# Cameron–Storvick theorem associated with Gaussian paths on function space

Jae Gil Choi<EMAIL_ADDRESS>School of General Education, Dankook
University, Cheonan 31116, Republic of Korea

###### Abstract

The purpose of this paper is to provide a more general Cameron–Storvick
theorem for the generalized analytic Feynman integral associated with Gaussian
process $\mathcal{Z}_{k}$ on a very general Wiener space $C_{a,b}[0,T]$. The
general Wiener space $C_{a,b}[0,T]$ can be considered as the set of all
continuous sample paths of the generalized Brownian motion process determined
by continuous functions $a(t)$ and $b(t)$ on $[0,T]$. As an interesting
application, we apply this theorem to evaluate the generalized analytic
Feynman integral of certain monomials in terms of Paley–Wiener–Zygmund
stochastic integrals.

###### keywords:

Cameron–Storvick theorem , generalized analytic Feynman integral , Gaussian
process , generalized Brownian motion process , Paley–Wiener–Zygmund
stochastic integral.

###### MSC:

[2010] Primary 46G12 , 60G15; Secondary 28C20 , 46B09

##  1 Introduction

Let $(C_{0}[0,T],\mathcal{W},\mathfrak{m})$ denote the classical Wiener space,
where $C_{0}[0,T]$ is the set of all $\mathbb{R}$-valued continuous functions
$x$ on $[0,T]$ with $x(0)=0$, $\mathcal{W}$ denotes the complete
$\sigma$-field of all Wiener measurable subsets of $C_{0}[0,T]$, and
$\mathfrak{m}$ denotes the Wiener measure characterized by

$\mathfrak{m}(\\{x:x(t)\leq\tau\\})=\frac{1}{\sqrt{2\pi
t}}\int_{-\infty}^{\tau}\exp\bigg{[}-\frac{u^{2}}{2t}\bigg{]}du.$

Using the Kolmogorov’s extension theorem (for instance, see [11, 19]), the
Wiener space $C_{0}[0,T]$ can be illustrated as the set of all (continuous)
sample paths of the standard Brownian motion process (SBMP). In [1], Cameron
provided an integration by parts formula for functionals on the classical
Wiener space $C_{0}[0,T]$. In [2], Cameron and Storvick developed the parts
formula for the analytic Feynman integral of functionals on $C_{0}[0,T]$. They
also applied their result to establish the evaluation formula for the analytic
Feynman integral of unbounded functionals on $C_{0}[0,T]$. The parts formula
on $C_{0}[0,T]$ introduced in [1] also was developed in [14, 17] to establish
various parts formulas for the analytic Feynman integral. The parts formula
for the analytic Feynman integral is now called the Cameron–Storvick theorem.

On the other hand, the concept of the generalized Wiener integral and the
generalized analytic Feynman integral on $C_{0}[0,T]$ were introduced in [9],
and further developed and used in [3, 15, 16]. In [3, 9, 15, 16], the
generalized Wiener integral was defined by the Wiener integral

$\int_{C_{0}[0,T]}F(\mathcal{Z}_{h}(x,\cdot))d\mathfrak{m}(x),$

where $\mathcal{Z}_{h}(x,\cdot)$ is a Gaussian path defined by the
Paley–Wiener–Zygmund (PWZ) stochastic integral [12, 13] as follows:

$\mathcal{Z}_{h}(x,t)=\int_{0}^{t}h(s)dx(s)\mbox{ with }h\in L_{2}[0,T].$

The parts formula on the function space $C_{a,b}[0,T]$, which is a
generalization of the Cameron–Storvick theorem was provided by Chang and Skoug
in [8] and further developed in [6]. The function space $C_{a,b}[0,T]$ can be
considered as the set of continuous sample paths of the generalized Brownian
motion process (GBMP) determined by continuous functions $a(t)$ and $b(t)$ on
$[0,T]$. A GBMP on a probability space $(\Omega,\mathcal{F},P)$ and a time
interval $[0,T]$ is a Gaussian process $Y\equiv\\{Y_{t}\\}_{t\in[0,T]}$ such
that $Y_{0}=0$ almost surely, and for any cylinder set
$I_{t_{1},\ldots,t_{n},B}$ having the form

$I_{t_{1},\ldots,t_{n},B}=\big{\\{}\omega\in\Omega:(Y(t_{1},\omega),\cdots,Y(t_{n},\omega))\in
B\big{\\}}$

with a set of time moments $0=t_{0}<t_{1}<\cdots<t_{n}\leq T$ and a Borel set
$B\subset\mathbb{R}^{n}$, the measure $P(I_{t_{1},\ldots,t_{n},B})$ of
$I_{t_{1},\ldots,t_{n},B}$ is equal to

$\displaystyle\bigg{(}(2\pi)^{n}\prod\limits_{j=1}^{n}\big{(}b(t_{j})-b(t_{j-1})\big{)}\bigg{)}^{-1/2}$
$\displaystyle\quad\times\int_{B}\exp\bigg{[}-\frac{1}{2}\sum\limits_{j=1}^{n}\frac{[(u_{j}-a(t_{j}))-(u_{j-1}-a(t_{j-1}))]^{2}}{b(t_{j})-b(t_{j-1})}\bigg{]}du_{1}\cdots
du_{n}$

where $u_{0}=0$, $a(t)$ is a continuous real-valued function on $[0,T]$, and
$b(t)$ is an increasing continuous real-valued function on $[0,T]$. For more
details, see [18, 19]. Note that choosing $a(t)\equiv 0$ and $b(t)=t$ on
$[0,T]$, one can see that the GBMP reduces a SBMP (or, Wiener process).

