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(1)國立高雄大學統計學研究所碩士論文. Analysis of Complex Brownian Functionals 複數布朗泛函之分析. 研究生：顏廣杰撰指導教授：李育嘉博士. 中華民國九十六年六月.

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(4) Analysis of Complex Brownian Functionals. by Kuang-Ghieh Yen Advisor Yuh-Jia Lee. Institute of Statistics, National University of Kaohsiung Kaohsiung, Taiwan 811 R.O.C. June 2007. 1.

(5) Contents. Z` zZ`. ii iii. 1 Introduction. 1. 2 White noise calculus. 2. 3 Test and generalized functions. 3. 4 Complex Brownian motion. 6. 5 Itˆ o formula. 9. References. 14. i.

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(7) Analysis of Complex Brownian Functionals Advisor: Dr. Yuh-Jia Lee Department of Applied Mathematics National University of Kaohsiung. Student: Kuang-Ghieh Yen Institute of Statistics National University of Kaohsiung. Abstract A theory of generalized functions based on the complex Brownian motion {Z(t) : t ∈ R}, for which each Z(t) is N (0, |t|) is established on the probability space (Sc0 , B(Sc0 ), ν(dz)), where S 0 is the dual of the Schwartz space S, Sc0 , the complexification of S 0 , identified as the product space S 0 × S 0 , B(Sc0 ) the Borel field of S 0 × S 0 and ν(dz) denotes the product measure µ1 (dx)µ1 (dy). Using the representation of the complex Brownian motion 1 Zt (x, y) = √ (hx, ht i + ihy, ht i) , 2 where ht =.   1(0,t] ,. t > 0,.  −1[t,0] , t < 0.. and employing the technique of white noise calculus initiated by Hida ( see, e.g. [2] and [4]), we analyze functionals of complex Brownian motion. To define generalized complex Brownian functionals, we adopt the space of CKS entire functionals as test functions. As applications, the stochastic integral with respect to a complex Brownian motion are defined and studied. The Itô formula for complex Brownian functionals is obtained and it is shown that the evaluation of stochastic integral with respect to a complex Brownian motion follows the rule of Stratonovich integral. Keyword : complex Brownian motion, white noise analysis iii.

(8) 1. Introduction In this paper we are devoted to a systematic study of complex Brownian func-. tionals, by which we mean functions of complex Brownian motion given by 1 Z(t, ω) = √ [B1 (t, ω) + iB2 (t, ω)], 2 where B1 and B2 are independent real-valued standard Brownian motion. Clearly Z(t) is normally distributed with mean zero and variance parameter |t|. As the calculus of the complex Brownian motion will play the main role, we need to represent Z(t) as a function on a certain probability space. In this paper, we choose (Sc0 , B(Sc0 ), ν(dz)) as the underlying probability space, where Sc0 is the complexification of S 0 which is identified as the product space S 0 × S 0 , B(Sc0 ), the Borel field of S 0 × S 0 and ν(dz) denotes the product measure µ1 (dx)µ1 (dy), where µt represents the Gaussian measure defined on S 0 with characteristic function given by. Z 2 /2. ei(x,ξ) µt (dx) = e−t|ξ|. C(ξ) =. .. S0. One sees easily that the complex Brownian motion on (Sc0 , B(Sc0 ), ν(dz)) may be represented by 1 Zt (x, y) = √ (hx, ht i + ihy, ht i) , 2 where ht =.   1(0,t]. , t > 0,.  −1[t,0]. , t < 0.. The calculus of complex Brownian functional is then performed with respect to the measure ν(dz). For example, let f : Hc → C be an entire function and Z(t) =. √1 hx 2. + iy, ht i the complex Brownian motion. Then we have Z Z 2. E[|f (Z(t))| ] = S0. S0. |f (hx + iy, ht i)|2 µ1/2 (dx)µ1/2 (dy).. The above example gives a connection between the calculus of complex Brownian motion and the Segal-Bargmann entire functional on H, denoted by SBt (H) (see §3 for definition). 1.

