数理解析研究所講究録 1227 巻 2001 年 154-169

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Functional It\^o Formula for Quantum
Semimartingales
UN CIG JI 1
DEPARTMENT Of BUSINESS ADMINISTRATIONHANYANG UNIVERSITYSEOUL, 133-791 KOREA
ucji@nuri.net

Introduction

1

The aim of this paper is to give apartial answer to the problem of deriving afunctional, wherequantum Ito formula for (unbounded) semimartingales, i.e., an It\^o formula forin acertain class of quantum semimartingales.
Since aquantum stochastic calculus ([21], [27]; also [24] for the white noise approach)of It\^o type first formulated by Hudson and Parthasarathy [12], the stochastic integral
representations of quantum martingales have been studied by many authors, see [10] $[11]$ .
In particular, Parthasarathy and Sinha [28] established astochastic integral representationof regular bounded quantum martingales in (Boson) Fock space with respect to the basic
martingales, namely the annihilation, creation and number processes. Anew proof of theParthasarathy and Sinha representation theorem has been discussed by Meyer in [22] inwhich he gives the special form of the number operator coefficient. The representation
theorem has been extended to regular bounded semimartingales by Attal [1] and the Itoformula for products of regular semimartingales belonging to acertain class has been
discussed which yields aquantum It\^o formula for polynomial [2]. In [30], by VincentSmith, afunctional quantum Ito formula for regular bounded semimartingales has been
widely studied with closed form of the It\^o correction term. For more discussions offunctional quantum It\^o formula, we refer to [4], [13].
In [16], we extended the quantum stochastic integral to awider class of adapted
quantum stochastic processes on Boson Fock space and aquantum stochastic integral
representation theorem has been proved for aclass of unbounded semimartingales. MO,
tivated by results in [16] and [30], we discuss afunctional quantum It\^o formula fora(unbounded) semimartingale such that andwhere is an entire function
certain conditions. Our approach is based on riggings of Fock space which areapplied in many fields of mathematics and mathematical physics, e.g., [3], [5], [8], [19],[20], [29], we also refer to [9], [18], [23] for nuclear riggings which are the fundamental
frameworks of white noise analysis.
$f(_{-}^{-}-)$$—\mathrm{i}\mathrm{s}$$)$$f(_{-}^{-}-)$$f$$\mathrm{a}\mathrm{n}\mathrm{d}---\mathrm{i}\mathrm{s}$$—\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{y}$lThis work was supported by the Brain Korea 21 Project

$f$

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2Riggings of Fock Spacebe the real Hilbert space of square integrable functions onwith norm
induced by the inner product , . The complexification ofis$H$
denoted bywhose norm is also denoted by. The (Boson) Fock space $??\equiv\Gamma(H)$over $H$ is defined byLet$H_{\mathbb{R}}=L^{2}(\mathbb{R}_{+}, dt)$$[0, \infty)$$\mathbb{R}_{+}=$$|\cdot|_{0}$$\langle\cdot$$\cdot\rangle$$H_{\mathbb{R}}$$|\cdot|_{0}$$\mathcal{H}=\{\phi=(f_{n})_{n=0}^{\infty}|f_{n}\in H^{\otimes n}\wedge$

where$H^{\otimes n}\wedge$

for all $n\geq 0$ and$||_{0}<\infty\}$$||\phi$is the -fold symmetric tensor power of $H$ and the norm$n$$||\phi$$||_{0}^{2}= \sum_{n=0}^{\infty}n!|f_{n}|_{0}^{2}<\infty$$||\cdot||_{0}$:is defined by.

the canonical -bilinear form on
defined through.
We denote by
be the -domain offor each $p\geq 0$ .
Let $N$ be the number operator and let. More precisely, for any
Then
is aHilbert space with norm$\mathbb{C}$$\langle\langle\cdot, \cdot\rangle\rangle$$\mathcal{H}$$\mathcal{G}_{p}$$||\cdot$$\mathcal{G}_{p}$$||\phi$$\langle\cdot, \cdot\rangle$$e^{pN}$$\mathcal{H}$$||_{p}=||e^{pN}\cdot||_{0}$$||_{p}^{2}= \sum_{n=0}^{\infty}n!e^{2pn}|f_{n}|_{0}^{2}$,$\phi$$p\underline{>}0$$=(f_{n})\in \mathcal{G}_{p}$.

(1)Then we naturally come to$\mathcal{G}\equiv \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}$$\lim \mathcal{G}_{p}\subset\cdots\subset \mathcal{G}_{q}\subset\cdots\subset \mathcal{G}_{p}\subset\cdots$$parrow\infty$$...\subset \mathcal{G}_{0}=74$$\subset\cdots\subset \mathcal{G}_{-p}\subset\cdots\subset \mathcal{G}_{-q}\subset\cdots\subset \mathcal{G}^{*}$,

is aand , respectively. Note thatare strong dual spaces of
where
and
countable Hilbert space equipped with the Hilbertian norms defined in (1) andis also denoted by , , and. The canonical -bilinear form on
we have$\mathcal{G}^{*}$$\mathcal{G}_{-p}$$\mathcal{G}_{p}$$\mathcal{G}$$\mathcal{G}$$\mathcal{G}^{*}=$$\mathbb{C}$$\mathcal{G}^{*}\cross \mathcal{G}$$\mathrm{i}\mathrm{n}\mathrm{d}\lim_{parrow\infty}\mathcal{G}_{-p}$$\langle\langle\Phi, \phi\rangle\rangle=\sum_{n=0}^{\infty}n!\langle F_{n}, f_{n}\rangle$,$\langle\langle\cdot$$\Phi=(F_{n})\in \mathcal{G}^{*}$,$(j)=(f_{n})\in \mathcal{G}$$\cdot\rangle\rangle$.

Moreover the Schwartz inequality takes the form:$|\langle\langle\Phi, \phi\rangle\rangle|\leq||\Phi$$||_{-p}||\phi||_{p}$

.

,
and
It is noted that for any.
for any ,, whereandFor each $\xi\in H$ , we write$B$ .
on
For notational convenience, we write $43=\xi[0,t]$ andwe have the decomposition$p\in \mathbb{R}$$p$$e^{-pN}\mathcal{G}_{-p}=\mathcal{H}$$e^{pN}\mathcal{H}=\mathcal{G}_{-p}$. Moreover,$e^{pN}\mathcal{G}_{q}=\mathcal{G}_{q-p}$$q\in \mathbb{R}$$\xi_{B}=\xi\chi_{B}$$B\subset \mathbb{R}_{+}$$\chi_{B}$is the indicator function
for any $t>0$ . Then$\xi[t=\xi[t,\infty)$$H=H_{s]}\oplus H[s,t]$ce$H[t,$$0<s<t<\infty$ ,where $H_{s]}=\{\xi_{s]}|\xi\in H\}$ , $H[s,t]$ $=\{\xi[s,t]|\xi\in H\}$ and $H[t=\{\xi[t|\xi\in H\}$ . Put$\mathcal{H}_{s]}=\Gamma(H_{s]})$,$\mathcal{H}_{[S,t]}=\Gamma(H[s,t])$and$f\ell[t=\Gamma(H[t)$.

