關於由-穩定噪音所導出之橢圓型方程式的隨機最佳化問題

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Stochastic Optimal Control Problems for Elliptic Equations

Driven by -Stable Noises in Hilbert Spaces

Kuo-Liang Su Associate Professor,

Department of Business, National Open University, Lu Chow, New Taipei City,

Taiwan

Cheng-Ying Wu Associate Professor,

Department of Mangement and Information,

National Open University, Lu Chow, New Taipei City,

Taiwan

Abstract

A mild solution of semilinear elliptic partial differential equation driven by a cylindrical -Stable noise is solved in an infinite dimensional Hilbert space. The relevant Hamilton-Jacobi-Bellman equation is also studied and obtained for applications to the infinite horizon stochastic optimal control problems. Moreover, it shows applications to some specific controlled stochastic differential equations. Impacts of this stochastic optimality and the corporate social responsibility on performance will also be studied.

Keywords: Ellptic equation; Stochastic optimal control; cylindrical -stable process; Hamilton-Jacobi-Bellman equation.

AMS Classification (2010). 60H15, 60H30, 93E20.

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1. Introduction

The problems of determining the minimal expected value of discounted losses

or the maximization of the present value of the profits which is usually written as

E{

0

λ ([ge t t, utX , su )+h( tu )]dt} can be equivalently expressed as stochastic optimal

control problems when { utX } is a stochastic process. In this paper we are dedicated

to study some abstract theory for the optimal control problems with cost functional

that are defined on a Hilbert space. Particularly, we are interested in the state

equation driven by -stable process noises [2, 15, 17, 26], although many authors

had already studied such problems with state equations perturbed with Wiener

noises [3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 23, 24]. In order to obtain our results we

need to employ different approaches when the state equations are associated with

non-Gaussian white noises. Since, for example, certain good characterizations such

as the change of measure approach and finite second moment [8, 9, 10, 11, 12, 13,

14, 22, 23, 24] are no guarantee to be applicable in the state equations with -stable

process noises. In the first part we investigate a class of semilinear elliptic differential equations in a separable Hilbert space of the form

u(x)= tL u(x)+(x, u, Du), xH. (1.1)

where the involved linear operator tL is defined by

tL v(x)=Ax+F(x), Dv(x)+ R

(β1

α

kk v(x+ ke z)v( ke z)) α1|| z

dz ,

where D= xD denotes the cheteFrˆ derivative for v: HR. The operator tL is also the generator of the Markov transition semigroup operator { tP } associated with the solution X=( x

tX ) to the following stochastic evolution equation

關於由-穩定噪音所導出之橢圓型方程式的隨機最佳化問題

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,,0,)(

0 HxX

tdZdtXFdtAXdX txt

xt

xt (1.2)

where A is a possibly unbounded operator which generates a compact 0C -semigroup ( tAe ), t0, on H, F is a Lipschitz continuous and bounded function

from H to H, tZ is a cylindrical -stable process taking values in a separable Hilbert space E usually greater than H. Under some suitable conditions, (1.2) admits a unique solution x

tX =X(t, x), t[0, T], xH. We also denote by tP the corresponding transition semigroup tP f(x)=E[f( x

tX )], for every bounded measurable function f: HR. One of the main purpose for this research in the nonlinear case is the connection with control theory, that is, the Hamilton-Jacobi-Bellman (HJB) equation for the value function of a stochastic optimal control problem is a solution to such a equation. We restrict our concern on the case of continuous solutions. The forms and the theory for the viscosity solution associated with HJB equation, which related to Wiener white noises in general, have been developed by many authors though. Hence, the results we wish to present are associated with a cylindrical -stable process which are not covered by the results of those papers that mentioned previously in particular. In this paper, we wish to obtain a mild solution of equation (1.1). A function is called a mild solution if it is cheteFrˆ differentiable and satisfies the equality [4, 10, 11, 12, 13, 24, 25, 26]

u(x)= dtxDuuPe tt )())] (), (, ([ψ

0

λ

, xH.

