数理解析研究所講究録 1227 巻 2001 年 114-123

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MULTI-DIMENSIONAL QUANTUM AZ\’EMA MARTINGALESUWE FRANZABSTRACT. We introduce a multi-dimensional generalization of the quantum
Az\’ema martingale.

1. INTRODUCTION

We will study the process defined by the quantum stochastic differential equations(1.1)$\mathrm{d}X:(t)$(1.2)$\mathrm{d}V^{:}(t)$$=A_{\dot{l}}^{+}(t)+ \sum_{j,k,\ell=1}^{n}\int_{0}^{t}Cj_{\ell}^{k}X_{j}(s)\mathrm{d}\Lambda_{k}^{\ell}(s)$,,$=A_{:}(t)+ \sum_{j,k,\ell=1}^{n}\int_{0}^{t}\overline{C_{jk}^{u}.}V^{j}(s)\mathrm{d}\Lambda_{k}^{\ell}(s)$,are the creation, annihilation, and conversation processes on the
where
’
,
boson Fock space$A_{\dot{1}}^{+}$$A_{:}$$\Lambda \mathrm{j}$$\Gamma(L^{2}(\mathbb{R}_{+},\emptyset))$$A_{\dot{1}}^{+}$$(t)=A^{+}(1[0,t[\otimes e:)$,A(t) $=A(1[0,t[\otimes e:),$$\Lambda_{j}^{\dot{1}}(t)=\mathrm{A}(1[0,t[\otimes|e_{i}\rangle\langle e_{j}|)$,and .are complex coefficients.
-lleads to the quantum Azema martingales
The special case $n=1$ anddefined by Parthasarathy [Par90] to get aquantum version of the classical Azema
martingales introduced by Emery in [Eme89]. Emery proved that the classical
Az\’ema martingales have the chaotic representation property, i.e. the iterated
integrals$C_{u}^{jk}$$C_{11}^{11}=q$$Q^{(0)}(t)=1$,$Q^{(1)}(t)=M_{t}$,$Q^{(n)}(t)= \int_{0}^{t}Q^{(n-1)}(s)\mathrm{d}M_{\partial}$,$n=2,3$ ,$\ldots$, whereis the probability space on which$F$
is defined and the -algebra generated by the process.
Other martingales having this property are, e.g., the Brownian motion and thecompensated Poisson process. The Az\’ema martingale provided the first example
of amartingale having the chaotic representation property that is not aLevy
process. The problem of the classification of all normal martingales having thechaotic representation property is still open

are total in$L^{2}(\Omega, F, P)$$\mathrm{c}\mathrm{r}$$(\Omega,\mathcal{F},P)$$(M_{t})_{t\in \mathbb{R}}+$

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Uwe Franz

Parthasarathy’s [Par90] construction of the quantum Az\’ema martingles leadto anew proof of their chaotic representation property.
It was shown by Schiirmann [Sch91, Sch93] that the the quantum Az\’ema martingales arise as acomponent of aLevy process on an involutive bialgebra.
The goal of this paper is to construct multi-dimensional analogues of the quantum Az\’ema martingales and to show that they lead to classical processes having the chaotic representation property. For some results on classical multidimensional Az\’ema martingales, see [AE94, AE96].
Roughly speaking, Parthasarathy’s proof of the chaotic representation property
relies on three properties of the quantum Az\’ema martingale defined by$X(t)$$=A^{+}(t)+ \int_{0}^{t}(q-1)X(s)\mathrm{d}\Lambda_{s}$$V(t)$$=$$A(t)+ \int_{0}^{t}(q-1)V(s)\mathrm{d}\Lambda_{s}$,.

He shows that the symmetric linear combinations$Z(t)=X(t)+V(t)$ ,$t\in \mathbb{R}_{+}$

are self-adjoint(even boundedshows that they commute for all times, i.e.
$\mathrm{f}\mathrm{o}\mathrm{r}-1\leq q\leq 1$,$\mathrm{f}\mathrm{o}\mathrm{r}-1\leq q<1$). Furthermore, he$[Z(t), Z(s)]=0$

for all ,
$s$

cess

$t\in \mathbb{R}_{\vdash}$. These two properties imply that there exists aclassical pr0-

