# Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift on a Function Space

Authors: Dong Hyun Cho
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

LetC[0,t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0,t] and define a stochastic process Y:C[0,t]×[0,t]→ℝ by Y(x,s)=∫0s‍h(u)dx(u)+a(s) for x∈C[0,t] and s∈[0,t], where h∈L2[0,t] with h≠0 a.e. and a is continuous on [0,t]. Let random vectors Yn:C[0,t]→ℝn and Yn+1:C[0,t]→ℝn+1 be given by Yn(x)=(Y(x,t1),…,Y(x,tn)) and Yn+1(x)=(Y(x,t1),…,Y(x,tn),Y(x,tn+1)), where 0<t1<⋯<tn<tn+1=t is a partition of [0,t]. In this paper we derive a translation theorem for a generalized Wiener integral and then prove that Y is a generalized Brownian motion process with drift a. Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions on C[0,t] with the drift and the conditioning functions Yn and Yn+1. As applications of these simple formulas, we evaluate the generalized conditional Wiener integrals of various functions on C[0,t].


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## Body

## 1. Introduction

LetC0[0,t] denote the Wiener space, the space of real-valued continuous functions x on [0,t] with x(0)=0. On the space, Yeh [1] introduced an inversion formula that a conditional expectation can be found by a Fourier-transform. But Yeh’s inversion formula is very complicated in its application when the conditioning function is vector-valued. In [2], Park and Skoug derived a simple formula for conditional Wiener integrals on C0[0,t] with a vector-valued conditioning function X:C0[0,t]→ℝn given by X(x)=(x(t1),…,x(tn+1)), where 0=t0<t1<⋯<tn<tn+1=t is a partition of the interval [0,t]. In their simple formula, they expressed the conditional Wiener integral directly in terms of an ordinary Wiener integral. Using the simple formula in [2], Chang and Skoug [3] investigated the effect that drift has on the conditional Fourier-Feynman transform, the conditional convolution product, and various relationships that occur between them.On the other hand, letC[0,t] denote the space of real-valued continuous functions on the interval [0,t]. Im and Ryu [4] introduced a probability measure wφ on C[0,t], where φ is a probability measure on the Borel class of ℝ. When φ=δ0, the Dirac measure concentrated at 0, wφ is exactly the Wiener measure on C0[0,t]. On the space C[0,t], the author [5, 6] derived two simple formulas for the conditional Wiener wφ-integral of functions on C[0,t] with the vector-valued conditioning functions Xn:C[0,t]→ℝn+1 and Xn+1:C[0,t]→ℝn+2 given by Xn(x)=(x(t0),x(t1),…,x(tn)) and Xn+1(x)=(x(t0),x(t1),…,x(tn),x(tn+1)) which generalize the Park and Skoug’s formula in [2]. Using these formulas with the conditioning functions Xn and Xn+1, he evaluated the conditional Wiener wφ-integral of function of the form Fm(x)=∫0t‍(x(s))mds for any positive integer m.Leth∈L2[0,t] with h≠0 a.e. on [0,t], and let a be a continuous function on [0,t]. Define a stochastic process Y:C[0,t]×[0,t]→ℝ by  Y(x,s)=∫0s‍h(u)dx(u)+a(s) for x∈C[0,t] and s∈[0,t]. Let Yn:C[0,t]→ℝn and Yn+1:C[0,t]→ℝn+1 be given by
(1)Yn(x)=(Y(x,t1),…,Y(x,tn)),Yn+1(x)=(Y(x,t1),…,Y(x,tn),Y(x,tn+1)).
In this paper, we derive a translation theorem for a generalized Wiener wφ-integral, and then prove that Y is a generalized Brownian motion process with drift a and variance ∫0s‍(h(u))2du for s∈[0,t]. Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions on C[0,t] with the drift and the conditioning functions Yn and Yn+1. As applications of these simple formulas, we evaluate the generalized conditional Wiener integrals of functions of the forms
(2)Fm(Y(x,·)),Y(x,s1)Y(x,s2),∫L2[0,t]‍exp{i(v,Y(x,·))}dσ(v)
for x∈C[0,t], where 0<s1<s2≤t and σ is a complex Borel measure on L2[0,t].