The aim of this paper is to provide a more general Cameron–Storvick theorem
for the generalized analytic Feynman integral associated with Gaussian paths
on the function space $C_{a,b}[0,T]$. As an application, we apply our general
Cameron–Storvick theorem to evaluate the generalized analytic Feynman integral
of certain monomials in terms of PWZ stochastic integrals.

In order to present our assertions, we assume that $a(t)$ is an absolutely
continuous real-valued function on $[0,T]$ such that $a(0)=0$,
$a^{\prime}(t)\in L^{2}[0,T]$, and

$\int_{0}^{T}|a^{\prime}(t)|^{2}d|a|(t)<+\infty,$ (1.1)

where $|a|(\cdot)$ denotes the total variation function of the function
$a(\cdot)$, and $b(t)$ is an increasing, continuously differentiable real-
valued function with $b(0)=0$ and $b^{\prime}(t)>0$ for each $t\in[0,T]$. We
also assume familiarity with [6, 8] and adopt the notation and terminologies
of those papers. The basic concepts and definitions of the function space
$(C_{a,b}[0,T],\mathcal{W}(C_{a,b}[0,T]),\mu)$, which forms a complete
probability space, the concept of the scale-invariant measurability on
$C_{a,b}[0,T]$, the Cameron–Martin space $C_{a,b}^{\prime}[0,T]$ and the PWZ
stochastic integral on $C_{a,b}[0,T]$ may also be found in [4, 5]. In
particular, we refer to the reference [7] for the definition and the
properties of the Gaussian processes $\mathcal{Z}_{k}$ used in this paper.
However, in order to propose our assertions in this paper, we shall introduce
the following terminologies:

(i) The Hilbert space: Let

$L_{a,b}^{2}[0,T]=\bigg{\\{}v:\int_{0}^{T}v^{2}(s)db(s)<+\infty\hbox{ and
}\int_{0}^{T}v^{2}(s)d|a|(s)<+\infty\bigg{\\}}.$

Then $L_{a,b}^{2}[0,T]$ is a separable Hilbert space with the inner product
given by

$(u,v)_{a,b}=\int_{0}^{T}u(t)v(t)dm_{|a|,b}(t)\equiv\int_{0}^{T}u(t)v(t)d[b(t)+|a|(t)],$

where $m_{|a|,b}$ denotes the Lebesgue–Stieltjes measure induced by
$|a|(\cdot)$ and $b(\cdot)$.

(ii) The Cameron–Martin space in $C_{a,b}[0,T]$: Let

$C_{a,b}^{\prime}[0,T]=\bigg{\\{}w\in
C_{a,b}[0,T]:w(t)=\int_{0}^{t}z(s)db(s)\hbox{ for some }z\in
L_{a,b}^{2}[0,T]\bigg{\\}}.$

Then $C_{a,b}^{\prime}\equiv C_{a,b}^{\prime}[0,T]$ with the inner product

$(w_{1},w_{2})_{C_{a,b}^{\prime}}=\int_{0}^{T}Dw_{1}(t)Dw_{2}(t)db(t)$

is a separable Hilbert space, where the (homeomorhic) operator
$D:C_{a,b}^{\prime}[0,T]\to L_{a,b}^{2}[0,T]$ is given by

$Dw(t)=z(t)=\frac{w^{\prime}(t)}{b^{\prime}(t)}.$ (1.2)

(iii) The PWZ stochastic integral: Let $\\{e_{n}\\}_{n=1}^{\infty}$ be a
complete orthonormal set in
$(C_{a,b}^{\prime}[0,T],\|\cdot\|_{C_{a,b}^{\prime}})$ such that the
$De_{n}$’s are of bounded variation on $[0,T]$. Then for $w\in
C_{a,b}^{\prime}[0,T]$ and $x\in C_{a,b}[0,T]$, we define the PWZ stochastic
integral $(w,x)^{\sim}$ as follows:

$(w,x)^{\sim}=\lim\limits_{n\to\infty}\int_{0}^{T}\sum_{j=1}^{n}(w,e_{j})_{C_{a,b}^{\prime}}De_{j}(t)dx(t)$

if the limit exists. For each $w\in C_{a,b}^{\prime}[0,T]$, the PWZ stochastic
integral $(w,x)^{\sim}$ exists for a.e. $x\in C_{a,b}[0,T]$.

##  2 Gaussian processes on $C_{a,b}[0,T]$

In order to present our Cameron–Storvick theorem on the function space
$C_{a,b}[0,T]$, we follow the exposition of [4, 5, 7].

Let $C_{a,b}^{*}[0,T]$ be the set of functions $k$ in $C_{a,b}^{\prime}[0,T]$
such that $Dk$ is continuous except for a finite number of finite jump
discontinuities and is of bounded variation on $[0,T]$. For any $w\in
C_{a,b}^{\prime}[0,T]$ and $k\in C_{a,b}^{*}[0,T]$, let the operation $\odot$
between $C_{a,b}^{\prime}[0,T]$ and $C_{a,b}^{*}[0,T]$ be defined by

$w\odot k=D^{-1}(DwDk),\,\mbox{ i.e., }D(w\odot k)=DwDk,$

where $DwDk$ denotes the pointwise multiplication of the functions $Dw$ and
$Dk$. Then $(C_{a,b}^{*}[0,T],\odot)$ is a commutative algebra with the
identity $b$.