(9) Being motivated by the (real) white noise analysis initiated by Hida (see e.g.[4], [2]), it is desirable to develop a theory of generalized complex functionals by white noise calculus approach. The main results are given as follows: (1) A theory of generalized Segal-Bargmann functional is established using CKS entire functionals [1] as test functionals. (2) A stochastic integral with respect to complex Brownian motion is defined and studied. (3) An Itô formula for complex Brownian functionals is derived. As an example, it is shown that, for any Segal-Bargmann entire function F , the Itô formula is given by Z. b. F (Z(b)) − F (Z(a)) =. F 0 (Z(t))dZ(t).. a. The formula is then extended to a generalized CKS entire function F by using Hitsuda-Shorohod integral given below: d hhF (Z(t)), φiic = hh∂t∗ F (Z(t)), φiic dt where ∂t = Dδt , ∂t∗ is its adjoint, and φ is any CKS entire functional.. 2. White noise calculus. The (Gaussian) white noise is generally understood as a stationary stochastic process with constant spectral density in the engineering. In mathematics, it can be shown ˙ that the white noise is the time derivative of Brownian motion {B(t) : t > 0}. Since almost all sample path of Brownian motion are differentiable nowhere, the (Gaussian) white noise should be defined in mathematics as a generalized process. In this paper, the Brownian motion denoted by {B(t) : t > 0} will be regarded as a regular generalized functional on S 0 , the dual of the Schwartz space. Since for almost all x (µt ) the white noise t 7→ B˙ t (x) is a generalized function in S 0 . Thus S 0 = S 0 (R) is regarded as the state space of white noise. 2.

(10) S 0 also has a nuclear space structure described as follows: Let A denote the operator A = 1 + t2 − (d/dt)2 with domain D(A) ⊂ L2 = L2 (R) and D(A) contains a CONS {en : n ∈ N} of L2 consisting of eigenfunctions of A with corresponding eigenvalues {2n + 2; n = 1, 2, ...}. {en } are known as Hermite functions. For p ≥ 0, let Sp = D(Ap ). Its dual space is given by Sp∗ = S−p . Then we have S = ∩p Sp which is provided with the projective limit topology and the dual space of S(R1 ) is given by S 0 = ∪p Sp which is equipped with the inductive topology. Moreover, the spaces S ⊂ L2 (R) ⊂ S 0 , form a Gel’fand triple. It is well-known that S 0 carries a standard Gaussian measure µt with variance parameter t. Then the calculus on S 0 is performed with respect to µt . Let µ = µ1 . Then (S 0 , B(S 0 ), µ) is called the white noise space which serves as the underlying probability space for white noise functionals (or generalized Brownian functionals).. 3. Test and generalized functions. Definition 3.1. Let x,y ∈ S 0 , and z = x + iy ∈ Sc0 ≡ S 0 × S 0 . Define the measure ν(dz) := µ1 (dx) × µ1 (dy). Definition 3.2. (Segal-Bargmann space[6]) Let H be a real separable Hilbert space and Hc the complexification of H. A single-valued function f defined on Hc is called a Segal-Bargmann entire function if it satisfies the following conditions: (i) f is analytic in Hc . (ii) The number. Z Z |f (P x + iP y)|2 nt (dx)nt (dy). Mf := sup P. H. H. is finite, where nt denotes as the Gaussian cylinder measure on H with variance parameter t > 0 (for details, we refer the reader to [5]) and P ’s run through all finite dimensional orthogonal projections on H. 3.