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Then we have the identification$\mathcal{H}$$=?t_{s]}\otimes \mathcal{H}_{[s,t]}\otimes?t_{[t}$via the following decomposition:$\phi_{\xi}=\phi_{\xi_{s\mathrm{l}}}\otimes\phi_{\xi_{[s,t]}}\otimes\phi_{\xi_{\mathrm{l}t}}$,$\xi\in H$,

is the exponential vector of $\xi\in H$ . Moreover, for any$0<s<t<\infty$ , we have
,

where

$\phi_{\xi}=(\xi^{\otimes n}/n!)$$p\in \mathbb{R}$and$\mathcal{G}_{\mathrm{P}}=\mathcal{G}_{p;s]}\otimes \mathcal{G}_{pj[s,l]}\otimes \mathcal{G}_{pj[t}$where$\mathcal{G}_{pjS]}=\mathcal{G}_{p}\cap \mathcal{H}_{s]}$,$\mathcal{G}_{pj[t]}"=\mathcal{G}_{p}\cap \mathcal{H}[s,t]$,$\mathcal{G}_{pj[t}=\mathcal{G}_{p}\cap \mathcal{H}$[$t$

and their completion for $p\leq 0$ .

3Operators on Fock Spacebe the space of all bounded linear operators from alocally convex Iinto
Let
another locally convex space . Let , be non-negative integers. Then for each
is definedwith kernel
the integral kernel operatorby$\mathcal{L}(X, \mathfrak{Y})$$K_{l,m}\in$

$m$

$l$

$\mathfrak{Y}$$K_{l,m}$$—_{l,m}(K_{l,m})\in \mathcal{L}(\mathcal{G}, \mathcal{G})$$\mathcal{L}(H^{\otimes m}, H^{\otimes l})$$\phi=(f_{n})\in \mathcal{G}$$–_{l,m}-(K_{l,m}) \phi=(\frac{(n+m)!}{n!}(K_{l,m}\otimes I^{\otimes n}f_{n+m})_{\epsilon \mathrm{y}\mathrm{m})},$In this case, we have for any$p\in \mathbb{R}$, $q>0$ and.

$\phi\in \mathcal{G}$$||_{-l,m}^{-}-(K_{l,m}) \phi||_{p}\leq C(e^{pl-(\mathrm{p}+q)m+q/2})l^{l/2}m^{m/2}(\frac{e^{q/2}}{qe})^{(l+m)/2}||\phi||_{p+q}$,. Moreover, the integral
where $C\geq 0$ satisfies that $|K_{l,m}f|_{0}\leq C|f|_{0}$ for any
has aunique extension to acontinuous linear operator fromkernel operator
itself (see [14], [15]).
for any $\xi\in H$ . For$\eta\in
H$ and let
be defined byLet, i.e.,, whereis the adjoint operator of
simple notation, we identify$t\geq0$, we put. For eachfor all$f\in H^{\otimes m}$$—\iota_{m},(K_{l,m})$$g*\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{o}$$K_{\eta}(\xi)=\langle\eta, \xi\rangle$$K_{\eta}\in \mathcal{L}(H, \mathbb{C})$$\eta=K_{\eta}=K_{\eta}^{*}$$K_{\eta}$$K_{\eta}^{*}$$a\in \mathbb{C}$$K_{\eta}^{*}(a)=a\eta$$A_{t-0,1}=--(\chi_{t]})$,$A_{t}^{*}=---_{1,0}(\chi_{t]})$,$\Lambda_{t-1,1}=--(\chi_{t]})$,(2)

and for the definition of , the indicator function is considered as the
where
,
and
for any $\xi\in H$ . For each
multiplication operator on $H$ , i.e.,
respectively.
operator,
are called the annihilation operator and the creation
We now mention the following Fock expansion theorem. For the proof, see [17].
$\Lambda_{t}$$\chi_{t]}\equiv\chi[0,t]$$t\in \mathbb{R}_{+}$$\chi_{t]}(\xi)=\xi_{t}]$Theorem 1Let ,
$p$

. For
$m\geq0$ , such that, ,$q\in \mathbb{R}$$K_{l,m}\in \mathcal{L}(H^{\otimes m}, H^{\otimes l})\wedge\wedge$$any—\in \mathcal{L}(\mathcal{G}_{p}, \mathcal{G}_{q})$$A_{t}$$A_{t}^{*}$

there exists a unique family of operators$l$$—= \sum_{l,m,n=0}^{\infty}\frac{(-1)^{n}}{n!}---n+l,n+m(I^{\otimes n}\otimes K_{l,m})$where the series converges in$\rho^{f}/(-r\log\rho)<1$ and $\rho^{s}/(-s\log\rho)<1$ .

$\mathcal{L}(\mathcal{G}(p\vee q)+f’ \mathcal{G}_{q-S})$,

for any $r>0$ and $s>0$ satisfying

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For agiven entire function , let
be the class of continuous linear operators
satisfying that for any $p\geq 0$ there exist $q\geq 0$ , $M\geq 0$ and $0<\gamma<1$ such that$f$

$A_{f}$$—\mathrm{i}\mathrm{n}$$\mathcal{L}(\mathcal{G}, \mathcal{G})$$\frac{|f^{(n)}(0)|||_{-}^{-n}-\phi||_{p}}{n!}\leq M\gamma^{n}||\phi||_{q}$Proposition 2For $any—\in A_{f}$ , we can define

,

.

,

$n\geq 0$$f(_{-}^{-}-)$

as a continuous operator on$f(_{-}^{-}-)= \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}---n$$\phi\in \mathcal{G}$(3)$\mathcal{G}$

.

by
(4)

Proof. By definition of, for any $p\geq 0$ there exist $q\geq 0$ , $M\geq 0$ and $0<\gamma<1$such that (3) holds. Therefore, for any$A_{f}$$\phi\in \mathrm{C}\mathrm{i}$$\sum_{n=0}^{\infty}||\frac{f^{(n)}(0)}{n!}---n\phi||_{p}\leq\sum_{n=0}^{\infty}M\gamma^{n}||\phi||_{q}\leq M(\sum_{n=0}^{\infty}\gamma^{n})||\phi||_{q}$Hence the series in the right hand side of (4) converges inproof.

$\mathcal{L}(\mathcal{G}, \mathcal{G})$.

. It then follows the$\bullet$. Also, if is the exponential function, thenIf is apolynomial, thenis called an equicontinuous generator, see [26].
an element of
$f$

$f$

$A_{f}=\mathcal{L}(\mathcal{G}, \mathcal{G})$$A_{f}$

4Equicontinuous Generators
onto itself. In thisdenote the group of all linear homeomorphisms fromLet
, i.e., for eachofsection we consider a(complex) one-parameter subgroupand,$GL(\mathcal{G})$$\mathcal{G}$$GL(\mathcal{G})$$\{\Omega_{z}\}_{z\in \mathbb{C}}$$z\in \mathbb{C}$$\Omega_{z}\in GL(\mathcal{G})$$\Omega_{0}=I$(identity operator);

Aone-parameter subgroup,such that for any$\phi$$\{\Omega_{z}\}_{z\in \mathbb{C}}$$\Omega_{z_{1}}\Omega_{z_{2}}=\Omega_{z_{1}+z_{2}}$,$z_{2}\in \mathbb{C}$.
$\mathrm{a}---\in \mathcal{L}(\mathcal{G})$$\in \mathrm{C}\mathrm{i}$for all $p\geq 0$ .