According to [5, 17, 26], a mild solution to the elliptic equation (1.1) can be extended to satisfy the equality

u(x)= {0

λ

te tR f(x)+

t

stR0

[ F(), Du()+(, u(), Du())](x)ds}dt, xH. (1.3)

where u is a function in 1bC (H) and ( tR ) is the transition semigroup determined by

Orstein-Uhlenbeck differential equation. We first study that for every bounded and continuous function f and a mild solution u is of the form (1.3) and admits cheteFrˆ derivative. When tR f(x) is also cheteFrˆ differentiable, u will satisfy

0)(uDx ≤ γct

0f , for some (0, 1). (1.4)

Next, we consider the applications to optimal control problems. We take a

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controlled stochastic evolution equation

,0 ,)()(

0 .HxX

tdZdtuXGdtXFdtAXdXu

ttut

ut

ut

ut (1.5)

Under this circumstance, Hamiltonian function is required and formulated in order that the HJB equation admits the form (1.1). We also present the synthesis of the optimal control problem by means of the so-called closed-loop equation. In order to show the efficient applications to some particular models, we consider the applications on the corporate social responsibility on performance with discounted cost functions driven by Brownian motions. Our methods and concerns for searching optiamlity are more general and profound because our stochastic evolution equations are derived by a cylindrical -stable process, though these results can also be obtained by applying the techniques in [5, 12, 13].

Let us describe the structure of this paper as follows: in section 2 we define the semilinear Kolmogorov equation (1.1) and introduce the concept of a mild solution. In section 3, we prove the existence and uniqueness for the mild solution of a semilinear elliptic equation assocated with stochastic differential equations. We apply our results to study the optimal control problems and the corporate social responsibility on performance.

2. Notations and Preliminaries

Let H denote a real separable Hilbert space with norm ||. ( ne ) is a fixed orthonormal basis in H. The space bC (H) (resp. bB (H)) denotes for the Banach space of all real, continuous (resp. Borel)) and bounded functions f : HR, endowed with the supremum norm

0f =

Hxsup

|f(x)|. The space kbC (H), k1, means

the set of all k-time differentiable functions f : HR, whose i-th order cheteFrˆ derivative fDi , 1ik, are continuous and bounded on H.

We start out with a predictable H-valued stochastic process X=( xtX ) with initial

datum x H which satisfies the stochastic linear equation

tdX = tAX dt +F( tX )dt+ tdZ , 0X =xH. (2.1)

We say that ( xtX ) is a mild solution of (2.1) if for any t > 0 the following equality

holds

xtX = xetA +

tAste

0

)( F( xtX )ds + )(tZ A , )(tZ A =

tAste

0

)(sdZ . (2.2)

關於由-穩定噪音所導出之橢圓型方程式的隨機最佳化問題

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Later it will list certain assumptions for A and F in order to guarantee the integrability for (2.2). The process ( tZ ) is a cylindrical -stable process, (0, 2), which is defined

tZ =

1n

nntn eZ , t0, (2.3)

where ( ne ) is an orthonormal basis in Hilbert space H, ( n ) is a given sequence of positive numbers, and ( n

tZ ) are i.i.d. one-dimensional normalized symmetric -stable processes defined on the same stochastic basis (, F, ( tF ), P), satisfying the usual assumptions. In [1, 20, and 21], we have for any nN, t0, the characteristic function of n

tZ is

E[ntZie ]=

||te , R.

We follow some assumptions and results in [18 and 19] and stated as below.

Hypothesis 2.1.

(i) The processes nZ are independent and -stable, (0, 2). (ii) A : D(A)HH is a self-adjoint operator such that the fixed basis ( ne ) of H

verifies: ( ne )D(A), A ne = nne with n >0, for any n1, and n .

(iii)

1n n

n <, where n >0, for any n1.

(iv) For any t>0,n

/n

t

n

nesup

1

1= tC <.

Hypothesis 2.2.

(i) F : HH is Lipschitz continuous and bounded.

(ii) There exists (0, 1) and 1C >0 satisfying )1(

1

nC n , for any n1.

Remark 2.3.

The condition (ii) in Hypothesis 2.1, we need that D(A)={x=( nx ) H: 122

n nnx <}

because of (2.1). Hence we denote the weighted space 2l ={ x=( nx ) R :

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122

n nnx <} with inner product x, y=

12

n nnn yx , where =( n ) of real

numbers, x=( nx ) and y=( ny ) are in 2l [18 and 26].