## 2. A Generalized Brownian Motion Process with Drift

In this section, we introduce a generalized Brownian motion process with drift which generalizes the generalized Wiener spaceCa,b[0,T] as given in [7].Letℂ and ℂ+ denote the sets of complex numbers and complex numbers with positive real parts, respectively. Let (C[0,t],ℬ(C[0,t]),wφ) be the analogue of Wiener space associated with a probability measure φ on the Borel class of ℝ, where ℬ(C[0,t]) denotes the Borel class of C[0,t] [4]. Let α∈ℝ, and let φα be a measure on (ℝ,ℬ(ℝ)) such that φα(B)=φ(B+α) for B∈ℬ(ℝ). For v in L2[0,t] and x in C[0,t], let (v,x)=∫0t‍v(s)dx(s) denote the Paley-Wiener-Zygmund integral of v according to x [4]. Let h∈L2[0,t] with h≠0 a.e. on [0,t], and let a be a continuous function on [0,t]. Define stochastic processes Z,Y:C[0,t]×[0,t]→ℝ by Z(x,s)=∫0s‍h(u)dx(u) and Y(x,s)=∫0s‍h(u)dx(u)+a(s) for x∈C[0,t] and s∈[0,t]. Let b(s)=∫0s‍(h(u))2du for 0≤s≤t.By (3.1) in [4, Theorem 3.1], we have the following translation theorem for a generalized Wiener integral on the analogue of Wiener space.Theorem 1.
Letg be of bounded variation on [0,t], let g0(s)=∫0s‍h(u)g(u)du for s∈[0,t], and let F be a measurable function on C[0,t]. Then, F(Y(x,·)+g0) is also measurable function of x on C[0,t] and
(3)∫C[0,t]‍F(Y(x,·))dwφ(x)=*e-∥g∥2/2∫C[0,t]‍F(Y(x,·)+g0)exp{-(g,x)}dwφα(x),
where =* means that if one side exists then both sides exist, and they are equal.Lettingg=χ[0,s]h, we have the following corollary by Theorem 1.Corollary 2.
Leth be of bounded variation on [0,t] and F a measurable function on C[0,t]. Then, for s∈[0,t]F(Y(x,·)+b(min{s,·})) is also measurable on C[0,t] and
(4)∫C[0,t]‍F(Y(x,·))dwφ(x)=*exp{-12b(s)}∫C[0,t]‍F(Y(x,·)+b(min{s,·}))×exp{-Z(x,s)}dwφα(x).LettingF≡1 in Theorem 1 and Corollary 2, we have the following corollary.Corollary 3.
Under the assumptions as given in Theorem1 and Corollary 2(5)∫C[0,t]‍exp{-(g,x)}dwφα(x)=exp{12∥g∥2}.
In particular for s∈[0,t](6)∫C[0,t]‍exp{-Z(x,s)}dwφα(x)=exp{12b(s)}.Using the same method as used in the proof of Corollary 3.3 in [4], we have the following corollary.Corollary 4.
Under the assumptions as given in Theorem1, one has for λ∈ℂ(7)∫C[0,t]‍exp{-λ(g,x)}dwφα(x)=exp{λ22∥g∥2}.
In particular for s∈[0,t](8)∫C[0,t]‍exp{-λZ(x,s)}dwφα(x)=exp{λ22b(s)}.Lettingλ=-iξ(ξ∈ℝ) in the equations of Corollary 4, we have the following corollary by Fourier transforms on (C[0,t],wφα).Corollary 5.
Under the assumptions as given in Theorem1, let X(x)=(g,x). Then, X is normally distributed with mean zero and variance ∥g∥2 on (C[0,t],wφα) if ∥g∥>0. Moreover, for s∈(0,t]Y(·,s) is normally distributed with mean a(s) and variance b(s) on (C[0,t],wφα). In particular, Z(·,s) is normally distributed on (C[0,t],wφα) with mean 0 and variance b(s).Using the same method as used in the proof of Theorem 3.5 in [4], we have the following theorem.Theorem 6.
Let{g1,…,gm} be an orthonormal subset of L2[0,t] such that each gj is of bounded variation. For j=1,…,m, let Xj(x)=(gj,x). Then, X1,…,Xm are independent, and each Xj has the standard normal distribution. Moreover if, f:ℝm→ℂ is Borel measurable, then
(9)∫C[0,t]‍f(X1(x),…,Xm(x))dwφα(x)=*(12π)m/2∫ℝm‍f(u1,u2,…,um)×exp{-12∑j=1m‍uj2}d(u1,u2,…,um).Theorem 7.
Leth be of bounded variation on [0,t], and let 0=s0<s1<⋯<sm≤t. Then, the random vector (Y(·,s1),…,Y(·,sm)) has a joint density function Wm with respect to wφα given by
(10)Wm(s1,…,sm;u1,…,um)=∏j=1m‍[12π(b(sj)-b(sj-1))]1/2×exp{-∑j=1m‍(uj-a(sj)-(uj-1-a(sj-1)))22(b(sj)-b(sj-1))},
where u0=0, a(0)=0, and (u1,…,um)∈ℝm.Proof.