. Afamily of operators
called the infinitesimal generator ofis said to be equicontinuous if for any $p\geq 0$ there exist $q\geq 0$ and $C\geq 0$ such$\mathrm{a}---\mathrm{i}\mathrm{s}$$\mathcal{L}(\mathcal{G}, \mathcal{G})$$z_{1}$

is said to be holomorphic if there exists$\lim_{zarrow 0}||\frac{\Omega_{z}\phi-\phi}{z}----\phi||_{p}=0$

Such

,

$\{_{-:}^{-}-\}_{i\in I}\subset$$\{\Omega_{z}\}_{z\in \mathbb{C}}$that$||\Xi_{i}\phi||_{p}\leq C||\phi||_{q}$,$\phi\in \mathcal{G}$,$i\in I$,see [26].
Theorem 3[26] Every equicontintous generator
ator of some holomorphic one-parameter subgroup$R\}$ is equicontinuous
for some $R>0$ . In this case,$—\in \mathcal{L}(\mathcal{G}, \mathcal{G})$is the infinitesimal genersuch that { ; $|z|<$$\{\Omega_{z}\}_{z\in \mathbb{C}}\subset GL(\mathcal{G})$$\Omega_{z}=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}---n$there the series converges in$\mathcal{L}(\mathcal{G}, \mathcal{G})$.

,$z\in \mathbb{C}$,$\Omega_{z}$158
Prom Theorem 3, for an equicontinuous generator EEC (Ci, (; the correspondingCi) is denoted by.
holomorphic one-parameter subgroup of$\{\mathrm{e}’ \mathrm{p}(\mathrm{z}\mathrm{E})\}_{\mathrm{z}\mathrm{C}\mathrm{C}}$$GL\ovalbox{\tt\small REJECT}$

Lemma 4[26]$Let—\in \mathcal{L}(\mathcal{G}, \mathcal{G})$. Then the following conditions are equivalent:(i) there exists some $R>0$ such that $\{(R_{-}^{-}-)^{n}/n!;n=0,1,2, \cdots\}$ is equicontinuous(ii) $\{(R_{-}^{-}-)^{n}/n!;n=0,1,2, \cdots\}$ is equicontinuous for any $R>0$ .

Lemma 5Let (, $\eta\in H$ andsuch that$B\in \mathcal{L}(H, H)$. Then there exists a unique operator$G_{\eta,B,\zeta}\in$$\mathcal{L}(\mathcal{G}, \mathcal{G})$$G_{\eta,B,\zeta} \phi=(\sum_{l+m=n}\sum_{k=0}^{\infty}\frac{(l+k)!}{l!k!m!}\zeta\otimes m^{\wedge}\otimes((e^{B})^{\otimes l}(\eta^{\otimes k^{\wedge}}\otimes_{k}f_{l+k})))_{n=0}^{\infty}$

for any$\phi=(f_{n})_{n=0}^{\infty}\in \mathcal{G}$, there$\otimes_{k}\wedge$

is the right contraction$[P\mathit{3}]$.

For the proof, see [15]. For each $\xi\in H$ , we can easily see that
$G_{\eta,B,\zeta}\phi_{\xi}=\exp\{\langle\eta, \xi\rangle\}\phi_{e^{B}\xi+\zeta}$(5).

Motivated by results in [6] and Theorem 3, we now consider aholomorphic onewith infinitesimal generator
parameter subgroup of
and $t>0$ .
for arbitrary , a2, ,For notational convenience, we put$a_{1}I+a_{2}A_{t}+a_{3}\Lambda_{t}+a_{4}A_{t}^{*}$$GL(\mathcal{G})$$a_{3}$$a_{1}$$a_{4}\in \mathbb{C}$,$G_{t_{j}\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}=\alpha_{1}G_{\alpha_{2}\chi_{t\mathrm{l}\prime}\alpha\epsilon\chi_{t\mathrm{l}\prime}\alpha_{4Xt\mathrm{l}}}$Let and C’
respectively.
$\mathbb{C}$

Theorem 6Let.
group of$=\mathbb{C}-\{0\}$$\alpha_{1}$,$\alpha_{2}$,$\alpha_{3}$,$\alpha_{4}\in \mathbb{C}$,$t>0$ .

be the additive and multiplicative group of complex numbers,$\otimes_{t}=\{G_{t_{j}\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}} :\alpha_{1}\in \mathbb{C}^{*}, \alpha_{2}, \alpha_{3}, \alpha_{4}\in \mathbb{C}\}$.

$Then\otimes_{t}$forms a sub-$GL(\mathcal{G})$Proof. For any $\xi\in H$ we have, by (5),$G_{t;1,0,0,0\emptyset\epsilon=\phi}\mathrm{f}$and,$G_{t_{j}\alpha_{1}’,\alpha_{\acute{2}’}\alpha_{\acute{3}’}\alpha_{\acute{4}}}G_{t_{j}\alpha_{1\prime}\alpha_{2},\alpha\epsilon,\alpha_{4}}\phi_{\xi}=G_{t_{j}\alpha_{\acute{1}}\alpha_{1}e^{\alpha_{\acute{2}}\alpha_{4}t},\alpha_{\acute{2}}e^{\alpha_{3}}+\alpha_{2\prime}e^{\alpha}\acute{\mathrm{s}}^{+\alpha_{3}},e^{\alpha}\acute{\S}\alpha_{4}+\alpha_{\acute{4}}}\phi_{\xi}$for any ,subspace of$\alpha_{1}$$\alpha_{1}’\in \mathbb{C}$$\mathcal{G}$

and

’

and

$\alpha_{2}$,spans adense. But
is continuous. Hence it follows that for any$\alpha_{2}’$,$\alpha_{3}$,$\alpha_{3}’$,$\alpha_{4}$,$\{\phi_{\xi} :$\alpha_{4}’\in \mathbb{C}$\xi\in H\}$$\phi\in \mathcal{G}$$G_{t_{j}\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}$and$G_{t_{j}1,0,0,0\emptyset=\emptyset}$,$G_{t_{j}\alpha_{\acute{1}’}\alpha_{\acute{2}},\alpha_{\acute{3}’}\alpha_{\acute{4}}}G_{t_{j}\alpha_{1\prime}\alpha_{2},\alpha_{3},\alpha_{4}}\phi=G_{t_{j}\alpha_{\acute{1}}\alpha_{1}e^{\alpha_{\acute{2}}\alpha_{4^{t}}},\alpha_{\acute{2}}e^{\alpha_{3}}+\alpha_{2\prime}e^{\alpha}\acute{\mathrm{a}}^{+\alpha_{3}},e^{\alpha_{\acute{3}}}\alpha_{4}+\alpha_{\acute{4}}}\emptyset$

Put$\alpha_{1}’=(1/\alpha_{1})\exp\{e^{-\alpha_{S}}\alpha_{2}\alpha_{4}t\}$$G_{t_{j}\alpha_{1}’,\alpha_{2}’,\alpha_{\acute{3}},\alpha_{\acute{4}}}$is the inverse of

,

, and
,
. This completes the proof.