Let tR : bB (H) bB (H) is the transition semi-group associated with the

Ornstein-Uhlenbeck process tdX = tAX dt + tdZ , 0Y = xH. When taking this equation as an infinite sequence of independent one-dimensional stochastic differential equation, it will be

ntdX = n

tn X dt + ntndZ , nX 0 = nx , nN, x=( nx )H = 2l .

The unique solution is a stochastic process X=( ntX ), where

ntX = n

t xe n + t

nsn

st ,dZe n

0

)( nN, t0.

Hence,

xtX =

1nn

nt eX = xetA + )(tZ A .

Considering the transition Markov semi-group ( tP ) associated to ( xtX ) which fitted

with (2.1) [4, 18, 19, 26],

tP f(x)=E[f( xtX )]=E{f( xetA +

tAste

0

)( (F( tX )dt+ sdZ ))},

where xH, f bB (H), t0. The following theorem is needed directly from [26].

Theorem 2.4.

Assume that Hypotheses 2.1 and 2.2 hold. Then for any t>0, the transition semigroup ( tR ) associated to ( x

tX ) mapping Borel bounded functions into Lipschitz

continuous functions. Moreover, there exists C~ =C

~ (, c , 1C ,0

F )>0 such that for any x, yH we have

(i) 0

fDRt 0

8f

t

cc , t0;

(ii) |{ tR f(x)+

t

stR0

[ F(), Du(s, )](x)ds}{ tR f(y)+

t

stR0

[ F(), Du(s, )](y)ds}|

關於由-穩定噪音所導出之橢圓型方程式的隨機最佳化問題

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C~

0f

11t

|xy|, f bB (H), t0,

where tR f(x)=E[f( xetA + )(tZ A )], c =

R

dzzp

zp

)()(

81 , p is the density of

normalized symmetric -stable distribution and p =dz

dp .

When x

tdX = xtAX dt +F( x

tX )dt+ tdZ , 0X = x, let ( tP ) be the Markov semigroup associated to X=( x

tX )

tP f(x)=E[f( xtX )], xH, f ),(HBb t0. (2.4)

then tL is the generator of the Markov transition semigroup operator { tP } given by

tL f(x)=Ax+F(x), Df(x)+ R

(β1

α

kk f(x+ ke z)f( ke z)) α1|| z

dz . (2.5)

The corresponding Kolmogorov equaiton is

, ),(),0φ(

, ,0 ),)( ,φ(),(φ

Hxxfx

Hxtxtxtt tL

and its mild solution is (t, x)= tP f(x), where f(x)UC(H). Follow [7, 9, 13], the mild solution to the elliptic equation u(x)= tL u(x) + f(x), xH, is

u(x)=

0

φte (t, x)dt=

0t

t Pe f(x)dt, xH, f )(HBb . (2.6)

Remark 2.5.

We only consider the properties of the mild solutions associated with semigroup ( tR ) and use the integration by parts to get

(x) =

0t

t P~

e f(x)dt ,

=

0

{te t

sR0

[F(), D()+f()](x)ds}dt, t > 0, xH. (2.7)

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where fUC(H). Since it had been showed in [26] that ntP f = n

tP~

f, t 0, for any f

bB (H) and nN, which depends on the first n coordinates. ( ntP ) and ( n

tP~ ) denote

the corresponding semigroups acting on bB ( nH ), where nH is the subspace of H

generated by { 1e , . . . , ne }.

3. Semilinear Elliptic Equation

Consider the elliptic equation with the initial value

u(x)=Lu(x)+(x, u, Du), xH. (3.1)

where tL is the generator of the Markov transition semi-group ( tP ) given by (2.5). We consider the space u 1,1W (H)={ 1

bC (H):1

)φ( =0

)φ( +0

)φ(xD <}. In order to study the Hamiltonian function later we need the following assumptions [8, 12, 13, 14, 15, 26] .

Hypothesis 3.1.

