2(X1,...,N ) e,fY π ∈P { } { }∈ ∈P { r} { }∈ 16 AURÉLIENDEYAANDRENÉSCHOTT j where Z stands for the natural reordering of the variables X j , namely { sm } for all a 1,..., 2r and b 1,...,m + p , ∈ { } ∈ { ja ja } j j Z = Z := X ja . 2[(mj +pj )+...+(mj +pj )]+b 2[(mj +pj )+...+(mj +pj )]+[2(mj +pj ) b] s 1 1 a−1 a−1 1 1 a−1 a−1 a a − b As a result, the double sum in (23) is bounded by ℓ 1 2rn 2 2r Cr(π1) − 2− ϕ X1 ) q 1 ia=ib | | 1 1 { } π 2(X1,...,2r ) i1,...,iX2r=k+1 a,bY π ∈P { } { }∈ o Cr(π2) Cr(π3) j j q q ϕ ZeZf(25). 2 3 ′ 3 π 2X( 1,...,2r ) j1,...,jX2r=1 π 2(X1,...,N ) e,fY π ∈P { } ∈P { r} { }∈ Now observe that the last sum in (25) actually corresponds to o o 2r Cr(π3) j j q ϕ ZeZf = ϕ UjVj , 3 ′ 3 j1,...,jX2r=1 π 2(X1,...,N ) e,fY π jX=1 ∈P { r} { }∈ (26) and for every fixed π1 ( 1,..., 2r ), ∈ P2 { } ℓ 1 − 1 ia=ib 1 { } i1,...,iX2r=k+1 a,bY π { }∈ ℓ 1 2(1 2γ) ℓ 1 4γ 1 − − − − = 1 ia=ib 1 ia=ib 1 { } 1 { } i1,...,iX2r=k+1 a,bY π i1,...,iX2r=k+1 a,bY π { }∈ { }∈ 2(1 2γ)r 2r(4γ 1) (ℓ (k + 1)) − (ℓ (k + 1)) − ≤ − − tn tn 4rγ 24rγn t s 4rγ24rγn . (27) ≤ | ℓ − k+1| ≤ | − | By injecting (26) and (27) into (25), we end up with the estimate ϕ S1,n[U] 2r | st | 2 o 2r 4rγ 2r(1 2γ)n 2 2r Cr(π) t s 2− − ϕ X1 ) q ϕ UjVj ≤ | − | | | π 2(X1,...,2r ) jX=1 ∈P { } o 2r 4rγ 2r(1 2γ)n 2 2r 2r 2 t s 2− − ϕ X1 ) ϕ X1 ϕ UjVj , ≤ | − | | | | | jX=1 and so ϕ S1,n[U] 2r 1/2r | st | o 2r 1/2r 2γ (1 2γ)n 2 2r 1/r t s 2− − ϕ X1 )ϕ X1 ϕ UjVj ≤ | − | | | | | jX=1 o 2γ (1 2γ)n 4 t s 2− − X1 UjVj ≤ | − | k k jX=1 o 2γ (1 2γ)n 4 t s 2− − X1 Uj Vj . (28) ≤ | − | k k jX=1 k kk k ON STOCHASTIC CALCULUS WITH RESPECT TO q-BROWNIAN MOTION 17 It is easy to see that the above arguments could also be applied to the more general situation where U := o U V with j=1 j ⊗ j P Kj Uj := αj,kXsj,k Xsj,k 1 · · · mj,k kX=0 and Lj C j,k j,ℓ Vj := βj,ℓXuj,ℓ Xuj,ℓ , αj,k, βj,ℓ , sa , ub [0,s] , 1 · · · pj,ℓ ∈ ∈ ℓX=0 leading in the end to the same bound (28). Therefore, this bound (28) can actually be extended to any U , V , which then entails that for every j j ∈ As U ˆ , ∈As⊗As 1,n 2r 1/2r 2γ (1 2γ)n 4 ϕ S [U] t s 2− − X U , | st | ≤ | − | k 1k k k and by letting r tend to infinity, we get by (1) that 1,n 2γ (1 2γ)n 4 S [U] t s 2− − X U . k st k ≤ | − | k 1k k k 2,n The very same reasoning can of course be used in order to estimate Sst [U] , with the same resulting bound. Going back to (21) and (22), wekhave thusk X2,n λ proved that is a Cauchy sequence in 2 ( T ( ⇀)), and by Lemma 3.3, we can therefore assert that it converges inC thisL spaceA to some element X2,S. The product Chen identity (10) for X2,S is readily obtained by passing to the limit (in a pointwise way) in the product Chen identity that is trivially satisfied by X2,n. Finally, in