Reproducing Kernel Hilbert Space of A Set Indexed Brownian Motion by IJMER

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Reproducing Kernel Hilbert Space of A Set Indexed Brownian Motion Arthur Yosef Tel Aviv-Yaffo Academic College, 2Rabenu Yeruham St., Tel Aviv-Yaffo, Israel ABSTRACT:

This

study

researches

representation

set

indexed

Brownian

motion

X  { X A : A  A} via orthonormal basis, based on reproducing kernel Hilbert space (RKHS). The RKHS associated with the set indexed Brownian motion X is a Hilbert space of real-valued functions 2 on T that is naturally isometric to L ( A ) . The isometry between these Hilbert spaces leads to useful spectral representations of the set indexed Brownian motion, notably the Karhunen-Loève (KL) representation: X A

  en E[ X Aen ] where {en } is an orthonormal sequence of centered Gaussian

variables. In addition, we present two special cases of a representation of a set indexed Brownian motion, when

A  A([0,1]d ) and A = A( Ls) . Keywords: Brownian motion, orthonormal basis, Hilbert Space, Karhunen-Loève.

INTRODUCTION

In this article, we present the representation of set indexed Brownian motion X  { X A : A  A} via orthonormal basis, based on reproducing kernel Hilbert space (RKHS). Set indexed Brownian motion is a natural generalization of planar Brownian motion where A is a collection of compact subsets of a fixed topological space (T , ) . The frame of a set-indexed Brownian motion is not only a new stepto generalize theclassical Brownian motion, but it was proven asa new look upon a Brownian motion (see [Yo09], [Yo15], [MeYo], [He], [IvMe], [Kh], [MeNu]). RKHS is a robust tooland can be used in a wide variety of areas such as curve fitting,signal analysis and processing, function estimation and model description, differential equations, probability, statistics, nonlinear Burgers equations, empirical risk minimization, fractals, machine learning and etc. (see [Par67], [Par],[Ha],[Sc], [Va], [ScSm], [Dan], [Be], [Cu], [Ge], [Ad]). Let’s assume we havea set indexed Brownian motion on topological and separable space T , with a continuous covariance kernel R : A  A   . We can associate a Hilbert space, which is the reproducing 2

kernel Hilbert space of real-valued functions on T that is naturally isometric to L ( A ) .The isometry between these Hilbert spaces leads to useful spectral representations of the set indexed Brownian motion, notably the Karhunen-Loève (KL) theorem.(The KL theorem is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval).In this work, the KL representation of a set indexed Brownian motion is:

X A   ei E[ X Aei ]

 n n 1

Where {e }

an orthonormal sequences of centered Gaussian variables. In addition, in this study we present

two special casesof a KL representationof a set indexed Brownian motion, when: (a) T  [0,1] and A  A([0,1] )  {[0, x] : x  [0,1] } d

(b) T  [0,1] and A = A( Ls) d

In the first case, the KL representation of a set indexed Brownian motion is:

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Reproducing Kernel Hilbert Spaceof A Set Indexed Brownian Motion 

n 1

i 1

X A  X [0, x1 ][0, x2 ]...[0, xd ]  ( 2 2 )d  (2 n11)d en  sin( 12 (2n  1) xi ) In the second case: 

X A  X m gm ( A)  X 

kd k1 k2 m g m ([0, m ][0, m ]...[0, m ]) 2 2 2

II.

 lim( 2 2 )d  (2 n11)d en  sin( 12 (2n  1) m

n 1

i 1

ki 2m

PRELIMINARIES

As in earlier set-indexed works (see [IvMe]), processes and filtrations will be indexed by a class A whose elements are compact subsets of a fixed  -compact metric space T . In addition, we assume A satisfies several natural conditions. We use the definition and notation from [IvMe] and all this section comefrom there. Definition 1.Let (T , ) be a non-void sigma-compact connected topologicalspace. A nonempty class A of compact, connected subsets of T is called an indexed collection if it satisfies the following:

  A . In addition, there is an increasing sequence ( Bn ) of sets in A s.t. T   n 1 Bn . 2. A is closed under arbitrary intersections and if A, B  A are nonempty, then A  B is nonempty. If 1.

( Ai ) is an increasing sequence in A and if there exists n such that Ai  Bn for every i , then i Ai A . 3.  ( A) = B where B is the collection of Borel sets of T . 4.