the same distribution as the joint spectral density of thecommuting family of self-adjoint operatorsin the vacuum
state. Then he shows that the iterated integrals of $Z(t)$ generate the same spacefrom the vacuum as the quantum Browian motion $Q_{t}=A^{+}(t)+A(t)$ . Since
we know that the Brownian motion has the chaotic representation property, thisallows one to show that the classical process
has this property.
In this paper we will look for the conditions on the coefficientsin thequantum stochastic differential equations (1.1) and (1.2), that guarantee thatthe multi-dimensional process defined by$(\tilde{Z}_{t})_{t\in \mathbb{R}}+\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{n}\mathrm{g}$$(Z(t))_{t\in \mathrm{R}}+\mathrm{e}\mathrm{v}\mathrm{a}1\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}$$(\tilde{Z}_{t})_{t\in \mathrm{R}}+\mathrm{a}1\mathrm{s}\mathrm{o}$$cj_{\ell}^{k}$$Z_{i}(t)=X_{i}(t)+V^{i}(t)$has similar properties. In the end we get amulti-dimensional version of Parthasarathy’s result by combining these conditions

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Multi-dimensional quantum Az\’ema martingales

Note that thedifferential equation$Z_{\dot{l}}(t)$can be defined directly by one single quantum stochastic
,

(1.3)

$Z_{i}(t)=A_{i}^{+}(t)+A:(t)+ \sum_{j,k,\ell=1}^{n}\int_{0}^{t}C_{j\ell}^{\iota k}Z_{j}(s)\mathrm{d}\Lambda_{k}^{\ell}(s)$if the coefficients satisfy$Cj_{\ell}^{k}=\overline{C\mathrm{j}_{k}^{\ell}}$

for all $1\leq i,j$ , ,$k$$\ell\leq n$.

2. MULTI-DIMENSIONAL AZ\’EMA PROCESSES AS NON-COMMUTATIVE L\’EvY
PROCESSES

Let

$B$

be the free algebra generated by Xi, as, $i,j=1$ ,$\Delta x:=x_{j}\otimes a_{\dot{l}}^{j}+1\otimes x:$,,$\Delta a_{j}^{j}=a_{k}.\cdot\otimes a_{j}^{k}$, . Setting$n$$\ldots$$\epsilon(x_{\dot{l}})=0$,$\epsilon(a\mathrm{j})=\delta_{\dot{\iota}j}$,as algebra homomorphisms, we: $Barrow B\otimes B$ , :and extending
get abialgebra. Taking the free -algebra generated by these elements, and, ::extendingas algebra homomorphisms, we get an
we haveinvolutive bialgebra . For the adjoints $v^{:}=(x:)^{*}$ and$\Delta$$B$$\epsilon$$arrow \mathbb{C}$$\mathrm{B}\sim$$*$$\Delta$$\tilde{B}arrow \mathbb{C}$$\tilde{B}arrow\tilde{B}\otimes\tilde{B}$$\epsilon$$\tilde{B}$$\dot{\mathrm{W}}_{\dot{1}}=(a_{j}^{1})^{*}$$\Delta v^{:}=qj$$\otimes bj$$+1\otimes v^{:}$,.
$\Delta b!|$$=b_{\dot{\iota}}^{k}\otimes\dot{\nu}_{k}$,$\epsilon(v^{:})=0$,$\epsilon(\dot{\theta}_{\dot{l}})=\delta_{\dot{l}j}$.

Recall the definition of aL\’evy process on an involutive bialgebra.
be aan involutive bialgebra. Aquantum
Definition 2.1. [Sch93] Leton over some quantum probability space $(A, \Phi)$
stochastic process
is called aLivy process on the involutive bialgebra , if the following four conditions are satisfied.
1. (Increment property) We have$(B, \Delta,\epsilon)$$\{j,t\}_{0\leq\delta\leq\iota\leq\tau}$

$B$

$B$

$j_{\tau s}\star j_{st}$$j_{tt}$$=j_{rt}$$=\epsilon 1_{A}$for all $0\leq r\leq s\leq t\leq T$,
for all $0\leq t\leq T$.

2. (Independence of increments) The family, i.e.

$\{j,t\}_{0\leq’\leq t\leq T}$

is independent$(\mathrm{w}.\mathrm{r}.\mathrm{t}$.