(11) Denote the class of Segal-Bargmann entire functions on H by SBt [H] and define p kf kSBt [H] = Mf . Then (SBt [H], k · kSBt [H] ) is a Hilbert space. It follows immediately from [6] that we have ||f ||2SBt [H]. = =. N X. ∞ X (2t)k k=0 ∞ X k=0. k!. ! |Dk f (0)ei1 · · · eik |2. i1 ,...,ik =1. (2t)k k kD f (0)k2HS k [H] . k!. where ||S||HS n (H) denotes the Hilbert-Schmidt norm of a n-linear operator S ∈ Ln (H) defined by ||S||HS n (H) :=. ∞ X. !1/2 |Sei1 · · · ein |2. i1 ,··· ,in =1. which is independent of the choice of CONS {ei } of H. Next, we introduce the CKS entire functionals [1]. Let α(n), n ≥ 0, be a sequence of real numbers satisfying the following conditions: (a) α(0) = 1 and inf n≥0 α(n) > 0, (b) limn→∞ n−1 α(n)1/n = 0, (c) γn+2 /γn+1 ≤ γn+1 /γn , for all n ≥ 0, where γn = α(n)/n!. Definition 3.3. (Infinite-dimensional CKS entire functionals) For each p ∈ R, we define |||φ|||α,p =. ∞ X. α(n). kDn φ(0)k2HS n [S−p ]. !1/2. n!. n=0. and set SBp,α = {φ ∈ SB1/2 (H) : |||φ|||α,p < ∞}. Let SBα be the projective limit of SBp,α for p ≥ 0 and let SBα0 be the dual space of SBα . Then SBα is a nuclear space and we have the following continuous inclusions: 0 SBα ⊂ SBp,α ⊂ SB[(L2 )] ⊂ SBp,α ⊂ SBα0 .. 4.

(12) The space SBα , which is referred to as the CKS entire functionals on Sc0 , will serve as test functionals and SBα0 is referred to as the generalized complex Brownian functionals. 0 The space SBp,α may be identified as the space of entire functions defined on. Sp,c such that : |||φ|||α−1 ,−p < ∞ and the pairing of SBα0 and SBα is defined by ∞ X 1 hhDn Φ(0), Dn ϕ(0)iiHS n , hhΦ, ϕii = n! n=0. where n. h i Dn Φ(0)ei1 · · · ein Dn ϕ(0)ei1 · · · ein .. ∞ X. n. hhD Φ(0), D ϕ(0)iiHS n :=. i1 ,...,in =1. Definition 3.4. (One-dimensional CKS entire functions) If φ(z) can be represented P n by a formal power series ∞ n=0 an z , we define |||φ|||α =. ∞ X. !1/2 α(n)n!|an |2. n=0. and let SBα (R) = {φ : |||φ|||α,p < ∞}. If φ(z) is a formal power series represented by |||φ|||α−1 =. P∞. n=0 bn z. ∞ X n!|bn |2 n=0. α(n). n. , we define. !1/2 .. Then the dual space of SBα is characterized by SB 0α (R) = {φ : |||φ|||α−1 < ∞}. Remark 3.5. Let k be any positive integer and let f (w) = wk (w ∈ C). Then, for any h ∈ H, the functional Φ = f (h·, hi) is clearly a Segal-Bargmann entire function defined on H. Φ is also defined on Sc a.e. (ν) and we have, for ϕ ∈ SBα , Z Z f (hx + iy, hi)ϕ(x + iy)µ 1 (dx)µ 1 (dy). hhΦ, ϕiic = S0. 2. S0. The above identity will be used in the proof of Theorem 5.7. 5. 2.

(13) 4. Complex Brownian motion. Definition 4.1. A stochastic process B(t, ω) is called a real Brownian motion. If it satisfies the following conditions: (i) P{ω ; B(0, ω) = 0}=1. (ii) For any 0 ≤ s < t, the random variable B(t) − B(s) is normally distribution with mean 0 and variance t − s and for any a<b Z b 1 exp(−x2 /2(t − s))dx. P {a ≤ B(t) − B(s) ≤ b} = p 2π(t − s) a (iii) B(t,ω) has independent increments. That is, for any 0 ≤ t1 < ... < tn , the random variables B(t1 ), B(t2 ) − B(t1 ), ..., B(tn ) − B(tn−1 ) are independent. (iv) Almost all sample paths of B(t,ω) are continuous functions, P {ω | B(· , ω) is continuous} = 1. Definition 4.2. Let {B1 (t, ω) : t ≥ 0} and {B2 (t, ω) : t ≥ 0} be two mutually independent real Brownian motions: Define Z = {Z(t, ω) : t ≥ 0} by Z(t, ω) =. [B1 (t, ω) + iB2 (t, ω)] √ . 2. Then Z is a complex Brownian motion process. Lemma 4.3. Recall that the functional representation of complex Brownian motion Z(t) is given by Z(t, x, y) =. hx + iy, ht i √ . 2. Then we have (i). Z Z hhZ(t), 1iic = S0. S0. hx + iy, ht iµ1/2 (dx)µ1/2 (dy) = 0.. (ii). where δm,n. hhZ(t)m , Z(t)n iic = n!δm,n tn ,   1, m = n, =  0, m = 6 n. 6.