For each $t\in[0,T]$, let
$\Phi_{t}(\tau)=D^{-1}\chi_{[0,t]}(\tau)=\int_{0}^{\tau}\chi_{[0,t]}(u)db(u)$,
$\tau\in[0,T]$, and for $k\in C_{a,b}^{\prime}[0,T]$ with $Dk\neq 0$
$m_{L}$-a.e. on $[0,T]$ ($m_{L}$ denotes the Lebesgue measure on $[0,T]$), let
$\mathcal{Z}_{k}(x,t)$ be the PWZ stochastic integral

$\mathcal{Z}_{k}(x,t)=(k\odot\Phi_{t},x)^{\sim}.$ (2.1)

Let $\gamma_{k}(t)=\int_{0}^{t}Dk(u)da(u)$ and let
$\beta_{k}(t)=\int_{0}^{t}(Dk(u))^{2}db(u)$. Then the stochastic process
$\mathcal{Z}_{k}:C_{a,b}[0,T]\times[0,T]\to\mathbb{R}$ is Gaussian with mean
function

$\int_{C_{a,b}[0,T]}\mathcal{Z}_{k}(x,t)d\mu(x)=\int_{0}^{t}h(u)da(u)=\gamma_{k}(t)$

and covariance function

$\displaystyle\int_{C_{a,b}[0,T]}\big{(}\mathcal{Z}_{k}(x,s)-\gamma_{k}(s)\big{)}\big{(}\mathcal{Z}_{k}(x,t)-\gamma_{k}(t)\big{)}d\mu(x)$
$\displaystyle=\int_{0}^{\min\\{s,t}\\{Dk(u)\\}^{2}(u)db(u)=\beta_{k}(\min\\{s,t\\}).$

In addition, by [19, Theorem 21.1], $\mathcal{Z}_{k}(\cdot,t)$ is
stochastically continuous in $t$ on $[0,T]$. If $Dk$ is of bounded variation
on $[0,T]$, then, for all $x\in C_{a,b}[0,T]$, $\mathcal{Z}_{k}(x,t)$ is
continuous in $t$. Of course if $k(t)\equiv b(t)$, then
$\mathcal{Z}_{b}(x,t)=x(t)$, the continuous sample paths of the GBMP $Y$,
which consist the function space $C_{a,b}[0,T]$. Furthermore, if $a(t)\equiv
0$ and $b(t)=t$ on $[0,T]$, then the function space $C_{a,b}[0,T]$ reduces to
the classical Wiener space $C_{0}[0,T]$ and the Gaussian process (2.1) with
$k(t)\equiv t$ is a standard Brownian motion process.

Given any $w\in C_{a,b}^{\prime}[0,T]$ and $k\in C_{a,b}^{*}[0,T]$, it follows
that

$(w,\mathcal{Z}_{k}(x,\cdot))^{\sim}=(w\odot k,x)^{\sim}$ (2.2)

for $\mu$-a.e $x\in C_{a,b}[0,T]$.

In order to establish our Cameron–Storvick theorem for functionals on
$C_{a,b}[0,T]$, we define a class $\mathrm{Supp}_{C_{a,b}^{*}}[0,T]$ as
follows:

$\mathrm{Supp}_{C_{a,b}^{*}}[0,T]=\\{k\in C_{a,b}^{*}[0,T]:Dk\neq
0\,\,\,\,{m}_{L}\mbox{-a.e. on }[0,T]\\}.$

###### Remark 2.1

(i) The space $(\mathrm{Supp}_{C_{a,b}^{*}}[0,T],\odot)$ forms a monoid. The
variance function $b(\cdot)$ of the GBMP $Y$ is the identity in the space
$(\mathrm{Supp}_{C_{a,b}^{*}}[0,T],\odot)$.

(ii) Given a function $k$ in $\mathrm{Supp}_{C_{a,b}^{*}}[0,T]$, the process
$\mathcal{Z}_{k}$ on $C_{a,b}[0,T]\times[0,T]$ is the GBMP determined by the
functions $\gamma_{k}$ and $\beta_{k}$.

For any $k\in\mathrm{Supp}_{C_{a,b}^{*}}[0,T]$, the Lebesgue–Stieltjes
integrals

$\|w\odot k\|_{C_{a,b}^{\prime}}^{2}=\int_{0}^{T}(Dw(t))^{2}(Dk(t))^{2}db(t),$

and

$(w\odot
k,a)_{C_{a,b}^{\prime}}=\int_{0}^{T}Dw(t)Dk(t)Da(t)db(t)=\int_{0}^{T}Dw(t)Dk(t)da(t)$

exist for all $w\in C_{a,b}^{\prime}[0,T]$. Throughout the remainder of this
paper, we thus require $k$ to be in $\mathrm{Supp}_{C_{a,b}^{*}}[0,T]$ for the
process $\mathcal{Z}_{k}$.

##  3 Parts formula for functionals in Gaussian paths

In [1], Cameron derived an integration by parts formula for functionals on the
Wiener space $C_{0}[0,T]$. The parts formula involved the first variation (a
kind of Gâteaux derivative) of functionals on $C_{0}[0,T]$. In this section we
establish an integration by parts formula for functionals in Gaussian paths on
the function space $C_{a,b}[0,T]$. To do this we first provide the definition
of the first variation of functionals on the function space $C_{a,b}[0,T]$.

###### Definition 3.1

Let $F$ be a ${\mathcal{W}}(C_{a,b}[0,T])$-measurable functional on
$C_{a,b}[0,T]$ and let $w\in C_{a,b}[0,T]$. Then given two functions $k_{1}$
and $k_{2}$ in $C_{a,b}[0,T]$,

$\delta_{k_{1},k_{2}}F(x|w)\equiv\delta
F(\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(w,\cdot))=\frac{\partial}{\partial\alpha}F(\mathcal{Z}_{k_{1}}(x,\cdot)+\alpha\mathcal{Z}_{k_{2}}(w,\cdot))\bigg{|}_{\alpha=0}$
(3.1)

(if it exists) is called the first variation of $F$ in the direction $w$.