$\alpha_{2}’=-\alpha_{2}e^{-\alpha \mathrm{s}}$$G_{t_{j}\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}$in$\otimes_{t}$$\alpha_{3}’=-\alpha_{3}$$\alpha_{4}’=-\alpha_{4}e^{-\alpha_{3}}$1. Then159
For each$\mathrm{a}=(a_{1}, a_{2}, a_{3}, a_{4})\in \mathbb{C}^{4}$, we define the functions$\alpha_{1}$,$\alpha_{2}$$a_{1}z+-a \simeq a_{3}at[\frac{1}{a_{3}}(e^{a_{3}z}-1)-z]\}$$\{\begin{array}{l}\alpha_{2}(z)=\mathrm{p}a(a_{3}e^{a}\alpha_{1}(z)=\mathrm{e}\mathrm{x}\mathrm{p}\{\alpha_{3}(z)=a_{3}z\alpha_{4}(z)=\lrcorner a(a_{3}e^{a}\end{array}$on$\mathbb{C}$

by

,
(6)

$-1)$

$sz$

$a_{3}\neq 0$$\alpha_{4}$$-1)$ ,

$3z$if

, a3 and;(7)$\{\begin{array}{l}\alpha_{1}(z)=\mathrm{e}\mathrm{x}\mathrm{p}\{a_{1}z+\underline{a}_{2}a\approx_{Zt\}}2\alpha_{2}(z)=a_{2}z\alpha_{3}(z)=0\alpha_{4}(z)=a_{4}z\end{array}$if $a_{3}=0$ . For each$\{\mathrm{R}_{\mathrm{a},t_{j}z}\}_{z\in \mathbb{C}}$$\mathrm{a}=$($a_{1}$, a2,$a_{3}$,$a_{4}$)$\in \mathbb{C}^{4}$by, we also define afamily of transforms$\mathrm{R}_{\mathrm{a},t_{j}z}=\alpha_{1}(z)2_{\tilde{\mathrm{a}},t_{j}z}=\alpha_{1}(z)G_{t;\alpha_{2}(z),\alpha_{3}(z),\alpha_{4}(z)}$where
and the functions
by direct computations using (5),$\overline{\mathrm{a}}=(a_{2}, a_{3}, a_{4})$$\mathrm{a}\mathrm{i}$,$\alpha_{2}$$\{\mathrm{R}_{\mathrm{a},t_{j}z}\}_{z\in \mathbb{C}}$

Lemma 7For each$\overline{\mathrm{a}}=(a_{2}, a_{3}, a_{4})\in \mathbb{C}^{3}$,$z\in \mathbb{C}$,andare given as in (6) or (7). Then,is aone-parameter subgroup of.
$\alpha_{3}$and for any$\alpha_{4}$$GL(\mathcal{G})$$\phi\in Ci$, we have$\lim_{zarrow 0}||\frac{2_{\tilde{\mathrm{a}},t_{j}z}\phi-\phi}{z}-(a_{2}A_{t}+a_{3}\Lambda_{t}+a_{4}A_{t}^{*})\phi||_{p}=0$

Proof. Let$p\in \mathbb{R}$and$\phi=(f_{n})\in \mathcal{G}$,,$p\in \mathbb{R}$be given. Then by definition of.

$2_{\tilde{\mathrm{a}},t_{j}z}$, we have$\frac{2_{\overline{\mathrm{a}},t_{j}z}\phi-\phi}{z}-(a_{2}A_{t}+a_{3}\Lambda_{t}+a_{4}A_{t}^{*})\phi$$=( \frac{(e^{a_{3}z_{Xt\mathrm{l}}})^{\otimes n}-1}{z}f_{n}-a_{3}n(\chi_{t]}\otimes I^{\otimes(n-1)}f_{n})_{\mathrm{s}\mathrm{y}\mathrm{m}})$$+(^{\wedge} \frac{\alpha_{4}(z)}{z}\chi_{t]}\otimes((e^{a_{3}z_{Xt\mathrm{l}}})^{\otimes n}f_{n})-a_{4}\chi_{t]}\otimes f_{n})\wedge$$+( \frac{(n+1)!}{n!}[\frac{\alpha_{2}(z)}{z}(e^{a_{3}z_{Xt\mathrm{l}}})^{\otimes n}-a_{2}]\chi_{t]}\otimes_{1}f_{n+1)}\wedge$$+(\begin{array}{l}1\overline{z}g_{n}\end{array})$

,

where
$g_{n}= \sum_{l+m=n}\sum_{k+m\geq 2}\frac{(l+k)!}{l!k!m!}$$\cross(\alpha_{4}(z)\chi_{t]})^{\otimes m}\wedge\otimes[(e^{a_{3}z\chi t\mathrm{l}})^{\otimes l}((\alpha_{2}(z)\chi_{t]})^{\otimes k_{\otimes_{k}}^{\wedge}}f_{l+k})]$Therefore we obtain that$|| \frac{2_{\tilde{\mathrm{a}},t_{j}z}\phi-\phi}{z}-(a_{2}A_{t}+a_{3}\Lambda_{t}+a_{4}A_{t}^{*})\phi||_{p}^{2}\leq 4\sum_{j=1}^{4}I_{j}(z)$

,

.

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,

$I_{1}(z)$$=$$\sum_{n=0}^{\infty}n!e^{2pn}|\frac{(e^{a_{3}z_{Xt\mathrm{l}}})^{\otimes n}-1}{z}f_{n}-a_{3}n(\chi_{t]}\otimes I^{\otimes(n-1)}f_{n})_{\mathrm{s}\mathrm{y}\mathrm{m}}|_{0}^{2}$$I_{2}(z)$$=$$\sum_{n=0}^{\infty}n!e^{2pn}|\frac{(n+1)!}{n!}[\frac{\alpha_{2}(z)}{z}(e^{a_{3}z_{Xt\mathrm{J}}})^{\otimes n}-a_{2}]\chi_{t]}\otimes_{1}f_{n+1}\wedge|_{0}^{2}$$I_{3}(z)$$=$$\sum_{n=0}^{\infty}(n+1)!e^{2p(n+1)}|^{\wedge}\frac{\alpha_{4}(z)}{z}\chi_{t]}\otimes((e^{a_{3}z_{Xt\mathrm{l}}})^{\otimes n}f_{n})-a_{4}\chi_{t]}\otimes f_{n}|_{0}^{2}\wedge$,

and$I_{4}(z)= \sum_{n=0}^{\infty}n!e^{2pn}$$|\begin{array}{l}1\overline{z}g_{n}\end{array}|02$

Then by simple modification of the proof of Proposition 5.4.5 in [23], we can easily seethat $\lim_{zarrow 0}I_{1}(z)=0$ . On the other hand, by similar arguments of those used in the proof
Lemma 3.4 in [14], we see that $\lim_{zarrow 0}(I_{2}(z)+I_{3}(z)+I_{4}(z))=0$. The proof follows.
$\bullet$,Theorem 8For each $t>0$ andgenerator
thewithparameter subgroup of
infinitesimal$\mathrm{a}=(a_{1}, a_{2}, a_{3}, a_{4})\in \mathbb{C}$$\{h_{t_{j}z},\}_{z\in \mathbb{C}}$$GL(\mathcal{G})$Proof. Let$p\in \mathbb{R}$and$\phi\in \mathcal{G}$is a holomorphic one-$a_{1}I+a_{2}A_{t}+a_{3}\Lambda_{t}+a_{4}A_{t}^{*}$.

be given. Then we have$|| \frac{\mathrm{R}_{\mathrm{a},t_{j}z}\phi-\phi}{z}-(a_{1}I+a_{2}\dot{A}_{t}+a_{3}\Lambda_{t}+a_{4}A_{t}^{*})\phi||_{p}$$\leq|\frac{\alpha_{1}(z)-1}{z}-a_{1}|||2_{\tilde{\mathrm{a}},t_{j}z}\phi||_{p}+||a_{1}(2_{\tilde{\mathrm{a}},t_{j}z}-I)\phi||_{p}$$+|| \frac{2_{\tilde{\mathrm{a}},t_{j}z}\phi-\phi}{z}-(a_{2}A_{t}+a_{3}\Lambda_{t}+a_{4}A_{t}^{*})\phi||_{p}$Prom Lemma 7, the proof follows.
Theorem 9The transform$\bullet$$G_{t_{j}\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}$has the following representation:$G_{t_{j}\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}=\alpha_{1}e^{\alpha_{4}A_{\dot{t}}}\circ e^{\alpha_{3}\mathrm{A}}{}^{t}\circ e^{\alpha_{2}A_{t}}$.