(14) Proof. (i) is clear and the proof of (ii) follows immediately from integration by parts formula (see, for example [6]). Example 4.4. If ξ ∈ S and φ ∈ SBα , then it follows immediately from the definition of the hh·, ·iic pairing, we have hhh·, ξi, φiic = Dξ φ(0) = Dφ(0)ξ. (4.1). Proof.. := = =. =. =. hhh·, ξi, φiic Z φ(z)hz, ξiν(dz) Sc0 Z Z φ(x + iy)(hx, ξi − ihy, ξi)µ1/2 (dx)µ1/2 (dy) 0 0 ZS ZS φ(x + iy)hx, ξi − iφ(x + iy)hy, ξiµ1/2 (dx)µ1/2 (dy) S0 S0 Z Z [U sing the f ormula hx, hiφ(x)µt (dx) = t Dφ(x)hµt (dx)] S0 S0 Z Z 1 (Dφ(x + iy)ξ + Dφ(x + iy)ξ)µ1/2 (dx)µ1/2 (dy) S0 S0 2 Z Z Z √ [U sing the f ormula φ(ax + by)µt (dx)µt (dy) = φ( a2 + b2 x)µt (dx)] S0 S0 S0 Z (Dφ(0)ξ)µ1/2 (dx) S0. = Dφ(0)ξ.. It is worth to note that the identity (4.1) remain holds for ξ ∈ L2 [R] and φ ∈ SB[L2 ], in this case Dξ φ(0) is interpreted as Fréchet differentiable in the direction of L2 (R). Now apply the identity (4.1) for ξ = (ht+ − ht )/ and φ ∈ SBα , we have hhh·, (ht+ − ht )/i, φiic = Dφ(0)[(ht+ − ht )/] Let → 0 in the above identity. We obtain the definition of the complex white noise ˙ Z(t) as follows. ˙ hhZ(t), φiic = Dφ(0)δt . ˙ Clearly Z(t) ∈ SBα0 . 7.

(15) Example 4.5. Given h ∈ L2 (R), we have hheh·,hi , φiic = φ(h). (4.2). Proof. hheZ(t) , φiic Z Z φ(x + iy)ehx−iy,ht i µ1/2 (dx)µ1/2 (dy) 0 0 ZS ZS ehx,ht i e−hiy,ht i φ(x + iy)µ1/2 (dx)µ1/2 (dy) S0 S0 Z Z t2 φ(x + iy + ht /2)e−ihy,ht i µ1/2 (dx)µ1/2 (dy) e4 S 0 ZS 0 Z t2 −t2 φ(x + iy + ht /2 + ht /2)µ1/2 (dx)µ1/2 (dy) e4e 4 0 0 S S Z √ φ( 1 + i2 x + ht )µ1/2 (dx). = = = = =. S0. = φ(ht ).. Example 4.6. Let h, k ∈ L2 [R] such that hh, ki = 0. Then we have hhh·, hieh·,ki , φiic = Dφ(k)h Proof. hhh·, hieh·,ki , φii Z Z = hx − iy, hiehx−iy,ki φ(x + iy)µ1/2 (dx)µ1/2 (dy) S0. S0. = (F ixed y, let φy (x) = hx − iy, hiφ(x + iy)) Z Z = ehx,ki−ihy,ki φy (x)µ1/2 (dx)µ1/2 (dy) S 0 ZS 0 Z t2 e−ihy,ki φy (x + k/2)µ1/2 (dx)µ1/2 (dy) = e4 0 0 ZS ZS t2 e−ihy,ki hx − iy + k/2, hiφ(x + iy + k/2)µ1/2 (dx)µ1/2 (dy) = e4 S0. S0. (F ixed x, let φx (iy) = hx + ht /2 − iy, hiφ(x + k/2 + iy)) Z Z t2 4 e−ihy,ki φx (iy)µ1/2 (dy)µ1/2 (dx) = e 0 0 Z SZ S = e0 φx (i(y − ik/2))µ1/2 (dy)µ1/2 (dx) S0. S0. 8. (4.3).