The function : H×R×H→R is continuous and satisfies the following: (i) there exists a constant L > 0 such that

|(x, 1y , 1z ) − (x, 2y , 2z )| ≤ L(| 1y − 2y | +H

zz 21 ),

for every xH, 1y , 2y R, 1z , 2z H; (ii) there exists L>0 such that

|(x, y, z)| ≤ L (1 + |y| +H

z ) ,

for every xH, yR, zH. By (2.6), a mild solution of the semilinear elliptic equation (3.1) can be

presented as u(x)=

0t

t Pe f(x)dt, xH. In accordance with [26], a mild solution to the

elliptic equation (3.1) can be extended as the following formulation

u(x) = {0

λ

te tR f(x)+

t

stR0

[ F(), Du()+(, u(), Du())](x)ds}dt, xH, (3.2)

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where F, , u, and Du are satisfied previous assumptions, is sufficiently large, and( tR ) is the transition semigroup determined by the following linear equation

tdX = tAX dt + tdZ , 0X =x.

We also need the following definition.

Definition 3.2.

Let (0, 1). We say u: H→R is a mild solution of nonlinear Kolmogorov equation (3.1) if the following conditions are satisfies: (i) u 1

bC (H), (ii) Du is continuous on H, that is, u is cheteFrˆ differentiable, (iii) equality (3.2) holds.

Note that by [17] we have, for any f ),(HBb

0fDRt

00 ftC , t>0, for some 0C >0. (3.3)

We only consider the mild solution u(t, x) which satisfies equation (3.2). Therefore, in order to show the estimation of

0) ( ,tuDx , we need to study the behavior of

0 0

λ [t

stt DRe (s, , u(s, ), Du(s, ))](x)dsdt, where u(x) is the mild solution of (3.2).

The next lemma can be obtained by following the similar arguments in [12, 13], however, the difference from both [8, 12, 13] is that it only considers a generalized sup-norm on the space 1,1W (H).

Lemma 3.3.

Let Hypothesis 3.1 be hold and 1u , 2u 1,1W (H) are mild solutions of (3.1), then

| 0 ( 11,vu ) 0 ( 22 ,vu )| 2λ1 [

021021 vvuu ],

where 0 (u,v)=

0 0

λ [t

stt Re (, u(), v())](x)dsdt and iv ()=D iu ().

Proof. By Hypothesis 3.1 and use some basic calculus [12]

| 0 ( 11,vu )(x) 0 ( 22 ,vu )(x)|

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= |

0 0

λ [t

stt Re (, (1u ), (1v ))(, (2u ), (2v ))](x)dsdt|

|

0 01

λt

t ue )()( 2 xux + H

xvxv )()( 21 dsdt |

Hx

sup )()( 21 xuxu + H

xvxv )()( 21 dtte t

0

λ

2λ)2(

021021 vvuu .

Hence

| 0 ( 11,vu ) 0 ( 22 ,vu )| 2λ1

021021 vvuu . (3.4)

Note that let iu ( ix ) 1,1W (H), i =1, 2. Since is bounded by conditions (i) and

(ii) of Hypothesis 3.1, for any >0 when 1x and 2x are closed, we have ),,ψ(),,ψ( 222111 vux-vux

),,ψ(),,ψ( 221111 vux-vux + ),,ψ(),,ψ( 222221 vux-vux

L(| 1u − 2u | +H

vv 21 ) + L (| 1x − 2x H| ) L| 1u − 2u | + .

In addition, tR 1bC (H) if bB (H) [6, 18], hence, the previous lemma and

condition (ii) of Hypothesis 3.1 admit that

||

0 0

λ [t

stt DRe (s, , u(s, ), Du(s, ))](x)dsdt|| 2λ

10

) ( ,tuDx

whenever 1x and 2x are closed enough.

Proposition 3.4.

Let Hypothesis 3.1 be hold and u 1,1W (H) be the mild solution of (3.1). For any t>0, there exists c=c(γ, αc , 1C ,

0F )>0 such that, for any f bB (H), t>0,

0)(uDx ≤c

0f .

Proof. Following (3.2), we have

Du(x)= [0

λ

te D tR f(x)+

t

stDR0

[ F(), Du()](x)ds

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+

t

stDR0

[(, u(), Du())](x)ds]dt, x H.

Using (3.3), (3.4), and lemme 3.3,

(1 2λ1 )

0) ,( tDu

00

γλ [ fCte t +0

F t

st0

γ)(0

) ,( sDu (x)ds]dt.

Hence

0) ,( tDu C γ1λ

)γ1(

1

0λ22 λ)γ2(

λ11

F0

f .

Next we will use the contraction mapping principle to study the uniqueness of the mild solution of (3.2). Before showing this we also need to prove that 0 is a contraction mapping on 1,1W (H).