$\Phi)$(i)$\Phi(j_{s_{1}t_{2}}(b_{1})\cdots j_{s_{n}t_{n}}(b_{n}))=\Phi(j_{\epsilon_{1}t_{1}}(b_{1}))\cdots$ $\Phi(j_{\epsilon_{n}t_{n}}(b_{n}))$, and all $0\leq s_{1}\leq t_{1}\leq s_{2}\leq\cdots\leq t_{n}\leq T$ and(ii)
for all , $b_{2}\in B$ and all , , ,, and]s, $t[\cap]s’$ ,, $0\leq s’\leq t$.
3. (Stationarity of increments) The distribution
only on the difference $t-s$ .

for all$b_{1}$,$\ldots$,$b_{n}\in$$B$$[j_{\epsilon t}(b_{1}),j_{s’t’}(b_{2})]=0$$T$$\leq T$$s$$b_{1}$

$t$

$s’$

$t’$

with $0\leq s\leq t\leq$$t’[=\emptyset$$\varphi_{st}=\Phi$$\circ j_{st}$of$j_{st}$depends

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Uwe Franz

4. (Weak continuity) The quantum random variables.
distribution for t$j_{st}$converge to$j_{ss}$in$[searrow] s$

is aLevy process on an involutive bialgebra, then the marginalIfform aconvolution semi-group of states, i.e.
distributions
for all , . It can be shown that there exists auniqueandsuch thathermitian linear functional :, and. Furthermore, is conditionally positive, i.e. positive on
kills the unit of , i.e. $\psi(1_{B})=0$ . Such functionals are called generators.
, we can define aconvolution semiConversely, given agenerator :and reconstruct aLevy process on from it.
group of states by
Let us briefly describe how one can construct arealization of the Levy pr0cess on aboson Fock space. By aGNS-type construction one can complete a, i.e. atriple consisting of aunital
generator to aSchiirmann triple-representation of on some pre-Hilbert space $D$ , alinear map : $Barrow D$that satisfies$\{j_{st}\}_{0\leq s\leq t\leq T}$$\varphi_{t-s}=\varphi_{st}=\Phi\circ j_{st}$$\varphi_{0}=\epsilon$$s$$\varphi_{s}\star\varphi_{t}=\varphi_{s+t}$$\psi$$\psi$$+\cdots$$t$$Barrow \mathbb{C}$$=\epsilon+t\psi$$\varphi_{t}=\exp_{\star}t\psi$$+ \frac{t^{2}}{2}\psi\star$$\mathcal{B}^{0}=\mathrm{k}\mathrm{e}\mathrm{r}\epsilon$$\psi$$B$$Barrow \mathbb{C}$$\psi$$B$$\varphi_{t}=\exp_{\star}t\psi$$(\rho, \eta, \psi)$$\psi$$*$$B$$\rho$(21)$\mathrm{r}]\{\mathrm{a}\mathrm{b}$$\eta$

and ahermitian linear functional(2.2)$\mathrm{r}]\{\mathrm{a}\mathrm{b}$)for all a, b $\in B$ ,) $=\rho(a)\eta(b)+\eta(a)\epsilon(b)$ ,$\psi$:B$arrow \mathbb{C}$that satisfies$=\epsilon(a)\psi(b)+\langle\eta(a^{*}), \eta(b)\rangle+\psi(a)\epsilon(b)$for all , $b\in B$ .

,$a$

The realization of the L\’evy process associated to on the boson Fock spaceis then given as solution of the quantum stochastic differential
equations$\psi$$\Gamma(L^{2}(\mathbb{R}_{+},\overline{D}))$(2.3)$jst(b)= \mathrm{e}(\mathrm{b})\mathrm{i}\mathrm{d}+(\int_{s}^{t}j_{s\tau}\otimes \mathrm{d}I_{\tau})\Delta(b)$where the integrator$\mathrm{d}I$for all $b\in B$ ,,is given by.

$\mathrm{d}I_{t}(b)=\mathrm{d}\Lambda_{t}(\rho(b)-\epsilon(b)\mathrm{i}\mathrm{d})+\mathrm{d}A_{t}^{+}(\eta(b))+\mathrm{d}A_{t}(\eta(b^{*}))+\psi(b)\mathrm{d}t$