###### Remark 3.2

Setting $k_{1}=k_{2}\equiv b$ on $[0,T]$, our definition of the first
variation reduces to the first variation studied in [6, 8]. That is,

$\delta_{b,b}F(x|w)=\delta F(x|w).$

Let $\mathcal{Z}_{k}$ be the Gaussian process given by (2.1) on
$C_{a,b}[0,T]\times[0,T]$. We define the $\mathcal{Z}_{k}$-function space
integral (namely, the function space integral associated with the Gaussian
paths $\mathcal{Z}_{k}(x,\cdot)$) for functionals $F$ on $C_{a,b}[0,T]$ by the
formula

$E_{x}[F(\mathcal{Z}_{k}(x,\cdot))]=\int_{C_{a,b}[0,T]}F(\mathcal{Z}_{k}(x,\cdot))d\mu(x)$

whenever the integral exists.

In order to establish an integration by parts formula for the function space
integral associated with Gaussian paths on $C_{a,b}[0,T]$, we need a
translation theorem for the function space integral. The following translation
theorem is due to Chang and Choi [4].

###### Theorem 3.3

Let $k_{1}$ be a function in $\mathrm{Supp}_{C_{a,b}^{*}}[0,T]$ and let $F$ be
a functional on $C_{a,b}[0,T]$ such that $F(\mathcal{Z}_{k_{1}}(x,\cdot))$ is
$\mu$-integrable over $C_{a,b}[0,T]$. Then for any $\theta\in
C_{a,b}^{\prime}[0,T]$ and $k_{2}\in\mathrm{Supp}_{C_{a,b}^{*}}[0,T]$,

$\displaystyle
E_{x}\big{[}F(\mathcal{Z}_{k_{1}}(x,\cdot)+\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\big{]}$ (3.2)
$\displaystyle=\exp\bigg{[}-\frac{1}{2}\|\theta\odot
k_{2}\|_{C_{a,b}^{\prime}}^{2}-(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}\bigg{]}$ $\displaystyle\quad\times
E_{x}\Big{[}F(\mathcal{Z}_{k_{1}}(x,\cdot))\exp\big{[}(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}\big{]}\Big{]}.$

We are now ready to present our integration by parts formula for functionals
in Gaussian paths on $C_{a,b}[0,T]$.

###### Theorem 3.4

Let $k_{1}$ and $k_{2}$ be functions in $\mathrm{Supp}_{C_{a,b}^{*}}[0,T]$,
let $\theta$ be a function in $C_{a,b}^{\prime}[0,T]$, and let $F$ be a
functional on $C_{a,b}[0,T]$ such that $F(\mathcal{Z}_{k_{1}}(x,\cdot))$ is
$\mu$-integrable over $C_{a,b}[0,T]$. Furthermore assume that

$E_{x}\big{[}\big{|}\delta
F(\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\big{|}\big{]}<+\infty.$ (3.3)

Then

$\displaystyle E_{x}\big{[}\delta
F(\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\big{]}$ (3.4)
$\displaystyle=E_{x}\big{[}(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}F(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}-(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}E_{x}\big{[}F(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}.$

###### Proof 1

By using (3.1) and (3.2), it follows that

$\displaystyle E_{x}\big{[}\delta
F(\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\big{]}$ (3.5)
$\displaystyle=E_{x}\bigg{[}\frac{\partial}{\partial\alpha}F(\mathcal{Z}_{k_{1}}(x,\cdot)+\alpha\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\bigg{|}_{\alpha=0}\bigg{]}$
$\displaystyle=\frac{\partial}{\partial\alpha}\Big{(}E_{x}\big{[}F(\mathcal{Z}_{k_{1}}(x,\cdot)+\mathcal{Z}_{\alpha
k_{2}}(\theta\odot k_{1},\cdot))\big{]}\Big{)}\bigg{|}_{\alpha=0}$
$\displaystyle=\frac{\partial}{\partial\alpha}\bigg{(}\exp\bigg{[}-\frac{\alpha^{2}}{2}\|\theta\odot
k_{2}\|_{C_{a,b}^{\prime}}-\alpha(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}\bigg{]}$ $\displaystyle\qquad\qquad\times
E_{x}\Big{[}F(\mathcal{Z}_{k_{1}}(x,\cdot))\exp\big{[}\alpha(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}\big{]}\Big{]}\bigg{)}\bigg{|}_{\alpha=0}$
$\displaystyle=E_{x}\big{[}(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}F(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}-(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}E_{x}\big{[}F(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}.$

The second equality of (3.5) follows from (3.3) and Theorem 2.27 in [10].

##  4 Cameron–Storvick theorem for the generalized analytic Feynman integral
associated with Gaussian paths

In this section, we establish the Cameron–Storvick theorem for the generalized
analytic Feynman integral of functionals $F$ on the function space
$C_{a,b}[0,T]$. We begin this section with the definition of the generalized
analytic Feynman integral associated with Gaussian process $\mathcal{Z}_{k}$
($\mathcal{Z}_{k}$-Feynman integral) on $C_{a,b}[0,T]$.

Throughout the remainder of this paper, let $\mathbb{C}_{+}$ and
$\mathbb{\widetilde{C}}_{+}$ denote the set of complex numbers with positive
real part, and non-zero complex numbers with nonnegative real part,
respectively. For each $\lambda\in\mathbb{C}$, $\lambda^{1/2}$ denotes the
principal square root of $\lambda$; i.e., $\lambda^{1/2}$ is always chosen to
have nonnegative real part, so that $\lambda^{-1/2}=(\lambda^{-1})^{1/2}$ is
in $\mathbb{C}_{+}$ for all $\lambda\in\widetilde{\mathbb{C}}_{+}$.