Lemma 3.5.

Let Hypothesis 3.1 be hold and 1u , 2u 1,1W (H) are mild solutions of (3.1), then

) ,( 110 vu 1220 ) ,( vu γ2-λ

)γ1( [021 uu +

021 vv ],

where 0 (u,v)=

0 0

λ [t

stt Re (, u(), v())](x)dsdt and iv ()=D iu ().

Proof. From lemma 3.3, we already have the behavior of ) ,( 110 vu 0220 ) ,( vu .

By Hypothesis 3.1 and (3.3), for any H, we can obtain that

) ,( 110 vuD )ξ() ,( 220 vuD

H

ξ

0 0

λ [|t

stt DRe (s, , (1u ), (1v ))(s, , (2u ), (2v ))](x)|dsdt

H

ξ

0 0

γλ )(t

t steHx

sup

[ )()( 21 xuxu + )()( 21 xvxv ]dsdt

H

ξ [021 uu +

021 vv ]

0

γ1λ

γ1t

e t dt

=H

ξ)γ1(λ

)γ2(γ2

-[

021 uu +021 vv ].

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Hence the desired results.

Remark 3.6.

According to lemmas 3.3 and 3.5 we have that 0 is a contraction mapping since

) ,( 110 vu 1220 ) ,( vu

121 uu if taking c = 2λ1 + γ2-λ

)γ1( <1.

Theorem 3.7.

Suppose that Hypotheses 2.1, 2.2, and 3.1 hold. Then equation (3.2) admits a unique mild solution u(x) 1,1W (H). Proof. Following the similar arguments in lemma 3.3, let be a mapping defined on

1,1W (H) bC (H), where (u, v) 1,1W (H) bC (H)and v(x)=Du(x). We need to show that is a contraction mapping then there exists a unique fixed point satisfying (u,v)=(u, v). We can adapt the proof and argument of [5, 18] to show that is a contraction mapping. First let the mapping 1 : 1,1W (H) 1,1W (H) be defined as

( 1 (u))(x) =

0 0

λ [t

stt Re F(), Du()](x)dsdt.

is a contraction mapping if 11 )(u 2λ

10

F1

u 1

u for large .

Next we need to prove that 2 is a contraction mapping. Let 2 = 0 + 1 : 1,1W (H) 1,1W (H) which is defined as

( 2 (u,v))(x) = ( 0 (u,v))(x) +( 1 (u,v))(x)

=

0 0

λ [t

stt Re (, u(), v())](x)dsdt+

0 0

λ [t

stt Re F(), Du()](x)dsdt.

Then, by Proposition 3.4 and Lemma 3.5, we have ) ,( 112 vu

1222 ) ,( vu ) ,( 110 vu 1220 ) ,( vu + ) ,( 111 vu

1221 ) ,( vu

c121 uu + 2λ

10

F021 vv

02λ1

Fc121 uu ,

where c = 2λ1 + γ2-λ

)γ1( and 121 uu =

021 uu +021 vv .

Therefore, 2 is a contraction mapping on 1,1W (H) if

γ2-2 λ

)γ1(λ1

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02λ1

F 1 when is large enough. Then equation (3.2) has a unique mild solution

u.

4. Applications to Optimal Control Problems.

In this section, it will apply the results in the previous section to the solutions of stochastic optimal control problems. We start with formulating the stochastic optimal control problems in the usual sense.

4.1 Solutions of Cost Functions

Let Z = ( tZ ) be a cylindrical -stable process with values in Hilbert space H, (0, 2), which is defined as (2.3) in section 2 and (, F, P) be a complete probability measure space with a filtration 0ttF associated with the process ( tZ ). The control u is an 0ttF -predictable process with values in fixed bounded set U of another real separable Hilbert space U and there exists a constant uK such that

Uu uK for any uU. We also denote dA the set of admissible controls. Let G: [0, T]HL(U, H), where L(U, H) denotes the linear operators from U to H, and consider the following controlled state equation

utdX = u

tAX dt +F( utX )dt+G( u

tX ) tu dt+ tdZ , t[0, T], uX 0 =x, (4.1)

where F: HH is a Lipschitz continuous and bounded function and G:[0, T]H L(U, H). The solution of this equation is denoted by uX (t, x), ux

tX , , uX (t), or simply by u

tX . X is also called the state, xH and t>0. Next, we define the cost function

J(x,u)=E{

0

λ ([ge t t, utX , su )+h( tu )]dt} (4.2)

where g is defined on [0, T]HU and h : U[0, ) is convex and lower semicontinuous. The goal of control problems is to minimize the cost functional over all admissible control u. We denote that J(x, *u )=

dv Ainf J(x,u) is the value

function of the problem, if it exists and *u is called the optimal which reaching the infimum of the problem.