For details and proofs see [Sch93].
Set$R_{k\ell}^{ij}=C_{k\ell}^{ij}+\delta_{ij}\delta_{k\ell}$and define aSchiirmann triple$(\rho, \eta, \psi)$

on , acting on$\tilde{B}$

, by

,

,

$\rho(x_{i})=0$$\eta(a_{j}^{i})=\eta(b_{j}^{i})=0$,$\eta(x_{i})=e_{i}$,$\psi(a_{j}^{i})=0$,$\psi(x_{i})=0$,$\rho(a_{j}^{i})e_{\ell}=\sum_{k=1}^{n}R_{jl}^{ik}e_{k}$$\alpha$$\eta(v^{i})=0$,

118
Multi-dimensional quantum Akma martingales, are abasis of . There exists aunique Schiirmann triple having
where ,
these values on the generators of . The quantum stochastic differential equations
for the associated L\’evy process are$e_{1}$$\alpha$$e_{n}$$\ldots$$\tilde{B}$:1st$(a_{j}^{i})$$=$$j_{st}(x:)$$=$,$\delta_{\dot{l}j}\mathrm{i}\mathrm{d}+\int_{s}^{t}\sum_{k,p,q}j_{s\tau}(a_{k}^{\dot{1}})(R_{jq}^{kp}-\delta_{kj}\delta_{pq})\mathrm{d}\Lambda_{p}^{q}(\tau)$$A_{\dot{1}}^{+}$$(t)-A_{\dot{l}}^{+}(s)+ \int_{s}^{t}\sum_{j}j_{s\tau}(x_{j})(R_{\dot{|}q}^{jp}-\delta_{\dot{l}j}\delta_{pq})\mathrm{d}\Lambda_{p}^{q}(\tau)$,

Comparing the second equation with (1.1), we see that we have$j_{0t}(x:)=j_{t}(x:)=X_{\dot{1}}(t)$,$j_{0t}(v^{:})=j_{t}(v^{:})=V^{:}(t)$.

The theory of L\’evy processes now implies that the domains of the,obtained by solving the quantum stochastic differential equations (1.1) and (1.2), where $N$ denotes the number operator on the boson Fockcontainspace.
Note finally that we can also define the same L\’evy process as asolution of the
backwards quantum stochastic differential equations$X_{\dot{1}}(t)$$V_{\dot{l}}(t)$$\bigcap_{a\in \mathrm{R}_{+}}\mathrm{d}\mathrm{o}\mathrm{m}\alpha^{N}$(2.4)$j_{st}(a_{j}^{\dot{1}})$$=$$\delta_{\dot{|}j}\mathrm{i}\mathrm{d}+\int_{s}^{t}\sum_{k,p,q}(R_{kq}^{\dot{|}p}-\delta_{\dot{|}k}\delta_{pq})\mathrm{d}\Lambda_{p}^{q}(\tau)j_{\tau t}(a_{j}^{k})$(2.5)$j_{\epsilon t}(_{X:})$$=$$\int_{s}^{t}\sum_{j}\mathrm{d}A^{+}.\cdot(\tau)j_{\tau t}(a_{\dot{1}}^{j})$,,cf. [Sch93].

3. WHEN ARE OUR MULTI-DIMENSIONAL AZ\’EMA MARTINGALESCOMMUTATIVE?
In this section we find conditions on the coefficients that quarantee that theprocesses commute. In the one-dimensional case[Par90], no conditions were necessary, but the commutativity already followed from the form of the quantum
stochastic differential equations.
Proposition 3.1.

(a): The$rwo$-sided ideal Iof B generated by the elements$i,j=1$ ,$x:x_{j}-XjXi$ ,$a_{\dot{1}}^{k}x_{j}+x:a_{j}^{k}-a_{\mathrm{j}}^{k}x:-x_{j}a_{\dot{1}}^{k}$$a_{\dot{1}}^{k}a_{j}^{\ell}-a_{j}^{k}a_{\dot{1}}^{\ell}$

is a coideal,,$\mathrm{i}\mathrm{j}$$\ldots$,$n$, $=1$ ,,, , $\ell=1$ ,,$k$$\ldots$$\mathrm{i}\mathrm{j}$