Proof. It can be easily shown that for any $\xi\in H$ , we have
$G_{t_{j}\alpha_{1\prime}\alpha_{2\prime}\alpha_{3\prime}\alpha_{4}}\phi_{\xi}=\alpha_{1}e^{\alpha_{4}A_{\dot{t}}}\mathrm{o}e^{\alpha_{3}\mathrm{A}}{}^{t}\circ e^{\alpha_{2}A}{}^{t}\phi_{\xi}$

We note that
. Since$\mathcal{G}$are continuous linear operators onandspans adense subspace of , the proof follows.
$\alpha_{1}e^{\alpha_{4}A_{t}^{l}}\circ e^{\alpha_{3}\mathrm{A}}{}^{t}\circ e^{\alpha_{2}A_{t}}$$G_{t_{j}\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}$$\{\phi_{\xi} :\xi\in H\}$.
$\bullet$$\mathcal{G}$By similar arguments of those used in the proof of Lemma 5, $\{h_{t_{j}z},;|z|<R\}$ isequicontinuous for any $R>0$ . Therefore, by Theorems 8and 3,. Hence by( , a2, , )is an equicontinuous generator for each $t>0$ andTheorem 9we have$a_{1}I+a_{2}A_{t}+a_{3}\Lambda_{t}+a_{4}A_{t}^{*}$$\mathrm{a}=$$\in \mathbb{C}^{4}$$a_{1}$$a_{3}$$a_{4}$$e^{z(a_{1}I+a_{2}A_{t}+a_{3}\Lambda_{t}+a_{4}A_{\dot{t}})}=\alpha_{1}(z)e^{\alpha_{4}(z)A_{t}^{*}}\circ e^{\alpha_{3}(z)\Lambda}{}^{t}\circ e^{\alpha_{2}(z)A_{t}}$,are given as in (6) or (7).
andwhere the functions , ,, let $Q_{a,b}(t)=aA_{t}+bA_{t}^{*}$ . Then by Theorems 8and 3weand
For each ,is an equicontinuous generator
also see that$\alpha_{1}$$a$$b\in \mathbb{C}$$Q_{a,b}(t)$$\alpha_{2}$$\alpha_{3}$$t\in \mathbb{R}_{+}$$\alpha_{4}$161
5Quantum Stochastic Processes
Afamily of operators
is called aquantum stochastic process if there
exists ,such thatfor all $t\geq 0$ and for eachthe map
is strongly measurable. Aquantum stochastic process
is said
to be adapted if for each $t\geq 0$ there exists
such that $—t=—t$]
,
:
where
is the inclusion map.
For each, the conditional expectation(see [5], [25]) is defined by the second
quantization operatorof , i.e., for each$\{_{-t}^{-}-\}_{t\geq 0}\subset \mathcal{L}(\mathcal{G}, \mathcal{G}^{*})$$p$$q\in \mathbb{R}$$–t-\in \mathcal{L}(\mathcal{G}_{p}, \mathcal{G}_{q})$$t\vdash+---t\emptyset$$\phi\in \mathcal{G}_{p}$$\{_{-t}^{-}-\}_{t\geq 0}\subset \mathcal{L}(\mathcal{G}_{p}, \mathcal{G}_{q})(p\geq q)$$—t\mathrm{I}\in \mathcal{L}(\mathcal{G}_{pj}t], \mathcal{G}_{qj}t])$$I_{[t}$$\otimes I_{[t}$$\mathcal{G}_{pj[t}\mathrm{c}arrow \mathcal{G}_{qj[t}$$t\in \mathbb{R}_{+}$$\mathrm{E}_{t}$$\Gamma(\chi_{t]})$$t\in \mathbb{R}_{+}$$\chi_{t]}$$\mathrm{E}_{t}\Phi=(\chi_{t]}^{\otimes n}f_{n})$Then for any

and$p\in \mathbb{R}$$\Phi=(f_{n})\in \mathcal{G}_{p}$,$\Phi=(f_{n})\in \mathcal{G}^{*}$.

, we have.

$|| \mathrm{E}_{t}\Phi||_{p}^{2}=\sum_{n=0}^{\infty}n!e^{2pn}|\chi_{t]}^{\otimes n}f_{n}|_{0}^{2}\leq||\Phi||_{p}^{2}$,
Hence for any
andMoreover,and
An adapted process of operators
martingale if for any $0\leq s\leq t$$p\in \mathbb{R}$$t\in \mathbb{R}_{+}$$\mathrm{E}_{t}\in \mathcal{L}(\mathcal{G}, \mathcal{G})$$\mathrm{E}_{t}\in \mathcal{L}(\mathcal{G}_{p}, \mathcal{G}_{p})$$\mathrm{E}_{t}\in \mathcal{L}(\mathcal{G}^{*}, \mathcal{G}^{*})$

and$\mathrm{E}_{t}$is an orthogonal projection.

.

$\{_{-t}^{-}-\}_{t\geq 0}$

in$\mathcal{L}(\mathcal{G}_{p}, \mathcal{G}_{q})(p\geq q)$$\mathrm{E}_{s-t-}^{-}-\mathrm{E}_{s}=\mathrm{E}_{s^{-}s}^{-}\mathrm{E}_{s}$is called aquantum.

,
The processes
anddefined in (2) are called the annihilation,
creation and number (or gauge) processes, respectively. The quantum stochastic process$Q_{t}=Q_{1,1}(t)=A_{t}+A_{t}^{*}$ is called the quantum Brownian motion or the position process.
For any non-negative integers , $m$ , the processes
andare quantum
martingales, where is the Wick product or normal-0rdered product [7]. In particular,, the creation process
the annihilation process
and the number process
are quantum martingales. These martingales are called the basic martingales.
Also, the basic martingales and the time process are called the basic processes.
An adapted process
is called aregular semimartingale inif for any$p\geq 0$ there exists $q\geq p$ and an absolutely continuous measure $m$ onsuch that for,
any $r<s<t$ and]$\{A_{t}\}_{t\geq 0}$$\{A_{t}^{*}\}_{t\geq 0}$$\{\Lambda_{t}\}_{t\geq 0}$$l$$\{(A_{t}^{*})^{l}A_{t}^{m}\}_{t\geq 0}$$\{\Lambda_{t}^{ol}\}_{t\geq 0}$$0$$\{A_{t}\}_{t\geq 0}$$\{A_{t}^{*}\}_{t\geq 0}$$\{\Lambda_{t}\}_{t\geq 0}$$\{_{-t}^{-}-\}_{t\geq 0}$$\mathcal{L}(\mathcal{G}, \mathcal{G})$$\mathbb{R}_{+}$$\phi$$\in \mathcal{G}_{q;\mathrm{r}]}$$\psi$$\in \mathcal{G}_{-pj}r$;$||(_{-t-s}^{--}---)\phi||_{p}^{2}$$\leq$$||\phi||_{q}^{2}m([s, t])$$||(_{-ts}^{-*-*}----)\psi||_{-q}^{2}$$\leq$$||\psi||_{-p}^{2}m([s, t])$$||(\mathrm{E}_{s-t-s}^{--}---)\phi||_{p}$$\leq$$||\phi||_{q}m([s, t])$;.