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The structure of (4.1) leads to a semilinear HJB equation which is a special case of Kolmogorov equation as following

v(x)= )(xvtL 0H (x, v(x),Dv(x))g(t, x, u), xH. (4.3)

where Dv(x) is the derivative of v with respect to x. Given a Lipschitz continuous and bounded function F: HH and a generator tL of a strongly continuous semigroup on H, tL is defined by

tL v(x)=Ax+F(x),Dv(x) + R

(β1k

k v(x+ ke z)v( ke z)) α1|| z

dz . (4.4)

We need some assumptions on the coefficients of equation (4.1).

Hypothesis 4.1.

(i) F : HH is Lipschitz continuous and bounded. (ii) G:[0, T]HL(U, H) is Borel measurable and follows the conditions: for every u U, G(,)u is a measurable mapping from [0, T]HH; moreover, there exists a suitable constant GK such that |G(t, x) ),(| HUL GK , for every t[0, T] and xH and for some constant C such that

|G(t, x)uG(t, y)u H| C|xy H| , t [0, T] and x, yH, uU;

besides, for any t[0, T] and pH, pG(t, ) 1C (H), (pG *U ), with | xD pG(t, x)|

C(1 + m

Hx ) for every t[0, T] and some m0.

Note that ( tP~ ) is a semigroup of bounded linear operator on bB (H), by

Theorem 3.7. Hence, under the conditions (i) and (ii) of Hypothesis 4.1 on the coefficients A, F, and G, (4.1) admits a unique mild solution by following the fixed point argument.

Hypothesis 4.2.

(i) The function : HR is uniformly continuous and bounded by a constant which denoted by K .

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(ii) g : [0, T]HUR is a Borel measurable. The map xg(t, x, u) is Lipschitz continuous, uniformly with respect to uU and to t[0, T]. Moreover, there exists a constant gK 0 such that |g(t, x, u)| gK for all t[0, T], xH, and uU. (iii) h : U[0, ) is convex and lower semicontinuous. Next, define the Hamiltonian function in a classical way associated to the previous problem and equation (4.3). For any t[0, T], xH, and p *U

0H (x,p)= {Uu

sup u, pG(x)+h(u)}, (4.5)

(x,p)= 0H (x, p)g(t, x, u). (4.6)

Remark 4.3.

When Hypotheses 2.1, 2.2, and 4.1 hold. If g and h also satisfy the conditions of Hypothesis 4.2, then satisfies Hypothesis 3.1. [26] Under Hypothesis 4.2, applying Lemma 4.3, we can obtain and omit the proof that the HJB equation associated to the above problem,

v(x)= )(xvtL +(x, v(x), Dv(x)), xH, (4.7)

admits a unique mild solution by Theorem 3.7. Then it had been showed in [7] and [10] that the optimal control *

tu related to the optimal state *tX = )( x,tX * by the

following feedback formula

G(t, *tX ) *

tu = 0HpD (x, Dv(t, *tX )), t[0, T],

where *tX is the solution of the closed loop equation below

.], ,0[ ,))],(,()([

0

0

xX

TtdZdtXtDvxDXFAXdX ttpttt H (4.8)

We are going to show, that is, to show that for every admissible control u it leads the following

J(x,u) = E{

0

λ ([ge s s, usX , su )+h( su )]ds }

= v(x) + ))(,([ψ0

λ ut

ut

t XDvXe

+ g(s, utX , tu )

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+h( tu )+ tu ,Dv( utX )G(t, u

tX )]dt, (4.9)

where v is the solution of (4.7). From the definition of Hamiltonian function and under some suitable conditions, we will prove that there exists an admissible control

*u such that J(x, *u ) = v(x). This leads that *u optimal and *J (x)=v(x). We can follow the Galerkin’s approximations and arguments which applied by [18 and 26] to prove (4.9) and its related formulations directly.