$k$

$n$

$\ldots$$n$

119
Uwe Franz(b): The twO-sided ideal Iof$\tilde{B}$generated by the elements$(x:+v^{:})(x_{j}+v^{j})-(x_{j}+v^{j})(x:+v^{:})$$a_{i}^{k}x_{j}+x:a_{\mathrm{j}}^{k}.-a_{j}^{k}.x:-x_{j}a_{\dot{1}}^{k}$$b_{k}^{i}x_{j}+x:b_{k}^{1}-b_{k}^{f}x:-x_{j}b_{k}^{\dot{1}}$$a_{i}^{k}a_{j}^{\ell}-a_{j}^{k}a_{\dot{*}}^{\ell}$,$a_{\dot{l}}^{k}\dot{\theta}_{\ell}-a_{j}^{k}bi$,$i,j=1$ ,$+a_{\dot{l}}^{k}v^{j}+v^{\dot{l}}a_{j}^{k}-a_{j}^{k}v^{:}-v^{j}a_{\dot{1}}^{k}$$+b_{k}^{}v^{j}+v^{:}b_{k}^{J}-\theta_{k}v^{:}.-v^{j}b_{k}^{\dot{l}}$,$b_{k}^{}a_{j}^{\ell}-b_{k}^{j}a_{}^{\ell}$

,

,

,

,

$b_{k}^{\dot{l}}b_{\ell}^{J}-\dot{\theta}_{k}bi$$\ldots$,$n$, $k=1$ ,,$=1$ ,$i,j$ ,,
$i,j$ , , $\ell=1$ ,,$i,j$$\ldots$$n$$\ldots$

$n$

$k$

$k$

$\ldots$$n$

is a coideal

Proof. We only show (a), the proof of (b) is similar.
We have to show$\Delta(\mathrm{I})\subseteq \mathrm{I}\otimes B+B\otimes \mathrm{I}$.

and $\epsilon(1)=\{0\}$ . It is sufficient, to verify this on the algebraic generators of X.
We get$\Delta(_{X_{i}x_{j}-x_{j^{X}:}})$$=$$(x_{k}\otimes a_{i}^{k}+1\otimes x:)(x_{\ell}\otimes a_{j}^{\ell}+1\otimes x_{j})-(x_{\ell}\otimes a_{j}^{\ell}+1\otimes x_{j})(x_{k}\otimes a_{i}^{k}+1\otimes x_{i})$$=$$x_{k}x_{\ell}\otimes(a_{\dot{l}}^{k}a_{j}^{\ell}-a_{j}^{k}a_{\dot{l}}^{\ell})+x_{k}\otimes(a_{\dot{l}}^{k}x_{j}+x:a_{j}^{k}-a_{j}^{k}x_{\dot{1}} -x_{j}a_{\dot{l}}^{k})+1\otimes \mathrm{C}(x_{i}x_{j}-x_{j}x:)$,similarly$\Delta(a_{i}^{k}x_{j}+x_{i}a_{j}^{k}-a_{j}^{k}x:-x_{j}a^{k}.\cdot)$$=$$a_{\ell}^{k}x_{m}\otimes(a_{\dot{l}}^{\ell}a_{j}^{m}-a_{j}^{\ell}a_{\dot{\iota}}^{m})+x_{m}a_{l}^{k}\otimes(a_{i}^{m}a_{j}^{\ell}-a_{j}^{m}a_{i}^{\ell})$$+a_{\ell}^{k}\otimes$$(a_{i}^{\ell}x_{j} \dagger x_{i}a_{j}^{\ell}-a_{j}^{\ell}x:-x_{j}a_{\dot{1}}^{\ell})$,and finally$=$$\Delta(a_{i}^{k}a_{j}^{\ell}-a_{j}^{k}a_{i}^{\ell})$$a_{f}^{k}a_{s}^{\ell}\otimes(a_{\dot{\iota}}^{rs}a_{j}-a_{j}^{f}a_{i}^{s})$.
$\square$Corollary 3.2.

(a): If the coefficients$R_{j\ell}^{ik}$satisfy$Rj_{\ell}^{k}=R_{\ell j}^{ik}$and$R_{j\ell}^{ks}R_{j\ell}^{lk}=R\mathrm{j}_{\ell}^{k}R_{j\ell}^{\dot{l}k}$

for all $i,j$ , then we have$[X_{\dot{\iota}}(s), X_{j}(t)]=0$(b): If the coefficients$R\mathrm{j}_{\ell}^{k}$satisfy$Rj_{\ell}^{k}=Ri_{j}^{k}$

and

,

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Multi-dimensional quantum Azma martingales

for all i,j, then we have$[Z_{\dot{l}}(s), Z_{j}(t)]=0$

appearing in the quantum
Remark 3.3. Expressed in terms of the coefficientsstochastic differential equations (1.1) and (1.2) the conditions become$C_{j\ell}^{\dot{l}k}$and

Proof.