###### Definition 4.1

Given a function $k\in\mathrm{Supp}_{C_{a,b}^{*}}[0,T]$, let $\mathcal{Z}_{k}$
be the Gaussian process given by (2.1) and let $F$ be a $\mathbb{C}$-valued
scale-invariant measurable functional on $C_{a,b}[0,T]$ such that the
generalized $\mathcal{Z}_{k}$-function space integral (namely, the function
space integral associated with the Gaussian paths $\mathcal{Z}_{k}(x,\cdot)$)

$J_{F}(\mathcal{Z}_{k};\lambda)=E_{x}[F(\lambda^{-1/2}\mathcal{Z}_{k}(x,\cdot))]$

exists and is finite for all $\lambda>0$. If there exists a function
$J_{F}^{*}(\mathcal{Z}_{k};\lambda)$ analytic on $\mathbb{C}_{+}$ such that
$J_{F}^{*}(\mathcal{Z}_{k};\lambda)=J_{F}(\mathcal{Z}_{k};\lambda)$ for all
$\lambda\in(0,+\infty)$, then $J_{F}^{*}(\mathcal{Z}_{k};\lambda)$ is defined
to be the analytic $\mathcal{Z}_{k}$-function space integral (namely, the
analytic function space integral associated with the Gaussian paths
$\mathcal{Z}_{k}(x,\cdot)$) of $F$ over $C_{a,b}[0,T]$ with parameter
$\lambda$, and for $\lambda\in\mathbb{C}_{+}$ we write

$E_{x}^{\mathrm{an}_{\lambda}}[F(\mathcal{Z}_{k}(x,\cdot))]\equiv\int_{C_{a,b}[0,T]}^{\mathrm{an}_{\lambda}}F(\mathcal{Z}_{k}(x,\cdot))d\mu(x)=J_{F}^{*}(\mathcal{Z}_{k};\lambda).$

Let $q$ be a non-zero real number and let $F$ be a scale-invariant measurable
functional whose analytic $\mathcal{Z}_{k}$-function space integral,
$E_{x}^{\mathrm{an}_{\lambda}}[F(\mathcal{Z}_{k}(x,\cdot))]$, exists for all
$\lambda$ in $\mathbb{C}_{+}$. If the following limit exists, we call it the
generalized analytic $\mathcal{Z}_{k}$-Feynman integral of $F$ with parameter
$q$, and we write

$E_{x}^{\mathrm{anf}_{q}}[F(\mathcal{Z}_{k}(x,\cdot))]\equiv\int_{C_{a,b}[0,T]}^{\mathrm{anf}_{q}}F(\mathcal{Z}_{k}(x,\cdot))d\mu(x)=\lim\limits_{\begin{subarray}{c}\lambda\to-
iq\\\
\lambda\in\mathbb{C}_{+}\end{subarray}}E_{x}^{\mathrm{an}_{\lambda}}[F(\mathcal{Z}_{k}(x,\cdot))].$
(4.1)

We are now ready to establish a Cameron–Storvick type theorem for our
generalized analytic Feynman integral. It will be helpful to establish the
following lemma before giving the main theorem.

###### Lemma 4.2

Let $k_{1}$, $k_{2}$, $\theta$, and $F$ be as in Theorem 3.4. For each
$\rho>0$, assume that $F(\rho\mathcal{Z}_{k_{1}}(x,\cdot))$ is
$\mu$-integrable over $C_{a,b}[0,T]$. Furthermore assume that for each
$\rho>0$,

$E_{x}\big{[}\big{|}\delta
F(\rho\mathcal{Z}_{k_{1}}(x,\cdot)|\rho\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\big{|}\big{]}<+\infty.$

Then

$\displaystyle E_{x}\big{[}\delta
F(\rho\mathcal{Z}_{k_{1}}(x,\cdot)|\rho\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\big{]}$ (4.2)
$\displaystyle=E_{x}\big{[}(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}F(\rho\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}-(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}E_{x}\big{[}F(\rho\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}.$

###### Proof 2

Let $G(x)=F(\rho x)$. Then

$G(\mathcal{Z}_{k_{1}}(x,\cdot)+\alpha\mathcal{Z}_{k_{2}}(w,\cdot))=F(\rho\mathcal{Z}_{k_{1}}(x,\cdot)+\rho\alpha\mathcal{Z}_{k_{2}}(w,\cdot))$

and

$\frac{\partial}{\partial\alpha}G(\mathcal{Z}_{k_{1}}(x,\cdot)+\alpha\mathcal{Z}_{k_{2}}(w,\cdot))\bigg{|}_{\alpha=0}=\frac{\partial}{\partial\alpha}F(\rho\mathcal{Z}_{k_{1}}(x,\cdot)+\rho\alpha\mathcal{Z}_{k_{2}}(w,\cdot))\bigg{|}_{\alpha=0}.$

Thus $\delta G(\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))=\delta
F(\rho\mathcal{Z}_{k_{1}}(x,\cdot)|\rho\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))$. Hence by equation (3.4) with $F$ replaced with $G$, we have

$\displaystyle E_{x}\big{[}\delta
F(\rho\mathcal{Z}_{k_{1}}(x,\cdot)|\rho\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\big{]}$ $\displaystyle=E_{x}\big{[}\delta
G(\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\big{]}$
$\displaystyle=E_{x}\big{[}(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}G(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}-(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}E_{x}\big{[}G(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}$
$\displaystyle=E_{x}\big{[}(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}F(\rho\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}-(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}E_{x}\big{[}F(\rho\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}$

which establishes (4.2).

Next we provide the Cameron–Storvick theorem for the generalized analytic
$\mathcal{Z}_{k}$-Feynman integral on the function space $C_{a,b}[0,T]$.

###### Theorem 4.3

Let $k_{1}$, $k_{2}$, $\theta$, and $F$ be as in Lemma 4.2. Then if any two of
the three generalized analytic Feynman integrals in the following equation
exist, then the third one also exists, and equality holds:

$\displaystyle E_{x}^{\mathrm{anf}_{q}}\big{[}\delta
F(\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\big{]}$
$\displaystyle=-iqE_{x}^{\mathrm{anf}_{q}}\big{[}(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}F(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}$
$\displaystyle\quad-(-iq)^{1/2}(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}E_{x}^{\mathrm{anf}_{q}}\big{[}F(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}.$

###### Proof 3

Given $\rho>0$ and $\theta\in C_{a,b}^{\prime}[0,T]$, let
$\theta_{\rho}=\frac{1}{\rho}\theta$. Then $\theta_{\rho}$ is a function in
$C_{a,b}^{\prime}[0,T]$, and $\theta\odot k_{1}=\rho\theta_{\rho}\odot k_{1}$.
By equation (4.2) with $\theta$ replaced with $\theta_{\rho}$,

$\displaystyle E_{x}\big{[}\delta
F(\rho\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\big{]}$ (4.3) $\displaystyle=E_{x}\big{[}\delta
F(\rho\mathcal{Z}_{k_{1}}(x,\cdot)|\rho\mathcal{Z}_{k_{2}}(\theta_{\rho}\odot
k_{1},\cdot))\big{]}$
$\displaystyle=E_{x}\big{[}(\theta_{\rho},\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}F(\rho\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}-(\theta_{\rho}\odot
k_{2},a)_{C_{a,b}^{\prime}}E_{x}\big{[}F(\rho\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}$
$\displaystyle=\rho^{-2}E_{x}\big{[}(\theta,\rho\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}F(\rho\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}$
$\displaystyle\quad-\rho^{-1}(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}E_{x}\big{[}F(\rho\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}.$

Now let $\rho=\lambda^{-1/2}$. Then equation (4.3) becomes

$\displaystyle E_{x}\big{[}\delta
F(\lambda^{-1/2}\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(w,\cdot))\big{]}$
(4.4) $\displaystyle=\lambda
E_{x}\big{[}(\theta,\lambda^{-1/2}\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}F(\lambda^{-1/2}\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}$
$\displaystyle\quad-\lambda^{1/2}(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}E_{x}\big{[}F(\lambda^{-1/2}\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}.$

Since $\rho>0$ was arbitrary, we have that equation (4.4) holds for all
$\lambda>0$. We now use Definition 4.1 to obtain our desired conclusions.

###### Corollary 4.4

Under the assumptions as given in Theorem 4.3, it follows that if any two of
the three generalized analytic Feynman integrals in the following equation
exist, then the third one also exists, and equality holds:

$\displaystyle
E_{x}^{\mathrm{anf}_{q}}\big{[}(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}F(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}$
(4.5) $\displaystyle=\frac{i}{q}E_{x}^{\mathrm{anf}_{q}}\big{[}\delta
F(\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\big{]}$ $\displaystyle\quad+(-iq)^{-1/2}(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}E_{x}^{\mathrm{anf}_{q}}\big{[}F(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}.$

###### Remark 4.5

As commented in Section 2 above, if $k\equiv b$ on $[0,T]$, then
$\mathcal{Z}_{b}(x,t)=x(t)$ for each $x\in C_{a,b}[0,T]$. In this case the
generalized analytic $\mathcal{Z}_{b}$-Feynman integral
$E_{x}^{\mathrm{anf}_{q}}[F(\mathcal{Z}_{b}(x,\cdot))]$ agrees with the
previous definition of the generalized analytic Feynman integral
$E_{x}^{\mathrm{anf}_{q}}[F(x)]$, see [6, 8].

In view of Remarks 3.2 and 4.5, we have the following corollary.

######  Corollary 4.6 ([6])

Let $\theta$ be a function in $C_{a,b}^{\prime}[0,T]$, and let $F$ be a
functional on $C_{a,b}[0,T]$ such that for each $\rho>0$, $F(\rho x)$ is
$\mu$-integrable over $C_{a,b}[0,T]$. Furthermore assume that for each
$\rho>0$,

$E_{x}\big{[}\big{|}\delta
F(\rho\mathcal{Z}_{b}(x,\cdot)|\rho\mathcal{Z}_{b}(\theta\odot
b,\cdot))\big{|}\big{]}\equiv E_{x}\big{[}\big{|}\delta F(\rho
x|\rho\theta)\big{|}\big{]}<+\infty.$

Then if any two of the three generalized analytic Feynman integrals in the
following equation exist, then the third one also exists, and equality holds:

$E_{x}^{\mathrm{anf}_{q}}[\delta
F(x|\theta)]=-iqE_{x}^{\mathrm{anf}_{q}}[(\theta,x)^{\sim}F(x)]-(-iq)^{1/2}(\theta,a)_{C_{a,b}^{\prime}}E_{x}^{\mathrm{anf}_{q}}[F(x)].$

The formulas and results in this paper are more complicated than the
corresponding formulas and results in [1, 2, 14, 17] because the Gaussian
process used in this paper is neither centered nor stationary in time.
However, by choosing $a(t)\equiv 0$ and $b(t)=t$ on $[0,T]$, the function
space $C_{a,b}[0,T]$ reduces to the Wiener space $C_{0}[0,T]$, and so the
expected results on $C_{0}[0,T]$ are immediate corollaries of the results in
this paper.

##  5 Generalized analytic Feynman integral of monomials in terms of PWZ
stochastic integrals

When we evaluate the following generalized analytic Feynman integral

$E_{x}^{\mathrm{anf}_{q}}\bigg{[}\prod_{j=1}^{m}(\theta\odot
k_{j},x)^{\sim}\bigg{]}$ (5.1)

we might not be able to use the change of variables theorem of the usual
measure theory, because the set of Gaussian random variables $(\theta\odot
k_{j},x)^{\sim}$, $j=1,\ldots,m$, are generally not independent. In this case,
to apply the change of variables theorem for the calculation of (5.1), we
might apply the Gram–Schmidt process for the set of functions $\\{\theta\odot
k_{1},\ldots,\theta\odot k_{m}\\}$.