Remark. 4.4.

Notice that for every nN, ( )(tZ n ) is a Levy process and also a normalized symmetric -stable process, hence, )(tZ n )(sZ n has the same law of )( stZ n .

The characteristic function of r.v. )( stZ n , E[exp(i )( stZ n , )] = exp[(ts) α ], 0s<tT, which has a zero mean by direct differentiation with respect to if 1<. Hence, when 1< and 0s<tT, E[ )( stZ n ]=0 [26].

Following the standard arguments of Proposition 13.1.4 in [7], we can conclude

the Proposition below.

Theorem 4.5.

Let Hypotheses 2.1 and 2.2 hold true, and 1<<2. If v(x) is the mild solution of (4.7) and u

tX is the mild solution of the controlled stochastic differential equation (4.1), then the following equations hold:

J(x,u)=E{

0

λ ([ge t t, utX , tu )+h( tu )]dt }, (4.10)

and

v( x)=du A

inf E{

0

λ ([ge t t, utX , tu )+h( tu )]dt }. (4.11)

Proof. We follow Ito formula and formulations (4.5) and (4.6), we have

d[ te v( tX )]

=

t

e t

v( tX )+ te Dv( tX ) tdX + te 2

21

D v( tX )( tdX 2)

關於由-穩定噪音所導出之橢圓型方程式的隨機最佳化問題

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= te v( tX )+ te [A tX +F( tX )+G( tX ) tu ]dt+ tdZ , Dv( tX )

+21 te tr( 2D v( tX ))dt

= te v( tX )+ te [A tX +F( tX )], Dv( tX )dt

+21 te tr( 2D v( tX ))dt+ te tdZ , Dv( tX )+ te G( tX ) tu , Dv( tX )dt

= te (L)v( tX )+ te tdZ , Dv( tX )+ te G( tX ) tu , Dv( tX )dt = te [ 0H (Dv)+g] + te tdZ , Dv( tX )+ te G( tX ) tu , Dv( tX )dt.

Adding te [ g( tX )+h( tu )] on both sides of the above equality and taking expectation, then integrating with respect to t from 0 to infinite, we have

E 0 )( t

t Xve = E

0

[te 0H (D)+g( tX )]dt

+E

0

te G( tX ) tu dt, Dv( tX ),

where 0H (Dv) = 0H (t, x, Dv) =G( tX ) tu , Dv( tX )dt + h( tu ). Hence

(x) +J(x,u)= E

0

[te 0H (Dv)+g( tX )]dt

+ E

0

te G( tX ) tu , Dv( tX ) dt +E

0

λte [ g( tX )+h( tu )]dt

= E

0

[te (t, x, Dv)+h( tu )+g( tX )]dt

+ E

0

te G( tX ) tu dt, Dv( tX )

This implies

J(x,u)=v(x)+E

0

[te (t, x, Dv)+h( tu )+g( tX )+G( xtX ) tu , Dv( tX )]dt.

The Theorem 4.6 proves not only the equations (4.10) and (4.11) but also the fundamental relation (4.9). Note that in the convention of mathematical control theory, equation (4.9) is also called the fundamental relation for the cost J by many authors [8, 9, 11, 12, 13, and 26]. We let

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(t, x, p)={uU: g(t, x, u)+u, pG(x)+h(u)= (t, x, p)}.

where has been defined in (4.6). We also assume that for any (t, x, p)[0, T]H *U , (t, x, p) is not empty. Furthermore, we assume that there exists a measurable selection 0 in , that is, for every (t, x, p)[0, T]H *U , there exists a Borel measurable mapping from [0, T]H *U to U such that 0 (t, x, p)(t, x, p).

Theorem 4.6.

Let Hypotheses 2.1, 2.2, 4.1 and 4.2 hold true, and (1, 2). Then the followings hold: 1. For each admissible control u we have

J(x, u)v(x).