$\square$

4. WHEN ARE OUR MULTI-DIMENSIONAL AZ\’EMA MARTINGALES BOUNDED?
Lemma 4.1. Suppose that there exists a constant $0\leq M<1$ such that$||( \sum_{k,\ell}\overline{a_{k}}b_{\ell}Rj_{\ell}^{k)_{1\leq k,\ell\leq n}1}|\leq M||a||||b||$for alltio$ns$$a=(\begin{array}{l}a_{1}\vdots a_{n}\end{array})$,$b=(\begin{array}{l}b_{1}\vdots b_{n}\end{array})$$\in \mathbb{C}^{n}$. Then the operators$j_{st}(a\mathrm{j})$are contrac-.

Proof. Using the first fundamental lemma [Par92, Proposition 25.1], we get the

integral equations$\langle \mathcal{E}(f),j_{t}(a\mathrm{j})\mathcal{E}(g)\rangle$$=$$\delta_{j}.\cdot\langle \mathcal{E}(f), \mathcal{E}(g)\rangle$$+ \int_{0}^{t}\sum_{k,q,p}\langle \mathcal{E}(f),j_{f}(a_{k}^{\dot{l}})\mathcal{E}(g)\rangle(R_{jq}^{kp}-\delta_{kj}\delta_{pq})\overline{f_{p}(t)}g_{q}(t)\mathrm{d}s$

for the matrix elements of the operatorsset$j_{\epsilon t}(a_{j}^{\dot{1}})$, where$f,g\in L^{2}(\mathrm{R}_{+}, \emptyset)$$A_{f,g}(t)=\langle \mathcal{E}(f),j_{t}(aj)\mathcal{E}(g)\rangle$

and,$R_{f,g}(t)$$=$$( \sum_{p,q}R\mathrm{j}_{q}^{p}\overline{f_{p}(t)}g_{q}(t))_{1\leq:i\leq n}$$\overline{R}_{f,g}(t)$$=$$( \sum_{p,q}(R_{jq}^{1p}-\delta_{\dot{l}j}\delta_{pq})\overline{f_{p}(t)}g_{q}(t))_{1\leq\dot{\iota},j\leq n}$,. If we121
Uwe Franz

then we get the matrix-valued differential equation$\frac{\mathrm{d}}{\mathrm{d}t}A_{f,g}(t)=A_{f,g}(t)\overline{R}_{f,g}(t)$,with the initial condition$A_{f,g}(0)=(\langle \mathcal{E}(f), \mathcal{E}(g)\rangle\delta_{\dot{l}j})_{1\leq i,j\leq n}$.

The unique solution of this equation is$A_{f,g}(t)=A_{f,g}(0) \exp(\int_{0}^{t}\overline{R}_{f,g}(s)\mathrm{d}s)$,and therefore we get(4.1)$\langle \mathcal{E}(f),j_{st}(a_{j}^{i})\mathcal{E}(g)\rangle=$$( \exp(\int_{s}^{t}R_{f,g}(s)\mathrm{d}s))_{ij}\exp(\int_{0}^{s}\langle f(\tau), g(\tau)\rangle \mathrm{d}\tau+\int_{t}^{\infty}\langle f(\tau),g(\tau)\rangle \mathrm{d}\tau)$

Using this identity, we get the desired estimate.

$\square$Proposition 4.2. Suppose that there exists a constant $0\leq M<1$ sttch that$||( \sum_{k,\ell}\overline{a_{k}}b_{\ell}R_{j\ell}^{ik})_{1\leq k,\ell\leq n}||\leq M||a||||b||$for all $a=$$(\begin{array}{l}a_{1}\vdots a_{n}\end{array})$$Z_{i}(t)=X_{i}(t)+V^{i}(t)$, $b=$$(\begin{array}{l}b_{1}\vdots b_{n}\end{array})$$\in \mathbb{C}^{n}$are bounded for all. then the operators$t\leq t$$X_{i}(t)$,$V^{i}(t)$and.