Forj=1,…,m, let hj=χ[sj-1,sj]h, where χ[sj-1,sj] denotes the indicator function on [sj-1,sj]. Let gj=(1/b(sj)-b(sj-1))hj. Then, g1,…,gm are orthonormal, and Y(·,sj)=∑l=1j(hl,·)+a(sj)=∑l=1jb(sl)-b(sl-1)(gl,·)+a(sj). Let B be a Borel subset of ℝm. By Theorem 6,
(11)wφα((Y(·,s1),…,Y(·,sm))∈B)=(12π)m/2∫ℝm‍χB(∑l=1m‍b(s1)-b(s0)w1+a(s1),…,ffggfffffffffff∑l=1m‍b(sl)-b(sl-1)wl+a(sm))ffffffffff×exp{-12∑j=1m‍wj2}d(w1,…,wm).
For j=1,…,m, let uj=∑l=1j‍b(sl)-b(sl-1)wl+a(sj). Then, by the change of variable theorem,
(12)wφα((Y(·,s1),…,Y(·,sm))∈B)=∫Bwm(s1,…,sm;u1,…,um)d(u1,…,um)
which completes the proof.Remark 8.
The conditiona(0)=0 in Theorem 7 does not mean that the mean function a is continuous on [0,t] with a(0)=0. In fact, it is only needed to express formally the exponential function exp{-(u1-a(s1))2/(2b(s1))} in Wm.By Theorem7 and the change of variable theorem, we have the following corollary.Corollary 9.
Y is a Gaussian process with respect to wφα, and its covariance function is given by
(13)E[(Y(·,s1)-a(s1))(Y(·,s2)-a(s2))]=b(min{s1,s2})fors1,s2∈[0,t],
so that
(14)E[Y(·,s1)Y(·,s2)]=b(min{s1,s2})+a(s1)a(s2).Using Fourier transforms on(C[0,t],wφα), we have the following corollary by Theorem 7.Corollary 10.
Let0≤s1<s2≤t. Then, Y(·,s2)-Y(·,s1) is normally distributed on (C[0,t],wφα) with mean a(s2)-a(s1) and variance b(s2)-b(s1).By Corollary10, we can prove the following corollary.Corollary 11.
Let0≤s1<s2≤s3<s4≤t. Then Y(·,s2)-Y(·,s1) and Y(·,s4)-Y(·,s3) are independent on (C[0,t],wφα).Theorem 12.
Y is a generalized Brownian motion process on (C[0,t],wφα) determined by the mean function a and variance function b.Remark 13.
(1) A generalized Brownian motion process determined by the mean functiona and variance function b is a Gaussian process whose mean and covariance functions are given by a and b(min{s1,s2})-b(0), respectively, for s1,s2∈[0,t] [1].
(2) In [8], the author proved that Y-a is a generalized Brownian motion process using Theorem 6. Comparing with [8], we proved the results in this paper using only the translation theorem [4, Theorem 3.1] which is another approach to prove Y being a generalized Brownian motion process.


## 3. Simple Formulas for Generalized Conditional Wiener Integrals

In this section, we derive two simple formulas for generalized conditional Wiener integrals on(C[0,t],wφα). For this purpose, we start with this section defining a conditional wφα-integral.LetF:C[0,t]→ℂ be integrable on wφα and X a random vector on C[0,t] assuming that the value space of X is a normed space equipped with the Borel σ-algebra. Then, we have the conditional expectation E[F∣X] of F given X from a well-known probability theory. Furthermore, there exists a PX-integrable ℂ-valued function ψ on the value space of X such that E[F∣X](x)=(ψ∘X)(x) for wφα a.e. x∈C[0,t], where PX is the probability distribution of X. The function ψ is called the conditional wφα-integral of F given X, and it is also denoted by E[F∣X]. Let 0=t0<t1<⋯<tn<tn+1=t be a partition of [0,t]. For j=1,…,n+1, let αj(s)=(b(tj)-b(s))/(b(tj)-b(tj-1)), βj(s)=(b(s)-b(tj-1))/(b(tj)-b(tj-1)) and γj(s)=αj(s)βj(s)(b(tj)-b(tj-1)), where s∈[tj-1,tj]. Define random vectors Yn:C[0,t]→ℝn and Yn+1:C[0,t]→ℝn+1 by Yn(x)=(Y(x,t1),…,Y(x,tn)) and Yn+1(x)=(Y(x,t1),…,Y(x,tn),Y(x,tn+1)) for x∈C[0,t]. For any x in C[0,t], define a polygonal function [Y(x,·)]a,b of Y(x,·) by
(15)[Y(x,·)]a,b(s)=∑j=1n+1‍χ(tj-1,tj](s)(αj(s)Y(x,tj-1)+βj(s)Y(x,tj))+χ{0}(s)a(0)
for s∈[0,t], where χ(tj-1,tj] and χ{0} denote indicator functions. For x∈C[0,t] and ξ→n+1=(ξ1,…,ξn,ξn+1)∈ℝn+1, define polygonal functions [Z(x,·)]b and [ξ→n+1]b of Z(x,·) and ξ→n+1 by (15), where Y(x,tj) is replaced by Z(x,tj) and ξj, respectively, for j=1,…,n+1 (ξ0=0 and a(0)=0 formally). Moreover, define a polygonal function [a]b of a by (15), where Y(x,tj) is replaced by a(tj). For ξ→n=(ξ1,…,ξn)∈ℝn, the symbol [ξ→n]b is understood as χ[0,tn][ξ→n+1]b on [0,t].By Theorems 2.4 and 2.9 in [8], we can easily prove the following theorem.Theorem 14.