order to show that X2,S actually belongs to λ 2 ( T ( )), fix s 4. Comparison with L2(ϕ)-constructions Our objective in this section is to compare the previous L∞(ϕ)-constructions (i.e., constructions based on the operator norm) with the L2(ϕ)-constructions exhibited by Donati-Martin in [7]. In brief, we shall see that, when studied in L2(ϕ), the previous rough constructions correspond to Stratonovich-type integrals, while the constructions in [7] are more of a Itô-type. This com- parison relies on an additional ingredient, the so-called second-quantization operator, whose central role in q-integration theory was already pointed out in Donati-Martin’s work. 18 AURÉLIENDEYAANDRENÉSCHOTT Since we intend to make specific references to some of the results of [7], we assume for simplicity that we are exactly in the same setting as in the latter study. Namely, for a fixed q [0, 1), we assume that the q-Bm Xt t 0 we will handle in this section is∈ constructed as the “canonical process”{ } on≥ the q-Fock space ( , ϕ) (see [7] for details on these structures). A As in the previous sections, we denote by the closure, with respect to At the operator norm, of the algebra generated by Xs s t. { } ≤ 4.1. Second quantization. Recall that the space L2(ϕ) is defined as the completion of as a Hilbert space through the product A U, V := ϕ(UV ∗) . (29) h i We will denote by . 2 the associated norm, to be distinguished from k kL (ϕ) the operator norm . . For every t 0, let be the von Neumann algebra k k ≥ Bt generated by Xs s t (observe in particular that t t ) and denote { } ≤ A ⊂B ⊂A by ϕ( t) the conditional expectation with respect to t. In other words, for every·|B U , ϕ(U ) stands for the orthogonal projectionB of U onto , ∈A |Bt Bt with respect to the product (29): Z = ϕ(U t) if and only if Z t and ϕ(ZW )= ϕ(UW ) for every W . |B ∈ B ∗ ∗ ∈Bt A possible way to introduce the second-quantization operator goes through the following invariance result: Lemma 4.1. [7, Theorem 3.1]. For all s0 In fact, we will essentially use the operator Γq through the following result, which offers a quite general tool to study Itô/Stratonovich correction terms (for the sake of clarity, we have postponed the proof of this proposition to Section 4.4): ε ˆ 3 Proposition 4.4. For every adapted triprocess 1([s,t]; ⊗ ) (s 2 Us♯dXs L (ϕ) . ZI ∈ Given a biprocess U : I ˆ and a partition ∆ of I, we denote by U∆ the step-approximation→ A⊗A ∆ U := Uti 1[ti,ti+1[ . tXi ∆ ∈ The following isometry property, to be compared with the classical Brownian Itô isometry, is the key ingredient to identify Itô-integrable processes: Proposition 4.6. [7, Proposition 3.3]. For every interval I R, all adapted biprocesses ⊂ U : I , V : I , →A⊗A →A⊗A and all partitions ∆1, ∆2 of I, it holds that ∆1 ∆2 ∞ ∆1 ∆2 SX (U),SX (V) L2(ϕ) = Uu , Vu q du , (32) h i Z0 hh ii where ., . is the bilinear extension of the application defined for all hh iiq U1, U2, V1, V2 t 0 t as ∈∪ ≥ A U1 U2, V1 V2 q := ϕ U1Γq(U2V2∗)V1∗ . hh ⊗ ⊗ ii Corollary 4.7. Let U : I ˆ be an adapted biprocess such that →A⊗A 2 ∆ 2 Uu ˆ du< and Uu Uu ˆ du 0 as ∆ 0 , ZI k kA⊗A ∞ ZI k − kA⊗A → | |→ for every partition ∆ of I. Then U is Itô integrable against X and 2 Uu♯dXu = Uu, Uu q du . (33) ZI L2(ϕ) ZI hh ii Proof. Let us just provide a few details, the procedure being essential stan- dard. Consider a sequence Un : I of adapted biprocesses such n →A⊗A that for every t I, Ut Ut 0. Then, given two partitions ∆1, ∆2 of I, one has by (32)∈ k − k→ ∆1 n ∆2 n 2 n,∆1 n,∆2 n,∆1 n,∆2 S (U ) S (U ) 2 = U U , U U q du . X X L (ϕ) Z u u u u − I hh − − ii 20 AURÉLIENDEYAANDRENÉSCHOTT By applying Cauchy-Schwarz inequality and then (30), it is readily checked that for all V , ∈As ⊗As V, V X 2 V 2 , hh iiq ≤ k 1k k k and so ∆1 n ∆2 n 2 2 n,∆1 n,∆2 2 S (U ) S (U ) 2 X1 U U du , X X L (ϕ) Z u u − ≤ k k I − which, by letting n tend to infinity, leads us to ∆1 ∆2 2 2 ∆1 ∆2 2 S (U) S (U) 2 X1 U U du . X X L (ϕ) Z u u − ≤ k k I − The conclusion easily follows. 4.3. Comparison with the rough integral. We now have all the tools to identify, as elements in L2(ϕ), the rough constructions arising from Sections 2 and 3. Let us first consider the situation at the level of the product Lévy area provided by Theorem 3.1. To this end, given 0 s t 2 (U♯δXsu)dXu L (ϕ) . Zs ∈ Proposition 4.8. For all 0 s Id Γq [U V ] := UΓq(V ) . × ⊗ n Proof. Fix s (and accordingly in L2(ϕ)). Now write t X2,D˜ n ˜ n ˜ n st [U] = (U♯δXsu)dXu (36) Zs n 1 ˆ − 1 tk+1 u tˆ = U♯ δX + k (δX) du (δX) stˆ − tˆk tˆk+1 tˆk tˆk+1 tˆ tˆ Ztˆ k tˆ tˆ kX=1 k+1 − k k k+1 − k n 1 n 1 − 1 − = (U♯δX ˆ ) δXˆ ˆ + U♯(δX)ˆ ˆ (δX)ˆ ˆ stk tk tk+1 2 tk tk+1 tk tk+1 kX=1 kX=1 t n 1 n 1 − = (U♯δXˆ )dXu + U♯(δX)ˆ ˆ (δX)ˆ ˆ . (37) Z su 2 tk tk+1 tk tk+1 s kX=1 Thanks to (33), it holds that t t ˆ n 2 (U♯δX )dXu (U♯δXsu)dXu 2 Z su Z L (ϕ) s − s t ˆn ˆn = (U♯[δXsu δXsu]) 1, (U♯[δXsu δXsu]) 1 q du Zs hh − ⊗ − ⊗ ii t ˆ n 2 = U♯[δX δXsu] 2 du Z su L (ϕ) s − n 1 tˆk+1 2 − 2 U Xtˆ Xu du ≤ k k Zˆ k k − k kX=1 tk n 1 tˆk+1 2 − 1 2 n U (u tˆk)du U 2− t s 0 . ≤ k k Zˆ − ≤ 2k k | − | → kX=1 tk Observe finally that the limit of the second term in (37) is immediately provided by Proposition 4.4, which achieves the proof of (34). Let us now extend the correction formula (34) to any adapted controlled biprocess, that is to the class of biprocesses introduced in Definition 2.4. Using again Corollary 4.7, it is easy to check that, as an adapted Hölder path in ˆ , any such controlled biprocess is Itô-integrable