Proof Using the first fundamental lemma [Par92, Proposition 25.1] and the back-ward stochastic differential equation (2.5) for $X_{i}(t)=j_{t}(x_{i})$ , we get$\langle \mathcal{E}(f), X_{i}(t)\mathcal{E}(g)\rangle=\int_{0}^{t}\langle \mathcal{E}(f),j_{st}(a_{i}^{j})\mathcal{E}(g)\rangle\overline{f_{j}(s)}\mathrm{d}s$

Using (4.1), this implies that theand the.
$Z_{i}(t)$[FSSOI]$X_{i}(t)$are bounded, and therefore also the$V_{i}(t)$$\square$122
Multi-dimensional quantum Az&na martingales

5. Chaos COMpLETENESS
Define the iterated integrals$Q_{:_{1},\ldots,:_{r}}(t)= \int_{0\leq t_{1}\leq\cdots t_{r}\leq t}\mathrm{d}Q_{\dot{l}_{1}}\cdots \mathrm{d}Q:_{1}$,where $Q_{i}(t)=A_{\dot{l}}^{+}(t)+A:(t)$ , and$I_{1}.

\cdot,\ldots,i_{r}(t)=\int_{0\leq t_{1}\leq\cdots t_{r}\leq t}\mathrm{d}Z_{\dot{1}1}\ldots \mathrm{d}Z_{i_{1}}$then we have
$Q_{\dot{1}1,\ldots,r}:(t)\Omega=I_{i_{1},\ldots,:,}(t)\Omega$,, $i_{r}\in\{1, \ldots, n\}$ , and $t\geq 0$ .
, ,for all
This implies that we can copy Parthasarathy’s proof of the chaos completenesswhenever the define abounded and commutative self-adjoint operator process.
$r\in \mathrm{N}$$i_{1}$$\ldots$$Z_{\dot{1}}$Theorem 5.1. Suppose that the coefficients satisfy ...
Then there eists a classical process($(\tilde{Z}_{1}(t), \ldots,\tilde{Z}_{n}(t)))_{t\in \mathrm{R}}+having$

joint distribution as the quantum stochastic process(

the same$(Z_{1}(t), \ldots, Z_{n}(t)))_{t\in \mathbb{R}}+and$

havin the chaotic representation property, $i.e$ . the iterated integrals$\tilde{I}_{1,\ldots r}.\cdot,:(t)=\int_{0\leq t_{1}\leq\cdots t_{r}\leq t}\mathrm{d}\tilde{Z}_{\dot{l}_{1}}\cdots \mathrm{d}\tilde{Z}_{\dot{l}_{1}}$and , $\ldots.i_{r}\in\{1, \ldots, n\}$ are total in the -space over the underlying
withprobability space (where we assume that the -algebra is generated by the process).
$r\in \mathrm{N}$$L^{2}$$i_{1}$$\sigma$

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.

de structure pour des martingales vectorielles. In
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[A\’E96] S. Attal and M. \’Emery. Martingales d’Az\’ema bidimensionnelles. Astirisque, (236)
21, 1996. Hommage \‘a P. A. Meyer et J. Neveu.
[Eme89] M. Emery. On the Azema martingales. In Siminaire de Probabilitis XXIII, volume1372 of Lecture Notes in Math. Springer-Verlag, Berlin, 1989.
[FSSOI] U. Franz, R. Schott, and M. Schiirmann. Levy processes and Brownian motion on
braided spaces. Preprint, 2001.
[Par90] K.R. Parthasarathy. Az\’ema martingales and quantum stochastic calculus. In R.R. Bahadur, editor, Proc. R.C. Bose Memorial Symposium, pages 551-569. Wiley Eastern,
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[Par92] K.R. Parthasarathy. An Introduction to Quantum Stochastic Calculus. Birkhauser,
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$\text{\’{E}}_{\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{y}}$$\text{\’{E}}_{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}}$123
Uwe Franz[Sch91] M. Schiirmann. The Az\’ema martingales as components of quantum independent increment processes. In J. Az&na, P.A. Meyer, and M. Yor, editors, Siminaire de Prvbabilitis XXV, volume 1485 of Lecture Notes in Math. Springer-Verlag, 1991.
[Sch93] M. Schiirmann. White Noise on Bialgebras, volume 1544 of Lecture Notes in Math.
Springer-Verlag, Berlin, 1993.

INSTITUT P\"UR MATHEMATIK UND INFORMATIK, ERNST-MORITZ-ARNDT-UNIVERSIT\"AT GREIFSWALD, JAHNSTRASSE 15a, D-l 487 GREIFSWALD, GERMANY, EMAIL: FRANZ@MAIL.UN1GREIFSWALD.DE