Lettj-1<s<tj  for some j∈{1,…,n+1}. Then, Y(·,s)-[Y(·,·)]a,b(s) is normally distributed on (C[0,t],wφα) with mean a(s)-[a]b(s) and variance γj(s).Theorem 15.
The process{Y(·,s)-[Y(·,·)]a,b(s):0≤s≤t} and Yn+1 are stochastically independent on (C[0,t],wφα).Proof.
Lettl-1<s<tl for some l∈{1,…,n+1}. Then, for j=1,…,n+1,
(16)E[(Y(·,s)-[Y(·,·)]a,b(s))Y(·,tj)]=E[(Z(·,tj)+a(tj))(Z(·,s)-[Z(·,·)]b(s)+a(s)ffgff-[a]b(s))(Z(·,tj)+a(tj))]=E[Z(·,s)-[Z(·,·)]b(s)]E[Z(·,tj)]+(a(s)-[a]b(s))a(tj)
by Corollary 2.4 and Theorems 2.9 and 2.10 in [8]. By Corollary 5 and Theorem 14 we have
(17)E[(Y(·,s)-[Y(·,·)]a,b(s))Y(·,tj)]=(a(s)-[a]b(s))a(tj)=E[Y(·,s)-[Y(·,·)]a,b(s)]E[Y(·,tj)].
Since Y(·,s)-[Y(·,·)]a,b(s) and Y(·,tj) are normally distributed on wφα, they are independent. The proofs of remainder cases follow easily.Theorem 16.
The processes{Y(·,s)-[Y(·,·)]a,b(s):tj-1≤s≤tj}, where j=1,…,n+1, are stochastically independent on (C[0,t],wφα).Proof.
Lets1∈[tl-1,tl] and s2∈[tj-1,tj] with l<j. If s1∈{0,t1,…,tn+1} or s2∈{0,t1,…,tn+1}, we can prove easily the independence of Y(·,s1)-[Y(·,·)]a,b(s1) and Y(·,s2)-[Y(·,·)]a,b(s2). Suppose that tl-1<s1<tl≤tj-1<s2<tj. By Theorems 3.1 and 2.11 in [8], we have
(18)E[(Y(·,s1)-[Y(·,·)]a,b(s1))(Y(·,s2)-[Y(·,·)]a,b(s2))]=E[(Z(·,s1)-[Z(·,·)]b(s1)+a(s1)-[a]b(s1))gggg×(Z(·,s2)-[Z(·,·)]b(s2)+a(s2)-[a]b(s2))]=E[Z(·,s1)-[Z(·,·)]b(s1)]E[Z(·,s2)-[Z(·,·)]b(s2)]+(a(s1)-[a]b(s1))(a(s2)-[a]b(s2))=E[Y(·,s1)-[Y(·,·)]a,b(s1)]E[Y(·,s2)-[Y(·,·)]a,b(s2)]
which completes the proof.Applying the same method as used in the proof of Theorem 2 in [2, page 383] with Problem 4 of [9, page 216], we have the following theorem by Theorem 15.Theorem 17.