(16) Z Z = S0. S0. hx − iy + k, hiφ(x + iy + k)µ1/2 (dx)µ1/2 (dy). = hhh·, hi, φ(· + k)iic + hhhk, hi, φ(· + k)iic = Dφ(k)h + hhhk, hi, φ(· + k)iic (By Equation (4.1)) = Dφ(k)h. Taking h = (ht+ − ht )/ and k = ht in Equation (4.3) and then letting → 0, ˙ exp(Z(t)) given in the following. we are led to the definition of Z(t) Definition 4.7. For φ ∈ SBα , we define Z(t) ˙ hhZ(t)e , φiic := Dφ(ht )δt .. (4.4). Z(t) ˙ Clearly Z(t)e ∈ SBα .. Example 4.8. If Z(t) ∈ SBα0 is a complex Brownian motion, then d Z(t) ˙ e = eZ(t) Z(t) dt Proof. The proof is immediately follows from Equation (4.3) and Definition 4.7. Example 4.9. (Composition of generalized functional with complex Brownian motion) Let f ∈ SBα0 (R) be an one dimensional generalized CKS entire function repP P∞ −1 n 2 resented by f (z) = ∞ n=0 bn z . Assume that n=0 α(n) n!|bn | < ∞. Then, for ψ ∈ SBα , we define hhf (Z(t)), ψiic =. ∞ X. bn Dn ψ(0)hnt .. (4.5). n=0. It will be proved in the next section that f (Z(t)) ∈ SBα0 .. 5. Itˆ o formula. We first prove that the composition f (Z(t)) defined in Example 4.9 is in fact a generalized Segal-Bargmann functional.. 9.

(17) Theorem 5.1. Let f ∈ SBα0 (R) be an one dimensional generalized CKS entire P n function represented by f (z) = ∞ n=0 bn z . Assume that that {bn } satisfies ∞ X. α(n)−1 n!|bn |2 < ∞.. n=0. Then f (Z(t)), defined by Equation 4.5, is a member of SBα0 . Proof. Recall that hhf (Z(t)), ψiic =. ∞ X. bn Dn ψ(0)hnt .. n=0. For each t > 0, there existsp such that |ht |−p ≤ 1. Thus 1/2 ! n ∞ X D ψ(0)hnt n! √ (α(n))1/2 | |hhf (Z(t)), ψiic | = | bn α(n) n! n=0 v u∞ uX |Dn ψ(0)|2 h2n ≤ |||f |||α−1 t t α(n) n! n=0 v ! u∞ uX kDn ψ(0)k2HS n [S−p ] |ht |2n ≤ |||f |||α−1 t −p α(n) n! n=0 v ! u∞ uX kDn ψ(0)k2HS n [S−p ] α(n)|ht |2n ≤ |||f |||α−1 t −p n! n=0 ≤ |||f |||α−1 |||ψ|||α,p . This proves that f (Z(t)) ∈ SBα0 . Now we are ready to derive the Itô formula. Let f ∈ SBα0 (R). Then we have ∞ X d hhf (Z(t)), φiic = bn nDn φ(0)hn−1 δt t dt n=0. = =. ∞ X n=0 ∞ X. bn nDn−1 (Dφ(0)δt )hn−1 t bn nDn−1 (∂t φ)(0)hn−1 t. n=0. = hh∂t∗ f (Z(t)), φiic , where ∂t = ∂δt and ∂t∗ is the adjoint operator of ∂t . It follows that d f (Z(t)) = ∂t∗ f (Z(t)). dt 10.