Using equation (4.5), we indeed see that the generalized Feynman integral of
functionals having the form (5.1) can be calculated very explicitly. In this
section we present interesting examples to which equation (4.5) can be
applied.

###### Example 5.1

Let $k_{1}$ and $k_{2}$ be functions in $\mathrm{Supp}_{C_{a,b}^{*}}[0,T]$,
and given a function $\theta$ in $C_{a,b}^{\prime}[0,T]$, set
$F(x)=(\theta,x)^{\sim}$ for $x\in C_{a,b}[0,T]$. Then using equations (3.1)
and (2.2), it follows that for any $w$ in $C_{a,b}^{\prime}[0,T]$,

$\displaystyle\delta
F(\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(w,\cdot))$
$\displaystyle=\frac{\partial}{\partial\alpha}\big{\\{}(\theta,\mathcal{Z}_{k_{1}}(x,\cdot))^{\sim}+\alpha(\theta,\mathcal{Z}_{k_{2}}(w,\cdot))^{\sim}\big{\\}}\bigg{|}_{\alpha=0}$
$\displaystyle=(\theta,\mathcal{Z}_{k_{2}}(w,\cdot))^{\sim}$
$\displaystyle=(\theta\odot k_{2},w)^{\sim}$ $\displaystyle=(\theta\odot
k_{2},w)_{C_{a,b}^{\prime}}.$

From this, we see that

$\delta F(\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))=(\theta\odot k_{2},\theta\odot k_{1})_{C_{a,b}^{\prime}}.$ (5.2)

Also using (4.1), it follows that

$\displaystyle
E_{x}^{\mathrm{anf}_{q}}\big{[}F(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}$
$\displaystyle=E_{x}^{\mathrm{anf}_{q}}\big{[}(\theta,\mathcal{Z}_{k_{1}}(x,\cdot))^{\sim}\big{]}$
(5.3) $\displaystyle=E_{x}^{\mathrm{anf}_{q}}\big{[}(\theta\odot
k_{1},x)^{\sim}\big{]}=(-iq)^{-1/2}(\theta\odot k_{1},a)_{C_{a,b}^{\prime}}.$

Next using equations (4.5), (5.2), and (5.3), we obtain the formula

$\displaystyle E_{x}^{\mathrm{anf}_{q}}\big{[}(\theta\odot
k_{2},x)^{\sim}(\theta\odot k_{1},x))^{\sim}\big{]}$ (5.4)
$\displaystyle\equiv\int_{C_{a,b}[0,T]}^{\mathrm{anf}_{q}}(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}(\theta,\mathcal{Z}_{k_{1}}(x,\cdot))^{\sim}d\mu(x)$
$\displaystyle=E_{x}^{\mathrm{anf}_{q}}\big{[}(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}F(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}$
$\displaystyle=\frac{i}{q}E_{x}^{\mathrm{anf}_{q}}\big{[}\delta
F(\mathcal{Z}_{k_{1}}(x,\cdot)|\mathcal{Z}_{k_{2}}(\theta\odot
k_{1},\cdot))\big{]}$ $\displaystyle\quad+(-iq)^{-1/2}(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}E_{x}^{\mathrm{anf}_{q}}\big{[}F(\mathcal{Z}_{k_{1}}(x,\cdot))\big{]}$
$\displaystyle=\frac{i}{q}(\theta\odot k_{2},\theta\odot
k_{1})_{C_{a,b}^{\prime}}+\frac{i}{q}(\theta\odot
k_{2},a)_{C_{a,b}^{\prime}}(\theta\odot k_{1},a)_{C_{a,b}^{\prime}}.$

In our next example, for any positive integer $m\in\\{3,4,\ldots\\}$, we
obtain a recurrence relation for the generalized analytic Feynman integral

$E_{x}^{\mathrm{anf}_{q}}\bigg{[}\prod_{j=1}^{m}(\theta\odot
k_{j},x)^{\sim}\bigg{]}.$

###### Example 5.2

For a positive integer $m\geq 3$, let $\\{k_{1},\ldots,k_{m-1},k_{m}\\}$ be a
finite set of functions in $\mathrm{Supp}_{C_{a,b}^{*}}[0,T]$, and given a
function $\theta\in C_{a,b}^{\prime}[0,T]$, set

$\displaystyle F(x)=\prod_{j=1}^{m-1}(\theta\odot
k_{j},x)^{\sim}=\prod_{j=1}^{m-1}(\theta\odot
k_{j},\mathcal{Z}_{b}(x,\cdot))^{\sim}.$

First, using equation (3.1), it follows that for all $w\in
C_{a,b}^{\prime}[0,T]$,

$\displaystyle\delta F($ $\displaystyle x|\mathcal{Z}_{k_{m}}(w,\cdot))=\delta
F(\mathcal{Z}_{b}(x,\cdot)|\mathcal{Z}_{k_{m}}(w,\cdot))$
$\displaystyle=\frac{\partial}{\partial\alpha}\prod_{j=1}^{m-1}\big{\\{}(\theta\odot
k_{j},\mathcal{Z}_{b}(x,\cdot))^{\sim}+\alpha(\theta\odot
k_{j},\mathcal{Z}_{k_{m}}(w,\cdot))^{\sim}\big{\\}}\bigg{|}_{\alpha=0}$
$\displaystyle=\sum_{l=1}^{m-1}\bigg{(}\prod\limits_{\begin{subarray}{1}j=1\\\
j\neq l\end{subarray}}^{m-1}(\theta\odot
k_{j},\mathcal{Z}_{b}(x,\cdot))^{\sim}\bigg{)}(\theta\odot
k_{l},\mathcal{Z}_{k_{m}}(w,\cdot))^{\sim}.$