LetF:C[0,t]→ℂ be a function, and let F(Y(x,·)) be wφα-integrable over the variable x. Then, for a Borel subset B of ℝn(19)∫Yn+1-1(B)‍F(Y(x,·))dwφα(x)=∫B‍E[F(Y(x,·)-[Y(x,·)]a,b+[ξ→n+1]b)]dPYn+1(ξ→n+1),
where the expectation is taken over wφα and PYn+1 is the probability distribution of  Yn+1 on (ℝn+1,ℬ(ℝn+1)). Moreover, for PYn+1 a.e. ξ→n+1∈ℝn+1 (hence for a.e. ξ→n+1∈ℝn+1)
(20)E[F(Y(x,·))∣Yn+1](ξ→n+1)=E[F(Y(x,·)-[Y(x,·)]a,b+[ξ→n+1]b)].Using the same method as used in the proof of Theorem 2.5 in [6], we can prove the following theorem.Theorem 18.
LetF:C[0,t]→ℂ be a function and F(Y(x,·))wφα-integrable over the variable x. Moreover, let PYn be a probability distribution of Yn on (ℝn,ℬ(ℝn)). Then, for PYn a.e. ξ→n=(ξ1,…,ξn)∈ℝn (hence for a.e. ξ→n∈ℝn)
(21)E[F∣Yn](ξ→n)=[12π(b(t)-b(tn))]1/2×∫ℝ‍E[F(Y(x,·)-[Y(x,·)]a,b+[ξ→n+1]b)]×exp{-(ξn+1-a(t)-(ξn-a(tn)))22(b(t)-b(tn))}dξn+1,
where  ξ→n+1=(ξ1,…,ξn,ξn+1) and the expectation is taken over wφα.Note that the conditioning functionsYn+1 and Yn describe the positions of paths at the times t1,…,tn,tn+1=t (the present time). Yn+1 contains the present position Y(x,t) of the path Y(x,·) for x∈C[0,t], while Yn does not. Moreover if we let h=1 a.e., α=0, and φ=δ0, the Dirac measure concentrated at 0, then we can obtain Theorems 1 and 2 in [10] by the translation theorem (Theorem 1). If a(0)=0, h=b′, φα=δ0, and t=T, then we can obtain the space Ca,b[0,T] in [7] by Theorem 12. Furthermore, if Y is replaced by the generalized Brownian motion process x(s) on Ca,b[0,T]×[0,T], and we let φα=δ0, then we can also obtain Theorem 3.4 in [7] by Theorem 17. If we let a≡0 and φα=δ0, then we can obtain Theorem 3 in [11] by Theorem 17. If we let a≡0, then we can obtain Theorem 2.12 in [8] by Theorem 17. If we let n=0 and φα=δ0, then we can obtain Remark 2.2 in [3] by Theorem 17. Finally, if we let h=1 a.e., φα=δ0 and a≡0, then we can obtain Theorem 2 in [2] by Theorem 17 which is among the first result expressing the conditional Wiener integrals of functions on C0[0,t] as ordinary Wiener integrals.Remark 19.
Note that Theorems17 and 18 are not generalizations of Theorem 2.9 in [5] and Theorem 2.5 in [6]. In Theorem 2.9 of [5] and Theorem 2.5 of [6], the conditioning functions have initial distributions φ, while Yn+1 and Yn in Theorems 17 and 18 have no initial distributions.


## 4. Evaluation Formulas Using the Simple Formulas

In this section, we derive evaluation formulas for the generalized conditional Wiener integrals of various functions which are of interest in Feynman integration theories and quantum mechanics. For a functionF:C[0,t]→ℂ let FY(x)=F(Y(x,·)) for x∈C[0,t].Lemma 20.
LetFm(x)=∫0t‍(x(s))mds(m∈ℕ) for x∈C[0,t]. Then, (Fm)Y is wφα-integrable and
(22)E[(Fm)Y]=∑l=0[m/2]‍m!2ll!(m-2l)!∫0t‍(a(s))m-2l(b(s))lds,
where [·] denotes the greatest integer function.Proof.
By Corollary5, the change of variable theorem and the binomial expansion,
(23)∫0t‍∫C[0,t]‍|Y(x,s)|mdwφα(x)ds≤(1π)1/2∑l=0m‍2l/2Γ(l+12)(ml)∫(0,t]‍|a(s)|m-l(b(s))l/2ds,
where Γ denote the gamma function, so that (Fm)Y is integrable over (C[0,t],wφα). Furthermore, we have
(24)∫C[0,t]‍∫0t‍(Y(x,s))mdsdwφα(x)=2∑l=0[m/2]‍(m2l)∫(0,t]‍(a(s))m-2l(12πb(s))1/2×∫0∞‍u2lexp{-u22b(s)}duds=∑l=0[m/2]‍m!2ll!(m-2l)!∫0t‍(a(s))m-2l(b(s))lds
which completes the proof.Theorem 21.