(18) This proves the Itô formula for complex Brownian motion. As a summary, we state the above results as a theorem. Theorem 5.2 (Itô formula). Let f ∈ SBα0 (R). Then we have d f (Z(t)) = ∂t∗ f (Z(t)). dt or in the integral form, Z. b. f (Z(b)) − f (Z(a)) = a. ∂t∗ f (Z(t))dt. As in the case of real Brownian motion, the term on the right hand side may be interpreted as a stochastic integral as shown below. Definition 5.3. Suppose that f ∈ SBα0 . Define the stochastic integral f (Z(t)) as follows: Z b hh f (Z(t))dZ(t), φiic := a. lim hh. k4n k→0. n X. f (Z(ti−1 ))(Z(ti ) − Z(ti−1 )), φiic ,. i=1. where a = t0 < t1 < t2 < · · · < tn = b and k4n k = maxj |tj − tj−1 |. Lemma 5.4. Let e h = hx + iy, hi and e k = hx + iy, ki, then hhe he k, φiic = hD2 φ(0), h ⊗ ki Proof. hhe he k, φiic Z Z = hx − iy, hihx − iy, kiφ(x + iy)µ1/2 (dx)µ1/2 (dy) S0 S0 Z Z = (hx, hihx, ki − ihy, hihx, ki − ihx, hihy, ki S0. S0. −hy, hihy, ki)φ(x + iy)µ1/2 (dx)µ1/2 (dy). Z Z = S0. S0. hD2 φ(x + iy), h ⊗ kiµ1/2 (dx)µ1/2 (dy). 2. = hD φ(0), h ⊗ ki.. 11.

(19) Corollary 5.5. Let het = hx + iy, ht i, then n. hhhet , φiic = hDn φ(0), h⊗n t i. Example 5.6. To compare the stochastic integrals for complex Brownian motion with the stochastic integral for real Brownian motion, we first show that Z b 1 Z(t)dZ(t)dt = (Z 2 (b) − Z 2 (a)). 2 a In fact, we have. Z. Z. b. b. Z(t)dZ(t) = a. a. ∂t∗ Z(t)dt. Proof. For any test functional φ, we have Z b hh Z(t)dZ(t), φiic a. = and. n Z Z X. lim. k4n k→0. 0 ZS. = 0. ZS ZS. =. =. =. S0. S0. hx − iy, hti−1 ihx − iy, hti − hti−1 iφ(x + iy)µ1/2 (dx)µ1/2 (dy),. Z Z 0 ZS. =. i=1. 0. hx − iy, hti−1 ihx − iy, hti − hti−1 iφ(x + iy)µ1/2 (dx)µ1/2 (dy) hD2 φ(x + iy), hti−1 ⊗ (hti − hti−1 )iµ1/2 (dx)µ1/2 (dy). 1 hD2 φ(x + iy), [(hti ⊗ hti − hti−1 ⊗ hti−1 ) − (hti−1 − hti )⊗2 ]iµ1/2 (dx)µ1/2 (dy) 2 0 0 ZS ZS 1 hD2 φ(x + iy), (hti ⊗ hti − hti−1 ⊗ hti−1 )iµ1/2 (dx)µ1/2 (dy) 2 0 0 SZ SZ 1 − hD2 φ(x + iy), (hti − hti−1 )⊗2 iµ1/2 (dx)µ1/2 (dy) 2 S0 S0 Z Z 1 ⊗2 hD2 φ(x + iy), (h⊗2 ti − hti−1 )iµ1/2 (dx)µ1/2 (dy) 2 0 0 SZ SZ 1 − hD2 φ(x + iy), (hti − hti−1 )⊗2 iµ1/2 (dx)µ1/2 (dy) 2 S0 S0 (I) + (II).. It is easy to see that 1 lim (I) = hh (Z 2 (b) − Z 2 (a)), φiic , k4n k→0 2 12.