2. The equality holds if and only if u and uX fulfill the feedback law

u(t)(t, utX , Dv(t, u

tX )G(t, utX )), P-a.s. for a.a. t>0. (4.15)

3. If there is a measurable selection 0 and if there is an admissible control u satisfying the feedback law

u(t) = 0 (t, utX , Dv(t, u

tX )G(t, utX )), P-a.s. for a.a. t[0, T], (4.16)

then u is optimal. Proof. By Theorem 4.6, for the cost functional J and by the definition of the Hamiltonian function H, it results in that

J(t, x, u)v(t, x)

for each admissible control u. Next follow the definition of , it can notice as well that equality holds if and only if

u(t)(t, utX , Dv(t, u

tX )G(t, utX )), P-a.s. for a.a. t[0, T].

Therefore, if there exists a measurable selection 0 , then by (4.14), we can obtain an admissible control filfulling the infimum of the cost function J and hence it is an optimal control.

Following the previous theorem and the similar arguments in [12] and [14], we can obtain the corollary below, hence the proof is omitted.

關於由-穩定噪音所導出之橢圓型方程式的隨機最佳化問題

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Corollary 4.7.

Let Hypotheses 2.1, 2.2, 4.1 and 4.2 hold true, and 1<<2. If the closed loop equation

.,0 ,)),(),(,,(),()(

0

0*

xX

tdZXtGXtDvXtXtGXFdtAXdX tttttttt (4.17)

has a mild solution, defining

*u (t)= 0 (t, tX , Dv(t,

tX )G(t, tX )),

then the pair ( *u , tX ) is optimal for the control problem.

Remark 4.8.

(, F, 0ttF , P, Z, u, uX ) is also called an admissible control system (a.c.s.) [12], shortly by (Z, u, uX ), when (, F, 0ttF , P, Z) is an admissible setup with u(t)U P-a.s. for a.a. t[0, T]. Then the optimal control problem associated the cost functional J is:

J(x, (Z, u, uX ))=E{

0

λ ([ge t t, utX , tu )+h( tu )]dt },

and the value function of the problem in the weak sense is only considered on the a.c.s. (Z, u, uX ) as

*J (x) =...),,(

infscaXuZ u

J(x, (Z, u, uX )).

Hence, the optimal control problem in the weak sense is to find an a.c.s. ( uXuZ ,, ) such that J(x, ( uXuZ ,, )) J(x, (Z, u, uX )) which can be obtained by the similar way as previously mentioned. Apparently, the closed-looped equation admits a solution in the weak sense which can construct the optimal pair for a control problem.

4.2 Stochastic Optimal Control for Cooperate Social Responsibilities

Here we wish to study the impacts of this stochastic optimality and the corporate social responsibility on performance. The goals of studying the corporate

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social responsibility on performances are equivalent to exploring the optimal strategies or controls that a corporate may employ to minimize its expected value of discounted losses or maximize its present value of the profits in the long run. In the real world, these optimal problems are usually associated with some stochastic dynamic systems which possess the structure of (4.1) with the Gaussian white noise ( tB 0) t = ( 0

1)

tn

nnt eB . The Gaussian white noise is, however, a special -stable

process ( tZ 0) t with =2. Therefore, we can apply our techniques, those obtained in the previous sections, to the stochastic control problems connected with the corporate social responsibility on performances. Consequently, when we focus on the optimal problems involved with both semilinear elliptic equation and its associated stochastic differential equation driven by a cylindrical Brownian motion instead of a cylindrical -stable process, the entire results in the section 4 will be held also. Namely, we can conclude the optimal strategies or controls

tu of the

form in (3.1) to optimize our expected cost or profit E{

0

λ [te g(t, utX , tu ) +h( tu )]dt }.

Hence, we shall omit to formulate tu again.

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關於由-穩定噪音所導出 之橢圓型方程式的隨機最佳化問題

蘇國樑 1，吳政穎 2 國立空中大學商學系副教授 1

國立空中大學管理與資訊學系副教授 2

摘 要

將解得在無限維 Hilbert 空間上由柱型-穩定噪音所導出之半線性橢圓型

方程式的溫和解；其次，將研究和導出關於和無窮範圍之隨機最佳化問題有關

的 Hamilton-Jacobi-Bellman 方程式；並且，將呈現對一些特殊隨機控制微分方

程式的應用；也將探討隨機最佳化和公司社會責任對績效之影響。

關鍵詞：橢圓方程式；隨機最佳化控制；柱型-穩定過程；Hamilton-Jacobi- Bellman方程式。

AMS 主題分類(2010)：60H15, 60H30, 93E20.