Let the assumptions be as given in Lemma20. Then, for a.e. ξ→n+1∈ℝn+1 (hence, PYn+1 a.e. ξ→n+1∈ℝn+1)
(25)E[(Fm)Y∣Yn+1](ξ→n+1)=∑j=1n+1‍∑l=0[m/2]‍m!2ll!(m-2l)!×∫tj-1tj‍(a(s)-[a]b(s)+[ξ→n+1]b(s))m-2l(γj(s))lds,
where the expectation is taken over wφα.Proof.
ForPYn+1 a.e. ξ→n+1∈ℝn+1, we have by Theorems 14 and 17(26)E[(Fm)Y∣Yn+1](ξ→n+1)=∑j=1n+1‍∫(tj-1,tj)‍(12πγj(s))1/2×∫ℝ‍(u+a(s)-[a]b(s)+[ξ→n+1]b(s))m×exp{-u22γj(s)}duds=∑j=1n+1‍∑l=0[m/2]‍m!2ll!(m-2l)!×∫tj-1tj‍(a(s)-[a]b(s)+[ξ→n+1]b(s))m-2l(γj(s))lds
by the same method as used in the proof of Lemma 20.Theorem 22.
Let the assumptions be as given in Theorem21, and for ξ→n=(ξ1,…,ξn)∈ℝn let
(27)Ξ(ξ→n)=∑j=1n‍∑l=0[m/2]‍m!2ll!(m-2l)!×∫tj-1tj‍(a(s)-[a]b(s)+[ξ→n]b(s))m-2l(γj(s))lds.
Then, for a.e. ξ→n∈ℝn (hence PYn a.e. ξ→n∈ℝn)
(28)E[(Fm)Y∣Yn](ξ→n)=Ξ(ξ→n)+m!×∑l=0[m/2]‍∑k=0m-2l‍∑p=0k‍∑q=0[p/2]‍ξnk-p(a(t)-a(tn))p-2q(b(t)-b(tn))q2l+ql!q!(k-p)!(p-2q)!(m-2l-k)!×∫tnt‍(γn+1(s))l(βn+1(s))p(a(s)-[a]b(s))m-2l-kds,
where the expectation is taken over wφα.Proof.
Forξ→n=(ξ1,…,ξn)∈ℝn, let ξ→n+1=(ξ1,…,ξn,ξn+1). By the binomial expansion, we have for tn≤s≤t(29)(a(s)-[a]b(s)+[ξ→n+1]b(s))m-2l=∑k=0m-2l‍∑p=0k‍ξnk-p(m-2l)!p!(k-p)!(m-2l-k)!(βn+1(s))pffffffff×(a(s)-[a]b(s))m-2l-k(ξn+1-ξn)p,
so that by Theorems 18 and 21, we have for a.e. ξ→n=(ξ1,⋯,ξn)∈ℝn(30)E[(Fm)Y∣Yn](ξ→n)=Ξ(ξ→n)+∑l=0[m/2]‍∑k=0m-2l‍∑p=0k‍∑q=0[p/2]‍m!ξnk-p(a(t)-a(tn))p-2q2l-1l!(2q)!(k-p)!(p-2q)!(m-2l-k)!×∫tnt‍(γn+1(s))l(βn+1(s))pffff×(a(s)-[a]b(s))m-2l-k[12π(b(t)-b(tn))]1/2×∫0∞‍u2qexp{-u22(b(t)-b(tn))}duds
by the change of variable theorem. Using the same method as used in the proof of Lemma 20, we have the desired result.From now on, we assume that every expectation is taken overwφα unless otherwise specified. By Corollary 9, Theorems 14 and 17, and Theorem 3.4 in [8], we have the following theorem.Theorem 23.
Let0<s1<s2≤t, s1∈[tl-1-tl] and s2∈[tj-1-tj]. For x∈C[0,t], let G(x)=x(s1)x(s2).  (1)
Ifl≠j, then for a.e. ξ→n+1∈ℝn+1 (hence, PYn+1 a.e. ξ→n+1∈ℝn+1)
(31)E[GY∣Yn+1](ξ→n+1)=(a(s1)-[a]b(s1)+[ξ→n+1]b(s1))×(a(s2)-[a]b(s2)+[ξ→n+1]b(s2)).(2)
Ifl=j, then for a.e. ξ→n+1∈ℝn+1 (hence, PYn+1 a.e. ξ→n+1∈ℝn+1)
(32)E[GY∣Yn+1](ξ→n+1)=αj(s2)(b(s1)-b(tj-1))+(a(s1)-[a]b(s1)+[ξ→n+1]b(s1))×(a(s2)-[a]b(s2)+[ξ→n+1]b(s2)).Theorem 24.