(20) and the term of (II), n Z Z X i=1 S n Z X. ≤. i=1. ≤ Cφ. 0. Z. S0 n X. 1 hD2 φ(x + iy), (hti − hti−1 )⊗2 iµ1/2 (dx)µ1/2 (dy) 2 S0. S0. kD2 φ(x + iy)kL2 (H) (ti − ti−1 )2 µ1/2 (dx)µ1/2 (dy). (ti − ti−1 )h −→ 0 as k 4n k→ 0.. i=1. Therefore we have Z b 1 hh Z(t)dZ(t), φiic = hh Z 2 (b) − Z 2 (a) , φiic . 2 a. (5.6). On the other hand. = = = = =. Z b ∂t∗ Z(t)dt, φiic hh a Z b hh∂t∗ Z(t), φiic dt (By Kubo-Takenaha formula [9]) a Z bZ Z hx + iy, ht iDδt φ(x + iy)µ1/2 (dx)µ1/2 (dy)dt a S0 S0 Z b hD2 φ(x + iy), ht ⊗ δt idt (By Equation (4.1)) a Z 1 b d hD2 φ(x + iy), ht ⊗ ht idt 2 a dt 1 hh (Z 2 (b) − Z 2 (a)), φiic . 2. (5.7). It follows from Equation (5.6) and (5.7) that Z b Z b Z(t)dZ(t) = ∂t∗ Z(t)dt a. a. Theorem 5.7. Let f ∈ SBα0 (R) and φ ∈ SBα then Z b Z b hh f (Z(t))dZ(t), φiic = hh ∂t∗ f (Z(t))dt, φiic . a. a. Proof. Since f can be represented by a formal power series, it is sufficient to prove the theorem for f (z) = z k for arbitrary non-negative integer k. By definition, 13.

(21) Z b hh f (Z(t))dZ(t), ϕiic = a. n X lim hhf (Z(ti−1 ))(Z(ti ) − Z(ti−1 )), ϕiic. k4n k→0. i=1. for any ϕ ∈ SBα . According to Remark 3.5, for each i, we have hhf (Z(ti−1 ))(Z(ti ) − Z(ti−1 )), ϕiic Z Z = f (hx − iy, hti i)hx − iy, htj − htj−1 iϕ(x + iy)µ 1 (dx)µ 1 (dy) 2 2 0 0 S S Z Z = f (hx − iy, hti i)hx − iy, htj − htj−1 iϕ(x + iy)µ 1 (dx)µ 1 (dy). (5.8) S0. 2. S0. 2. Next, apply integration by parts formula, Equation (5.8) becomes Z Z f (hx − iy, hti−1 i)hx − iy, hti − hti−1 iϕ(x + iy)µ 1 (dx)µ 1 (dy) 2 2 S 0Z S 0Z = f (hx − iy, hti−1 i)hx, hti − hti−1 iϕ(x + iy)µ 1 (dx)µ 1 (dy) 2 2 0 S 0 SZ Z −i f (hx − iy, hti−1 i)hy, hti − hti−1 iφ(x + iy)µ 1 (dx)µ1/ 1 (dy) 2 2 S0 S0 Z Z 1 hf (hx − iy, hti−1 i)hDϕ(x + iy), hti − hti−1 iµ 1 (dx)µ 1 (dy) = 2 2 2 S0 S0 Z Z i hif (hx − iy, hti−1 i)hDϕ(x + iy), hti − hti−1 iµ 1 (dx)µ 1 (dy) − 2 2 2 0 0 Z Z S S = f (hx − iy, hti−1 i)hDϕ(x + iy), hti − hti−1 iµ 1 (dx)µ 1 (dy). S0. 2. S0. 2. Then we have hhf (Z(ti−1 ))(Z(ti ) − Z(ti−1 )), ϕiic Z Z = f (hx − iy, hti−1 i)hDϕ(x + iy), hti − hti−1 iµ 1 (dx)µ 1 (dy) 2 2 S0 S0 Z Z ht − hti−1 i(ti − ti−1 )µ 1 (dx)µ 1 (dy). (5.9) = f (hx − iy, hti−1 i)hDφ(x + iy), i 2 2 ti − ti−1 S0 S0 Finally, by Equation (5.9), we have n X hhf (Z(ti−1 ))(Z(ti ) − Z(ti−1 )), ϕiic i=1 n X ht − hti−1 i(ti − ti−1 )iic hhf (Z(ti−1 ))hDφ(x + iy), i t − t i i−1 i=1 Z b → hh ∂t∗ f (Z(t))dt, φiic as k4n k → 0.. =. a. 14.