be the space of all locally bounded square integrable functions on
Let
and
by
spannedexponential,
subspaceofallvectors.
adenseof adapted processin
The spaceadmitting the integral representation:$L_{1\mathrm{b}}^{2}(\mathbb{R}_{+})$$\mathbb{R}_{+}$$\mathcal{H}$$\mathcal{E}_{1\mathrm{b}}$$S(\mathcal{G})$$\phi_{\xi}$$\{_{-t}^{-}-\}_{t\geq 0}$$\xi\in L_{1\mathrm{b}}^{2}(\mathbb{R}_{+})$$\mathcal{L}(\mathcal{G}, \mathcal{G})$$\Xi_{t}=\lambda I+\int_{0}^{t}(Ed\Lambda+FdA+GdA^{*}+Hds)$onand adapted processes $(E, F, G, H)$ inwith asatisfying that for$q\geqp$$p\geq0$
anythere exists
such thatandare locally square
integrable,
is locally bounded andis locally integrable, where.
is the operator norm on$\mathcal{E}_{1\mathrm{b}}$$\mathrm{A}\in \mathbb{C}$$\mathcal{L}(\mathcal{G}, \mathcal{G})$$s\vdasharrow||F_{s}||_{qjp}$$s\vdasharrow||H_{s}||_{qjp}$$s\vdasharrow||E_{s}||_{qp}$$||\cdot||_{q,p}$$s\vdash+||G_{s}||_{qjp}$$\mathcal{L}(\mathcal{G}_{q}, \mathcal{G}_{p})$162
Theorem 10 An adapted process
if \yen is a regular semimartingale$\{\mathrm{E}_{1}\}_{1>0}$in C ( C;, Ci) is an element of $S(’ i)$ if and only

For the proof, we refer to [16].

6Functional Quantum Ito Formula
be the class of quadruples $\mathrm{F}=(E, F, G, H)$ of adapted processes inLetandsatisfying that for any $p\geq 0$ there exists $q\geq p$ such thatis locallyandlocallyboundedisintegrable,
are locally square
integrable.
$\mathcal{L}(\mathcal{G}, \mathcal{G})$$\mathrm{L}_{2}(\mathcal{G})$$s\vdash+||G_{s}||_{qjp}$$s\vdash+||F_{s}||_{qjp}$$s\mapsto||H_{s}||_{qjp}$$s\vdasharrow||E_{s}||_{qjp}$

Theorem 11 Let
sentations:$\{_{-t}^{-}-\}_{t\geq 0}\in S(\mathcal{G})$$=$$—\prime t$$\mathcal{E}_{1\mathrm{b}}$for some$\mathrm{F}\in \mathrm{L}_{2}(\mathcal{G})$tions can be extended to,$–t–t–$$=$$\{_{-t}^{-\prime}-\}_{t\geq 0}\in S(\mathcal{G})$with the following integral repre-$= \int_{0}^{t}(Ed\Lambda+FdA+GdA^{*}+Hds)$ ,

$—t$

on

and

$\mathcal{G}$$\int_{0}^{t}(E’d\Lambda+F dA+G’dA^{*} +H’ds)$

andand we have$\mathrm{F}’\in \mathrm{L}_{2}(\mathcal{G})$, respectively. Then both integral representa-$\int_{0}^{t}(E_{-}^{-\prime}-d\Lambda+F_{-}^{-\prime}-dA+G_{-}^{-\prime}-dA^{*}+H_{-}^{-\prime}-ds)$$+ \int_{0}^{t}(_{-}^{-}-E’d\Lambda+---F’dA+---G’dA*+---H’ds)$$+ \int_{0}^{t}(EE’d\Lambda+FE’ dA +EG’dA^{*} +FG’ds)$.

(8)

Proof. By the similar arguments of those used in the proof of Theorems 6.1 and 6.2 in[16], the proof is straightforward.
$\bullet$

The equation (8) is sometimes written in the shorter differential form:$d(_{--}^{--\prime}--)=(d_{-}^{-}-)_{-+}^{-}-’---(d_{-}^{-\prime}-)+(d_{-}^{-}-)(d_{-}^{-\prime}-)$

(9)

,

where
$(d_{-}^{-}-)_{-}^{-\prime}-$$—(d_{-}^{-\prime}-)$,$=—E’d\Lambda+---F’dA+---G’dA^{*}+---H’ds$ ,$=$$(d_{-}^{-}-)(d_{-}^{-}-,)=$

Prom now on we consider$E_{-}^{-\prime}-d\Lambda+F_{-}^{-\prime}-dA+G_{-}^{-\prime}-dA^{*}+H_{-}^{-\prime}-ds$$EE’ dA$ $+FE’ dA$ $+EG’dA^{*}$ $+FG’ds$$—\in S(\mathcal{G})$.

with the integral representation:$—t=—0+ \int_{0}^{t}(Ed\Lambda+FdA+GdA^{*}+Hds)$ .

163
By similar arguments of those used in the proof of Theorem 5in [1] with Remark 7.4 in[16], we can easily prove that for each $p\geq 0$ there exists $q\geq 0$ the map
islocally bounded. Therefore, by (8) we see that for each positive integer ,and we have.
(10)
It follows the following lemma. For the proof, see the proof of Lemma 4.1 in [30].
$sarrow||_{-s}^{-}-||_{qp}$$—n+1\in \mathrm{S}\{\mathrm{Q})$$n$$d(_{-}^{-n+1}-)=(d_{-}^{-}-)_{-+}^{-}-n---(d_{-}^{-n}-)+(d_{-}^{-}-)(d_{-}^{-n}-)$Lemma 12 We have(11)$–_{t}-n=---n0+ \int_{0}^{t}(E_{n}d\Lambda+FndA+G_{n}dA^{*}+Hnds)$ ,where$E_{n}=(\Xi+E)^{n-n}---$ ,$F_{n}=\mathrm{I}---i+-1:F(_{-}^{-}-+E)^{j}$, $G_{n}= \sum_{i+j=n-1}(_{-}^{-}-+E)^{i}G_{-}^{-j}-$and$H_{n}= \mathrm{I}^{-}--^{i-j}H--+\sum_{ii+-1+j+k=n-2}---F(\Xi+E)^{j}G_{-}^{-k}i-$.

, where is apolynomial
From Lemma 12 we have aquantum Ito formula for, where is an entire. Now we consider aquantum Ito formula forandfunction on satisfying certain condition.
$p(_{-}^{-}-)$

$p$

$f$

$f(_{-}^{-}-)$$—\in S(\mathcal{G})$$\mathbb{C}$

Lemma 13 Let

such that$\{a_{n}\}_{n=0}^{\infty}\subset \mathbb{C}$$|a_{n}| \leq\frac{1}{n!}MR^{n}$for some $M\geq 0$ and $R>0$ .