Let the assumptions be as given in Theorem23.  (1)
Ifl≤n, j≤n and l≠j, then for a.e. ξ→n∈ℝn (hence, PYn a.e. ξ→n∈ℝn)
(33)E[GY∣Yn](ξ→n)=(a(s1)-[a]b(s1)+[ξ→n]b(s1))×(a(s2)-[a]b(s2)+[ξ→n]b(s2)).(2)
Ifl≤n, j≤n and l=j, then for a.e. ξ→n∈ℝn (hence PYn a.e. ξ→n∈ℝn)
(34)E[GY∣Yn](ξ→n)=αj(s2)(b(s1)-b(tj-1))+(a(s1)-[a]b(s1)+[ξ→n]b(s1))×(a(s2)-[a]b(s2)+[ξ→n]b(s2)).(3)
Ifl=n+1 and j≤n, then for a.e. ξ→n=(ξ1,…,ξn)∈ℝn (hence PYn a.e. ξ→n∈ℝn)
(35)E[GY∣Yn](ξ→n)=(a(s1)-[a]b(s1)+ξn+βn+1(s1)(a(t)-a(tn)))×(a(s2)-[a]b(s2)+[ξ→n]b(s2)).(4)
Ifl≤n and j=n+1, then for a.e. ξ→n=(ξ1,…,ξn)∈ℝn (hence PYn a.e. ξ→n∈ℝn)
(36)E[GY∣Yn](ξ→n)=(a(s2)-[a]b(s2)+ξn+βn+1(s2)(a(t)-a(tn)))×(a(s1)-[a]b(s1)+[ξ→n]b(s1)).(5)
Ifl=n+1 and j=n+1, then for a.e. ξ→n=(ξ1,…,ξn)∈ℝn (hence PYn a.e. ξ→n∈ℝn)
(37)E[GY∣Yn](ξ→n)=ξn2+ξn(a(t)-a(tn))×(a(s1)-[a]b(s1)+a(s2)-[a]b(s2)fffff+βn+1(s1)+βn+1(s2))+(a(s1)-[a]b(s1)+βn+1(s1)(a(t)-a(tn)))×(a(s2)-[a]b(s2)+βn+1(s2)(a(t)-a(tn)))+βn+1(s1)βn+1(s2)(b(t)-b(tn))+αn+1(s2)(b(s1)-b(tn)).Proof.
Forξ→n=(ξ1,…,ξn)∈ℝn, let ξ→n+1=(ξ1,…,ξn,ξn+1). (1) and (2) follow immediately from Theorem 23. If s∈[tn,t], then
(38)[12π(b(t)-b(tn))]1/2×∫ℝ‍[ξ→n+1]b(s)fffff×exp{-(ξn+1-ξn-(a(t)-a(tn)))22(b(t)-b(tn))}dξn+1=[12π(b(t)-b(tn))]1/2×∫ℝ‍(ξn+βn+1(s)(ξn+1-ξn))×exp{-(ξn+1-ξn-(a(t)-a(tn)))22(b(t)-b(tn))}dξn+1=ξn+βn+1(s)(a(t)-a(tn))
by which (3) and (4) follow. If s1,s2∈[tn,t], then
(39)[12π(b(t)-b(tn))]1/2×∫ℝ‍[ξ→n+1]b(s1)[ξ→n+1]b(s2)×exp{-(ξn+1-ξn-(a(t)-a(tn)))22(b(t)-b(tn))}dξn+1=[12π(b(t)-b(tn))]1/2×∫ℝ‍(ξn+βn+1(s1)(ξn+1-ξn))fffff×(ξn+βn+1(s2)(ξn+1-ξn))fffff×exp{-(ξn+1-ξn-(a(t)-a(tn)))22(b(t)-b(tn))}dξn+1=ξn2+ξn(a(t)-a(tn))(βn+1(s1)+βn+1(s2))+βn+1(s1)βn+1(s2)((a(t)-a(tn))2+b(t)-b(tn))
by which we have (5).Forj=1,…,n+1, let hj=χ(tj-1,tj]h and gj=hj/b(tj)-b(tj-1). Let V be the subspace of L2[0,t] generated by {g1,…,gn+1}, V⊥ the orthogonal complement of V, and 𝒫⊥:L2[0,t]→V⊥ the orthogonal projection. Let ℳ(L2[0,t]) be the class of all ℂ-valued Borel measures of bounded variation over L2[0,t], and let 𝒮wφα be the space of all functions F which for σ∈ℳ(L2[0,t]) have the form
(40)F(x)=∫L2[0,t]‍exp{i(v,x)}dσ(v)
for wφα a.e. x∈C[0,t]. Note that 𝒮wφα is a Banach algebra [4].Now, we have the following theorems which are our final results.Theorem 25.