when considered on anA interval⊗A I of finite Lebesgue measure. This puts us in a position to state the formula: Corollary 4.9. For all 0 s As another spin-off of Formula (38), we can finally derive an expression t XS 2 of the rough Stratonovich integral s Uu♯d u as the L (ϕ)-limit of “mean- value” Riemann sums. The result,R which emphasizes the analogy between the rough construction and the classical (commutative) Stratonovich inte- gral, can be stated as follows: Corollary 4.10. For all 0 s = Uti ♯δXtiti+1 1 X,1 X,2 ♭ + (δXti ti+1 ♯ ti )♯δXti ti+1 + ( ti ♯δXtiti+1 )♯δXti ti+1 + Utiti+1 ♯δXti ti+1 , 2 U U and observe that, with the notations of Section 2.3, we have U♭ ♯δX t t 2γ+1/2 X [U♭; 2γ ([s,t])] . k titi+1 ti ti+1 k ≤ | i+1 − i| k 1kN C2 Taking the sum over i and then letting ∆ tend to 0, we get by Proposition 4.4 that the sum in (39) converges in L|2(ϕ| ) to the right-hand side of (38), which leads us to the conclusion. 4.4. Proof of Proposition 4.4. When t = Ut Vt Wt, the convergence property (31) has been shown in the proofU of [7,⊗ Theorem⊗ 3.2]. However, since we want the formula to hold for general adapted triprocesses here, we n 3 need to exhibit additional controls. Let : [s,t] ⊗ be a sequence of n U → A adapted triprocesses such that u u 0 for every u [s,t], and fix a subdivision ∆ = (t ) of [s,t]. Then U set−U successively → Y := δX∈ , i i titi+1 S ( ) := (Y ♯ )♯Y (t t ) Id Γ Id ( ) ∆ U i Uti i − i+1 − i × q × Uti (tXi) ∆ ∈ and Sn ( ) := (Y ♯ n)♯Y (t t ) Id Γ Id ( n) . ∆ U i Uti i − i+1 − i × q × Uti (tXi) ∆ ∈ n n n n 3 n If t := ℓ Ln Ut,ℓ Vt,ℓ Wt,ℓ t⊗ , S∆( ) thus corresponds to U ≤ t ⊗ ⊗ ∈A U P n n S∆( )= Mi,ℓ , U n (tXi) ∆ ℓXL ∈ ≤ ti with M n := U n Y V n Y W n (t t )U n Γ (V n )W n , i,ℓ ti,ℓ i ti,ℓ i ti,ℓ − i+1 − i ti,ℓ q ti,ℓ ti,ℓ ON STOCHASTIC CALCULUS WITH RESPECT TO q-BROWNIAN MOTION 23 so n 2 n n S∆( ) L2(ϕ) = ϕ Mi1,ℓ1 (Mi2,ℓ2 )∗ . (40) k U k n n (tiX) ∆ (tiX) ∆ ℓ1 XLt ℓ2 XLt 1 ∈ 2 ∈ ≤ i1 ≤ i2 n n n n n n For more clarity, let us set Ui,ℓ := Uti,ℓ, Vi,ℓ := Vti,ℓ, Wi,ℓ := Wti,ℓ, and consider then the expansion n n ϕ Mi1,ℓ1 (Mi2,ℓ2 )∗ n n n n, n, n, = ϕ Ui1,ℓ1 Yi1 Vi1,ℓ1 Yi1 Wi1,ℓ1 Wi2,ℓ∗2 Yi2 Vi2,ℓ∗2 Yi2 Ui2,ℓ∗2 n n n n, n, n, (t t )ϕ U Y V Y W W ∗ Γ (V ∗ )U ∗ − i2+1 − i2 i1,ℓ1 i1 i1,ℓ1 i1 i1,ℓ1 i2,ℓ2 q i2,ℓ2 i2,ℓ2 n n n n, n, n, (t t )ϕ U Γ (V )W W ∗ Y V ∗ Y U ∗ − i1+1 − i1 i1,ℓ1 q i1,ℓ1 i1,ℓ1 i2,ℓ2 i2 i2,ℓ2 i2 i2,ℓ2 n n n n, n, n, ∗ ∗ ∗ +(ti1+1 ti1 )(ti2+1 ti2 )ϕ Ui1,ℓ1 Γq(Vi1,ℓ1 )Wi1,ℓ1 Wi2,ℓ2 Γq(Vi2,ℓ2 )Ui2,ℓ(41)2 . − − Step 1: Non-diagonal terms (i = i ). Observe first that if for instance 1 6 2 i1 < i2, we have, by combining Lemma 4.1 and Definition 4.2, n n n n, n, n, ϕ Ui1,ℓ1 Yi1 Vi1,ℓ1 Yi1 Wi1,ℓ1 Wi2,ℓ∗2 Yi2 Vi2,ℓ∗2 Yi2 Ui2,ℓ∗2 n n n n, n, n, = ϕ U Y V Y W W ∗ ϕ Y V ∗ Y U ∗ i1,ℓ1 i1 i1,ℓ1 i1 i1,ℓ1 i2,ℓ2 i2 i2,ℓ2 i2 Bti2 i2,ℓ2 n n n n, n, n, = (ti2+1 ti2 )ϕ Ui1,ℓ1 Yi1 Vi1,ℓ1 Yi1 Wi1,ℓ1 Wi2,ℓ∗2 Γq (Vi2,ℓ∗2 )Ui2,ℓ∗2 , − and with the same conditioning argument n n n n, n, n, ϕ Ui1,ℓ1 Γq(Vi1,ℓ1 )Wi1,ℓ1 Wi2,ℓ∗2 Yi2 Vi2,ℓ∗2 Yi2 Ui2,ℓ∗2 n n n n, n, n, = (t t )ϕ U Γ (V )W W ∗ Γ (V ∗ )U ∗ , i2+1 − i2 i1,ℓ1 q i1,ℓ1 i1,ℓ1 i2,ℓ2 q i2,ℓ2 i2,ℓ2 n n so that, going back to (41), one has ϕ Mi1,ℓ1 (Mi2,ℓ2 )∗ = 0. Similar argu- ments lead to the same conclusion when i2 < i1. Step 2: Diagonal terms (i1 = i2 = i). First, observe that with the same conditioning argument as above, decomposition (41) actually reduces to n n n n n n, n, n, ϕ Mi,ℓ1 (Mi,ℓ2 )∗ = ϕ Ui,ℓ1 YiVi,ℓ1 YiWi,ℓ1 Wi,ℓ2∗YiVi,ℓ2∗YiUi,ℓ2∗ 2 n n n n, n, n, ∗ ∗ ∗ (ti+1 ti) ϕ Ui,ℓ1 Γq(Vi,ℓ1 )Wi,ℓ1 Wi,ℓ2 Γq(Vi,ℓ2 )Ui,ℓ2 . − − Now, on the one hand, using (5) and the Cauchy-Schwarz inequality, n n n n, n, n, ϕ Ui,ℓ1 YiVi,ℓ1 YiWi,ℓ1 Wi,ℓ2∗YiVi,ℓ2∗YiUi,ℓ2∗ Y 4 U n V n W n W n V n U n ≤ k ik k i,ℓ1 kk i,ℓ1 kk i,ℓ1 kk i,ℓ2 kk i,ℓ2 kk i,ℓ2 k (t t )2 X 4 U n V n W n W n V n U n . ≤ i+1 − i k 1k k i,ℓ1 kk i,ℓ1 kk i,ℓ1 kk i,ℓ2 kk i,ℓ2 kk i,ℓ2 k On the other hand, using the definition of Γq(Vj1 ), n n n n, n, n, ∗ ∗ ∗ ϕ Ui,ℓ1 Γq(Vi,ℓ1 )Wi,ℓ1 Wi,ℓ2 Γq(Vi,ℓ2 )Ui,ℓ2 n n n n, n, n, = ϕ Ui,ℓ1 (δX)ti ,ti+1Vi,ℓ1 (δX)ti ,ti+1W i,ℓ1 Wi,ℓ2∗Γq(Vi,ℓ2∗)Ui,ℓ2∗ n, n n n n, n, U ∗U (δX) V (δX) W W ∗ Γ (V ∗) , i,ℓ2 i,ℓ1 ti ,ti+1 i,ℓ1 ti ,ti+1 i,ℓ1 i,ℓ2 L2(ϕ) q i,ℓ2 L2(ϕ) ≤ which, by (30), entails that n n n n, n, n, ϕ Ui,ℓ1 Γq(Vi,ℓ1 )Wi,ℓ1 Wi,ℓ2∗Γq(Vi,ℓ2∗)Ui,ℓ2∗ X 4 U n V n W n W n V n U n . ≤ k 1k k i,ℓ1 kk i,ℓ1 kk i,ℓ1 kk i,ℓ2 kk i,ℓ2 kk i,ℓ2 k 24 AURÉLIENDEYAANDRENÉSCHOTT Going back to (40), we have thus shown that 2 n 2 4 2 n n n S∆( ) L2(ϕ) X1 (ti+1 ti) Ui,ℓ Vi,ℓ Wi,ℓ , k U k ≤ k k − n k kk kk k (tXi) ∆ ℓXL ∈ ≤ ti and so we can assert that n 2 4 2 n 2 S ( ) 2 X (t t ) k ∆ U kL (ϕ) ≤ k 1k i+1 − i kUti k (tXi) ∆ ∈ 4 2 n 2 2 2 X1 (ti+1 ti) ti ti + sup u t s ∆ . ≤ k k − kU −U k u [s,t] kU k | − || | (tXi) ∆ ∈ ∈ 2 By letting n tend to infinity first, we can conclude that S ( ) 2 0 k ∆ U kL (ϕ) → as the mesh ∆ tends to 0. 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[13] Victoir, Nicolas: Levy area for the free Brownian motion: existence and non-existence. J. Funct. Anal. 208 (2004), no. 1, 107–121. ON STOCHASTIC CALCULUS WITH RESPECT TO q-BROWNIAN MOTION 25 [14] D. Voiculescu: Limit laws for random matrices and free products. Invent. Math. 104 (1991), no. 1, 201–220. (A. Deya) Institut Elie Cartan, University of Lorraine B.P. 239, 54506 Vandoeuvre- lès-Nancy, Cedex France Email address: [email protected] (R. Schott) Institut Elie Cartan, University of Lorraine B.P. 239, 54506 Vandoeuvre-lès-Nancy, Cedex France Email address: [email protected]