,are equicontinuous generators for all $t\geq 0$ ,$and—_{t}+E_{t}$$If_{-t}^{-}-$$n\geq 0$

then the series$\sum_{n=0}^{\infty}a_{n}E_{n}$converge in$\mathcal{L}(\mathcal{G}, \mathcal{G})$where$E_{n}$,,$\sum_{n=0}^{\infty}a_{n}F_{n}$$F_{n}$,$G_{n}$,

and,$\sum_{n=0}^{\infty}a_{n}G_{n}$$H_{n}$$\sum_{n=0}^{\infty}a_{n}H_{n}$are given as in Lemma 12.

since theconverges in
Proof. We will prove only that the series$n\geq0$$p\geq0$,
we haveandproofs of convergence of other series are very similar. For any$\sum_{n=0}^{\infty}a_{n}H_{n}$

$|$

$|$

$H_{n}\phi$

$|$

$|p$

$\leq$$(i \sum_{+j=n-}$

$|$

$|$

$—i$ $H$ $–j-$ $\phi$

$|$

$|p$

$+$

$|$

$|$

$\mathcal{L}(\mathcal{G}, \mathcal{G})$$—i$ $F$ $(—$ $+$ $E$ $)j$ $G$ $—k$ $\phi$—and the continuity of $H$ , there exist
By the equicontinuity of, , $s\geq 0$ such that for any$r$$|p$ $)$

$i$

$1$

$q$

$|$

$+j \sum_{+k=n}-2$$C_{1}$,$\epsilon>0$$\leq$$C_{1} \sum_{i+j=n-1}\epsilon^{i}i!||H_{-}^{-j}-\phi||_{q}$$\sum_{i+j=n-1}||_{-}^{-i}-H_{-}^{-j}-\phi||_{p}$$\leq$$C_{1}C_{2} \sum_{i+j=n-1}\epsilon^{i}i!||_{-}^{-j}-\phi||_{q+f}$$\leq$$C_{1}C_{2}C_{3} \epsilon^{n-1}(\sum_{i+j=n-1}i!j!)||\phi||_{s}$$\leq$$C_{1}C_{2}C_{3}\epsilon^{n-1}n!||\phi||_{s}$.

$C_{2}$, $C_{3}\geq 0$ and164
Similarly, by the equicontinuity ofand q 0 such that$\ovalbox{\tt\small REJECT}+E$

and the continuity of F and G, there exist C$\ovalbox{\tt\small REJECT}$$0$$\ovalbox{\tt\small REJECT}$$\sum_{:+j+k=n-2}||_{-}^{-i}-F(_{-}^{-}-+E)^{j}G_{-}^{-k}-\phi||_{p}\leq C\epsilon^{n-2}n!||\phi||_{q}$Therefore, by choosing$\epsilon>0$$\sum_{n=0}^{\infty}||a_{n}H_{n}\phi||_{p}$such that$R\epsilon<1$.

we have$MR(C_{1}C_{2}C_{3}+RC)( \sum_{n=0}^{\infty}(R\epsilon)^{n})||\phi||_{q’}$$\leq$

where $q’=s\vee q$ . It proves that the series$\sum_{n=0}^{\infty}a_{n}H_{n}$converges in$\mathcal{L}(\mathcal{G}$,1, (;).

For the following theorem we assume that there exists $R>0$ such that $\{(R_{-t}^{-}-)^{n}/n!;t\in$, $n=0,1,2$ ,and $\{(R(_{-t}^{-}-+E_{t}))^{n}/n!;t\in K, n=0,1,2, \cdots\}$ are equicontinuous, i.e., for any $p\geq 0$ there exists , $C’\geq 0$ andfamilies for any bounded interval, $q’\geq 0$ such that for any bounded interval$K$$\cdots\}$$C$$K\subset \mathbb{R}_{+}$$K\subset \mathbb{R}_{+}$$q$$\sup_{t\in K}||\frac{(R_{-t}^{-}-)^{n}}{n!}\phi||_{p}\leq C||\phi||_{q}$for all $n=0,1,2$ ,$\cdots$

Theorem 14 Let$f$,$\sup_{t\in K}||\frac{(R(_{-t}^{-}-+E_{t}))^{n}}{n!}\phi||_{p}\leq C’||\phi||_{q’}$$\phi$$\in \mathcal{G}$.
be an entire function with Taylor expansion$f(z)= \sum_{n=0}^{\infty}a_{n}z^{n}$$and—\in S(\mathcal{G})$,,$z\in \mathbb{C}$admit the integral representation$—t=—0+ \int_{0}^{t}(Ed\Lambda+FdA+GdA^{*}+Hds)$.

Assume that there exist $M\geq 0$ and $R>0$ such that$|a_{n}| \leq\frac{1}{n!}MR^{n}$,$n\geq 0$.

Then we have$f(_{-t}^{-}-)=f(_{-0}^{-}-)+ \int_{0}^{t}(E’d\Lambda+F dA+G’dA^{*} +H’ds)$

(12)

,

where
$E’= \sum_{n=0}^{\infty}a_{n}E_{n}$,$F’= \sum_{n=0}^{\infty}a_{n}F_{n}$,$G’= \sum_{n=0}^{\infty}a_{n}G_{n}$,$H’= \sum_{n=0}^{\infty}a_{n}H_{n}$.

165
Proof. By similar arguments of those used in the proof of Lemma 13, we can easily
prove that for any p
0 there exists q p such that s $*$is locally bounded,and sHare locally square integrable and s $+$is locallyintegrable. Therefore, by Lemma 12, for a1H t 0 and
we have$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$||E\ovalbox{\tt\small REJECT}||_{q\ovalbox{\tt\small REJECT} p}$$||G\mathrm{s}||_{q;p}$$s’\ovalbox{\tt\small REJECT}||\mathrm{f}^{\mathrm{f}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}||_{q\ovalbox{\tt\small REJECT} p}$$||H\ovalbox{\tt\small REJECT}||_{q\ovalbox{\tt\small REJECT} p}$$f|\ovalbox{\tt\small REJECT}\rangle$$\ovalbox{\tt\small REJECT}$$\langle\langle f(_{-t}^{-}-)\phi_{\xi}, \phi_{\eta}\rangle\rangle$$=$$7/\mathrm{E}$$L\mathrm{r}(\mathrm{R}_{+})$$\sum_{n=0}^{\infty}a_{n}\langle\langle_{-0}^{-n}-\phi_{\xi}, \phi_{\eta}\rangle\rangle$$+ \sum_{n=0}^{\infty}a_{n}\int_{0}^{t}\langle\langle(\xi\eta E_{n}+\xi F_{n}+\eta G_{n}+H)\phi_{\xi}, \phi_{\eta}\rangle\rangle ds$$=$$\langle\langle f(_{-0}^{-}-)\phi_{\xi}, \phi_{\eta}\rangle\rangle$$+ \int_{0}^{t}\sum_{n=0}^{\infty}a_{n}\langle\langle(\xi\eta E_{n}+\xi F_{n}+\eta G_{n}+H)\phi_{\xi}, \phi_{\eta}\rangle\rangle ds$,

where for the last equality we used the dominated convergence theorem. It follows theproof.
$\bullet$be an equicontinuous generator and let
Lemma 15 For each
. Then thebe the corresponding holomorphic one-parameter subgroup of
following conditions are equivalent:$t\in \mathbb{R}_{+},$$let–t-\in \mathcal{L}(\mathcal{G}, \mathcal{G})$$GL(\mathcal{G})$$\{\Omega_{t;z}\}_{z\in \mathbb{C}}$(i) there exists $R>0$ such that $\{(R_{-t}^{-}-)^{n}/n!;t\in K, n=0,1,2, \cdots\}$ is equicontinuous
for any bounded interval$K\subset \mathbb{R}_{+},\cdot$(ii) $\{\Omega_{t;Z}; t\in K, |z|<R\}$ is equicontinuous for some $R>0$ and any bounded interval$K\subset \mathbb{R}_{+}$$\mathrm{P}$

roof.

.