Leth be of bounded variation and a absolutely continuous on [0,t]. Let F∈𝒮wφα and σ∈ℳ(L2[0,t]) be related by  (40). Then,
(41)E[FY]=∫L2[0,t]‍exp{-12∥vh∥2}dσa(v),
where (dσa/dσ)(v)=exp{i(v,a)} for v∈L2[0,t]. Moreover for a.e. ξ→n+1∈ℝn+1 (hence for PYn+1 a.e. ξ→n+1∈ℝn+1) E[FY∣Yn+1](ξ→n+1) is given by
(42)E[FY∣Yn+1](ξ→n+1)=∫L2[0,t]‍exp{i(v,a-[a]b+[ξ→n+1]b)-12∥𝒫⊥(vh)∥2}dσ(v).Proof.
By Corollary5 and the Fubini’s theorem
(43)E[FY]=∫L2[0,t]‍∫C[0,t]‍exp{i(v,Y(x,·))}dwφα(x)dσ(v)=∫L2[0,t]‍exp{i(v,a)}exp{-12∥vh∥2}dσ(v),
where the last equality follows from the following integral formula:
(44)∫ℝ‍exp{-au2+ibu}du=(πa)1/2exp{-b24a}
for a∈ℂ+ and any real b. Furthermore, for a.e. ξ→n+1∈ℝn+1, we have by Theorems 3.4 and 3.6 in [8]
(45)E[FY∣Yn+1](ξ→n+1)=∫L2[0,t]‍exp{i(v,a-[a]b+[ξ→n+1]b)}×∫C[0,t]‍exp{i(v,Z(x,·)-[Z(x,·)]b)}fffffffdwφα(x)dσ(v)=∫L2[0,t]‍exp{i(v,a-[a]b+[ξ→n+1]b)}fffffff×exp{-12∥𝒫⊥(vh)∥2}dσ(v)
which completes the proof.Theorem 26.
Let the assumptions be as given in Theorem25 and for ξ→n∈ℝn let
(46)A(v,ξ→n)=exp{i(v,a-[a]b+[ξ→n]b)-12∥𝒫⊥(vh)∥2}.
Then, for a.e. ξ→n∈ℝn (hence for PYn a.e. ξ→n∈ℝn) E[FY∣Yn](ξ→n) is given by
(47)E[FY∣Yn](ξ→n)=∫L2[0,t]‍A(v,ξ→n)exp{i〈vh,gn+1〉(a(t)-a(tn))b(t)-b(tn)dgggggggggggggggdd-12〈vh,gn+1〉2〈vh,gn+1〉(a(t)-a(tn))b(t)-b(tn)}dσ(v),
where 〈·,·〉 denotes the inner product on L2[0,t].Proof.
Letξ→n=(ξ1,…,ξn)∈ℝn and let ξ→n+1=(ξ1,…,ξn,ξn+1). By the definition of the Paley-Wiener-Zygmund integral it is not difficult to show
(48)(v,[ξ→n+1]b)=(v,[ξ→n]b)+〈vh,gn+1〉b(t)-b(tn)(ξn+1-ξn)
so that we have for a.e. ξ→n=(ξ1,…,ξn)∈ℝn(49)E[FY∣Yn](ξ→n)=[12π(b(t)-b(tn))]1/2×∫L2[0,t]‍A(v,ξ→n)×∫ℝ‍exp{i〈vh,gn+1〉b(t)-b(tn)(ξn+1-ξn)ggggggggggg-(ξn+1-ξn-(a(t)-a(tn)))22(b(t)-b(tn))〈vh,gn+1〉b(t)-b(tn)}dξn+1dσ(v)=∫L2[0,t]‍A(v,ξ→n)exp{i〈vh,gn+1〉(a(t)-a(tn))b(t)-b(tn)ffffffffffffgggg-12〈vh,gn+1〉2〈vh,gn+1〉(a(t)-a(tn))b(t)-b(tn)}dσ(v)
by Theorem 25 and the change of variable theorem.



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*Source: 469840-2013-11-02.xml*