Then, in particular, it follows that

$\displaystyle\delta F($
$\displaystyle\mathcal{Z}_{b}(x,\cdot)|\mathcal{Z}_{k_{m}}(\theta\odot
b,\cdot))=\delta
F(\mathcal{Z}_{b}(x,\cdot)|\mathcal{Z}_{k_{m}}(\theta,\cdot))$ (5.5)
$\displaystyle=\sum_{l=1}^{m-1}\bigg{(}\prod\limits_{\begin{subarray}{1}j=1\\\
j\neq l\end{subarray}}^{m-1}(\theta\odot
k_{j},\mathcal{Z}_{b}(x,\cdot))^{\sim}\bigg{)}(\theta\odot
k_{l},\mathcal{Z}_{k_{m}}(\theta,\cdot))^{\sim}$
$\displaystyle=\sum_{l=1}^{m-1}\bigg{(}\prod\limits_{\begin{subarray}{1}j=1\\\
j\neq l\end{subarray}}^{m-1}(\theta\odot
k_{j},\mathcal{Z}_{b}(x,\cdot))^{\sim}\bigg{)}(\theta\odot k_{l},\theta\odot
k_{m})_{C_{a,b}^{\prime}}.$

Next, using (4.5) with $k_{1}$ and $k_{2}$ replaced with $b$ and $k_{m}$,
respectively, and with $F(x)=\prod_{j=1}^{m-1}(\theta\odot k_{j},x)^{\sim}$,
(5.5), and the equation $\mathcal{Z}_{b}(x,\cdot)=x$, it follows that

$\displaystyle E_{x}^{\mathrm{anf}_{q}}\bigg{[}\prod_{j=1}^{m}(\theta\odot
k_{j},x)^{\sim}\bigg{]}$ (5.6)
$\displaystyle=E_{x}^{\mathrm{anf}_{q}}\bigg{[}(\theta,\mathcal{Z}_{k_{m}}(x,\cdot))^{\sim}\prod_{j=1}^{m-1}(\theta\odot
k_{j},\mathcal{Z}_{b}(x,\cdot))^{\sim}\bigg{]}$
$\displaystyle=E_{x}^{\mathrm{anf}_{q}}\big{[}(\theta,\mathcal{Z}_{k_{m}}(x,\cdot))^{\sim}F(\mathcal{Z}_{b}(x,\cdot))\big{]}$
$\displaystyle=\frac{i}{q}E_{x}^{\mathrm{anf}_{q}}\big{[}\delta
F(\mathcal{Z}_{b}(x,\cdot)|\mathcal{Z}_{k_{m}}(\theta\odot b,\cdot))\big{]}$
$\displaystyle\quad+(-iq)^{-1/2}(\theta\odot
k_{m},a)_{C_{a,b}^{\prime}}E_{x}^{\mathrm{anf}_{q}}\big{[}F(\mathcal{Z}_{b}(x,\cdot))\big{]}$
$\displaystyle=\frac{i}{q}\sum_{l=1}^{m-1}(\theta\odot k_{l},\theta\odot
k_{m})_{C_{a,b}^{\prime}}E_{x}^{\mathrm{anf}_{q}}\bigg{[}\prod\limits_{\begin{subarray}{1}j=1\\\
j\neq l\end{subarray}}^{m-1}(\theta\odot k_{j},x)^{\sim}\bigg{]}$
$\displaystyle\quad+(-iq)^{-1/2}(\theta\odot
k_{m},a)_{C_{a,b}^{\prime}}E_{x}^{\mathrm{anf}_{q}}\bigg{[}\prod_{j=1}^{m-1}(\theta\odot
k_{j},x)^{\sim}\bigg{]}.$

###### Remark 5.3

Letting $m=3$ in equation (5.6) and applying equations (5.3) and (5.4) allows
us to easily and completely calculate the generalized Feynman integral

$\displaystyle E_{x}^{\mathrm{anf}_{q}}\big{[}(\theta\odot
k_{1},x)^{\sim}(\theta\odot k_{2}x)^{\sim}(\theta\odot k_{3},x)^{\sim}\big{]}$
$\displaystyle=E_{x}^{\mathrm{anf}_{q}}\big{[}(\theta,\mathcal{Z}_{k_{1}}(x,\cdot))^{\sim}(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}(\theta,\mathcal{Z}_{k_{3}}(x,\cdot))^{\sim}\big{]}.$

Then setting $m=4$ in equation (5.6) allows us to completely evaluate the
generalized Feynman integral

$\displaystyle E_{x}^{\mathrm{anf}_{q}}\big{[}(\theta\odot
k_{1},x)^{\sim}(\theta\odot k_{2}x)^{\sim}(\theta\odot
k_{3},x)^{\sim}(\theta\odot k_{4},x)^{\sim}\big{]}$
$\displaystyle=E_{x}^{\mathrm{anf}_{q}}\big{[}(\theta,\mathcal{Z}_{k_{1}}(x,\cdot))^{\sim}(\theta,\mathcal{Z}_{k_{2}}(x,\cdot))^{\sim}(\theta,\mathcal{Z}_{k_{3}}(x,\cdot))^{\sim}(\theta,\mathcal{Z}_{k_{4}}(x,\cdot))^{\sim}\big{]},$

since we already have complete evaluation formulas for

$E_{x}^{\mathrm{anf}_{q}}\bigg{[}\prod_{j=1}^{l}(\theta\odot
k_{j},x)^{\sim}\bigg{]},\,\,l=1,2,3.$

Then we can evaluate

$E_{x}^{\mathrm{anf}_{q}}\bigg{[}\prod_{j=1}^{5}(\theta\odot
k_{j},x)^{\sim}\bigg{]},$

since we have already evaluated

$E_{x}^{\mathrm{anf}_{q}}\bigg{[}\prod_{j=1}^{l}(\theta\odot
k_{j},x)^{\sim}\bigg{]}$

for $l=1,2,3$ and $4$; etc.

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