$(\mathrm{i})\Rightarrow(\mathrm{i}\mathrm{i})$By assumption,$\phi\in \mathcal{G}$$\Omega_{t_{j}z}\phi=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}---t\phi n$converges in (;. Moreover, for any$\phi\in \mathcal{G}$,$|z|<R$and $|z|<R’<R$ we have$\sup_{t\in K}||\Omega_{t_{jZ}}\phi||_{p}\leq C(\frac{R}{R-|z|})||\phi||_{q}\leq C(\frac{R}{R-R’})||\phi||_{q}$$(\mathrm{i}\mathrm{i})\Rightarrow(\mathrm{i})$For each fixed$\Phi\in \mathcal{G}^{*}$

and$\phi\in \mathcal{G}$.

, we have$\langle\langle\Phi,---t\emptyset n\rangle\rangle=\frac{n!}{2\pi i}\int_{|z|=\mathrm{r}}\langle\langle\Phi, \Omega_{t_{j}z}\phi\rangle\rangle\frac{dz}{z^{n+1}}$,$r>0$ .

(13)

On the other hand, by assumption for any $p\geq 0$ there exists $C\geq 0$ and $q\geq 0$ such thatfor any bounded interval$K\subset \mathbb{R}_{+}$$\sup_{t\in K}||\Omega_{t;z}\phi||_{p}\leq C||\phi||_{q}$,$|z|<R$ ,$\phi\in(i$.

166
Therefore, by (13) we have$|\langle\langle\Phi, ---tn\emptyset\rangle\rangle|$

It follows the proof.

$\leq$$\frac{n!}{2\pi}\int_{|z|=\mathrm{r}}||\Phi||_{-p}||\Omega_{t_{j}z}\phi||_{p}\frac{|dz|}{|z|^{n+1}}$$\leq$$C \frac{n!}{r^{n}}||\Phi||_{-p}||\phi||_{q}$,$t\in K$,$0<r<R$ .

$\bullet$. Then by direct computation, we see that.
is equicontinuous for any $R>0$ and bounded interval
Therefore, by Lemma 15, $\{(R_{-t}^{-}-)^{n}/n!;t\in K, n=0,1,2, \cdots\}$ is equicontinuous for any.
, where$R>0$ and bounded intervalLet$t\in \mathbb{R}_{+}$

and$\mathrm{a}_{t}=(a_{1}t, a_{2}, a_{3}, a_{4})\in \mathbb{C}^{4}$$\{h_{t},t_{j}z;t\in K, |z|<R\}$$K\subset \mathbb{R}_{+}$$—_{t}=a_{1}t+a_{2}A_{t}+a_{3}\Lambda_{t}+a_{4}A_{t}^{*}$$K\subset \mathbb{R}_{+}$

and $let—t=a_{1}\Lambda_{t}+a_{2}A_{t}+a_{3}A_{t}^{*}+a_{4}t$ .
, ( , a2, , )Theorem 16 Let
is a regular semimartingale and. ThenAssume that$t\in \mathbb{R}_{+}$$a_{1}\neq 0$$e^{-t}\overline{-}$$=$$\in \mathbb{C}^{4}$$a_{1}$$a_{3}$$a_{4}$$e^{-t}--$$I+ \int_{0}^{t}((e^{a_{1}}-1)e^{-s}\overline{-}d\Lambda_{s}+\frac{a_{2}}{a_{1}}(e^{\alpha_{1}}-1)e^{-s}\overline{-}dA_{t}$$+ \frac{a_{3}}{a_{1}}(e^{a_{1}}-1)e^{\overline{\underline{-}}_{s}}dA_{s}^{*}+(a_{4}+\frac{a_{2}a_{3}}{a_{1}}[\frac{1}{a_{1}}(e^{a_{1}}-1)-1])e^{\underline{\overline{-}}_{\mathrm{S}}}ds)$.

Proof. By Theorem 14, we have$e^{\overline{\underline{-}}t}=I+ \int_{0}^{t}$$(E’d\Lambda+F’ dA +G’dA^{*} +H’ds)$ ,

(14)

where
$E’= \sum_{n=0}^{\infty}\frac{1}{n!}[(_{-}^{-}-+a_{1})^{n-n}---]=e^{\overline{-}+a_{1}}--e^{-}--=(e^{a_{1}}-1)e^{\overline{-}}-$$F’=a_{2} \sum_{n=0}^{\infty}\frac{1}{(n+1)!}\sum_{:+j=n}---(_{-}^{-}:-+a_{1})^{j}$,,$G’=a_{3} \sum_{n=0}^{\infty}\frac{1}{(n+1)!}\sum_{\dot{|}+j=n}(_{-}^{-}-+a_{1})^{i-j}--$

and
$H’$

$=$

:

$\sum_{n=0}^{\infty}\frac{1}{n!}(a_{4}n_{-}^{-n-1}-+a_{2}a_{3}\sum_{:+j+k=n-2}---(_{-}^{-}-+a_{1})_{-}^{j-k)}-$.

$=a_{4}e^{\overline{-}}-+a_{2}a_{3} \sum_{n=0}^{\infty}\frac{1}{(n+2)!}\sum_{:+j=n}---(---:+a_{1})^{j}$

By similar arguments of those used in the proof of Proposition 5.1 in [30], we see that$F’= \frac{a_{2}}{a_{1}}(e^{a_{1}}-1)e^{-}--$,$G’= \frac{a_{3}}{a_{1}}(e^{a_{1}}-1)e^{-}--$

and$H’=(a_{4}+ \frac{a_{2}a_{3}}{a_{1}}[\frac{1}{a_{1}}(e^{a_{1}}-1)-1])e^{-}--$

This completes the proof from (14). I

The following result is immediate from Theorem 16.

167
Corollary 17 Let
$—t$

$=$

$(a, b, c, d)\in \mathbb{C}^{4}$. Assume that $a\neq 0,$ -1. TAen$\exp\{\ln(a+1)\Lambda_{t}+\frac{b}{a}\ln(a+1)A_{t}$$+ \frac{c}{a}\ln(a+1)A_{t}^{*}+[(d-\frac{bc}{a})+\frac{bc}{a^{2}}\ln(a+1)]t\}$is a solution,$t\in \mathbb{R}_{+}$of the following quantum stochastic differential equation:$d_{-t-t}^{--}-=-(a\Lambda_{t}+bA_{t}+cA_{t}^{*}+ddt)$

Theorem 18 Let ,$a$$b\in \mathbb{C}$and let$f$,$—0=1$ .

be an entire function with Taylor expansion$f(z)= \sum_{n=0}^{\infty}a_{n}z^{n}$,$z\in \mathbb{C}$.

Assume that there exist $M\geq 0$ and $R>0$ such that$|a_{n}| \leq\frac{1}{n!}MR^{n}$,$n\geq 0$.

TAen ate have$f(Q_{a,b}(t))=f(0)+ \int_{0}^{t}f’(Q_{a,b}(s))dQ_{a,b}(t)+\frac{ab}{2}\int_{0}^{t}f’(Q_{a,b}(s))ds$.

Proof. Since $Q_{a,b}(t)= \int_{0}^{t}(adA_{s}+bdA_{s}^{*})$ , by Lemma 12 we see that
$E_{n}=0$ ,$F_{n}=anQ_{a,b}^{n-1}$,$G_{n}=bnQ_{a,b}^{n-1}$and$H_{n}=ab \sum_{\alpha+\beta+\gamma=n-2}Q_{a,b}^{\alpha+\beta+\gamma}=\frac{ab}{2}n(n-1)Q_{a,b}^{n-2}$

for each $n=1,2$ ,$\cdots$. Hence by Theorem 14 we complete the proof.

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