(22) hti − hti−1 converges to δti uniformly with ti − ti−1 respect to the norm of Sp for p > 21 . We complete the proof.. The last step follows from the fact that. Remark 5.8. Conclusion: We have shown that the stochastic integral with respect to a complex Brownian motion behaves exactly in the same manner as Stratonovich integral. Since the Segal-Bargmann entire functions play the same role as the Ufunctionals in white noise analysis, one can take the inverse Segal-Bargmann transform S −1 to obtain the corresponding white noise functional, for example, Z b Z b −1 n S Z(t) dZ(t) = B(t)n dB(t), a. a. where. Z S. −1. √ ϕ( 2(x + iy))µ(dy).. ϕ(x) = S0. It is worthwhile to mention here that S −1 ϕ(x) is nothing but the conditional √ expectation of ϕ( 2z) with respect to the real Brownian motion. As an application, we consider the linear Itô stochastic differential equation Z t Z t Xs dBs , t ∈ [0, T ] (5.10) Xs ds + σ Xt = X0 + c 0. 0. for given constants c and σ > 0. Let us first consider the corresponding stochastic differential equation with Bt being replaced by the complex Brownian motion √ 2Z(t), i.e. √ dXtc ˙ = cXtc + 2σXtc Z(t), X0c = X0 (5.11) dt Solve (5.11) using the method of usual ordinary differential equation, we obtain Xtc = X0 ect+. √ 2σZt. .. Finally, take conditional expectation for Xtc with respect to real Brownian motion, we obtain. Z eσhx+iy,ht i µ(dy). ct. Xt = X0 e. S0 1. 2 )t+σhx,h i t. 1. 2 )t+σB. = X0 e(c− 2 σ = X0 e(c− 2 σ. t. t ∈ [0, T ].. Xt is a geometric Brownian motion which solves the stochastic differential equation (5.10). 15.

(23) References [1] Cochran,W.G., Kuo, H.-H. and Sengupta, A.: A New Class of White Noise Generalized Functions, Infinite Dim. Anal. Quantum Probab. Related Topics 1 (1998), 43-67. [2] Hida, T.: Brownian Motion, Springer-Verlag, 1980. [3] Hitsuda, M.: Formula for Brownian partial Derivatives, Proc.Fac. of Integrated Arts and Sciences, Hiroshima University, Series III, Vol.4 (1978) 1-15. [4] Kuo, H.-H.: White Noise Distribution Theory, CRC Press, 1996. [5] Kuo, H.-H.: Gaussian Measure in Banach Spaces, LNM 463, Springer-Verlag, 1975. [6] Lee, Y.-J.: Analytic Version of Test Functionals, Fourier Transform, and a Characterization of Measures in White Noise Calculus, J. Funct. Anal. 100(1991), 359-380. [7] Lee, Y.-J.: Transformation and Wiener-Ito Decomposition of White noise Functionals, Bulletin of the Institute of Mathematics Academia Sinica, Vol.21(1993), No.4., 279-291. [8] Lee, Y.-J.: Generalized White Noise Functionals on Classical Wiener Space, J. Korean Math. Soc. 35(1998), No.3, 613-635 [9] Lee, Y.-J.: Calculus of Generalized White Noise Functionals, Lecture Notes (privately circulated). [10] Rudin, W.: Principles of Mathematical Analysis. 3rd edtion, McGraw-Hill, 1976.. 16.

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