Separability from above: There exist an increasing sequence of finite sub-classes An

 {A1n ,..., Aknn }  A

closed under intersection with  , Bn  A n (u) ( A n (u ) is the class of union of sets in A n ), and a sequence of functions g n : A  A n (u)  T such that: (i) g n preserves arbitrary intersections and finite unions. (ii) For each A  A, A  g n ( A) and A  n g n ( A) , g n ( A)  g m ( A) if n  m . 

(iii) g n ( A)  A '  A if A, A ' A and g n ( A)  A '  A n if A  A and A '  A n . (iv) g n ()   for all n . Examples. a.

The

classical

example

T  d

T  [0,1]d

and A = A( )  {[0, x] : x  } (or d

and

A = A([0,1]d ) ). This example can be extended to T  d and A = A(d )  {[0, x] : x d } , which will give rise to a sort of b.

2d -sides process.

The example (a) may be generalized as follows. Let T   or T   or T  [0,1] and take A to be d

the class of compact lower sets, i.e. the class of compact subsets A of T satisfying t  A [0, t ]  A . (We denote the class of compact lower sets by A( Ls) ). c.

implies

Additional examples have been given when T is a “continuous” rooted tree (see [Sl]) and T a subspace of the Skorokhod space, D[0,1] (see [IvMe]). We will require other classes of sets generated by A . The first is A(u) , which is the class of finite

unions of sets in A . We note that A(u) is itself a lattice with the partial order induced by set inclusion. Let C consists of all the subsets of T of the form C  A \ B, A  A, B  A(u) . Any A -indexed function which has a (finitely) additive extension to C will be called additive (and is easily seen to be additive on C(u) as well). For stochastic processes, we do not necessarily require that each sample path be additive, but additivity will be imposed in an almost suresense:

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Reproducing Kernel Hilbert Spaceof A Set Indexed Brownian Motion A set-indexed stochastic process X  { X A : A  A} is additive if ithas an (almost sure) additive extension to

C : X   0 and if C , C1 , C2 C with C  C1  C2 and C1  C2   then almost surely X C  X C1  X C2 . In particular, if C  C and C  A \  i 1 Ai , A, A1 ,..., An  A then almost surely n

X C  X A   i 1 X A Ai   i  j X A Ai  Aj  ...  (1) n X An n

i 1 Ai

We shall always assume that our stochastic processes are additive. We note that a process with an (almost sure) additive extension to C also has an (almost sure) additive extension to C(u) .

 on (T , B) is called strictly monotone on A if:   '  0 and  A   B for all A  B , A, B  A . The collection of these measures is denoted by M ( A) . (  '   AA , A A , note Definition 2. A positive measure that  '   ) The classical examples for definition is Lebesgue measure or Radon measure when T   d

and

A = A( )  {[0, x] : x  } d 

d 

Definition 3. Let   M (A) . We say that the A -indexed process X is a Brownian motion with variance

 if X can be extended to a finitely additive process on C(u) and if for disjoint sets C1 ,..., Cn C , X C ,..., X C are independent mean-zero Gaussian random variables with variances  C ,...,  C , respectively. (For any   M (A) , there exists a set-indexed Brownian motion with variance  [IvMe]). 1

III.

RKHS OF A SET INDEXED BROWNIAN MOTION

L ( A ) is a separable space,and thusit must have a countable orthonormal basis. Then every set indexed stochastic process X  { X A : A  A} has the representation X A



  aii , ai  where  i are i 1

random variables. Our purpose is to find the representation forthe set indexed Brownian motion via orthonormal basis. We restart with the RKHS of a set indexed Brownian motion. A RKHSis a Hilbert Space of functions. It can be thought of as a space containing smoother function than the general Hilbert space. Let A  A and X  { X A : A  A} be a set indexed Brownian motion. We define the function

R( A, ) : A   by R( A, )  Cov( X A , X  )  E[ X A X  ] . Now, we define the set   { f : A   : f () 

 a R( A , ), A  A, a , n  1} . Define an inner i 1

product on  by: n

i 1

j 1

 f , g    ai b j R ( Ai , B j ) when f ()   ai R( Ai , ), g ()   b j R( Bi , ) (1) i 1 j 1

The fact that R is non-negative definite implies  f , f  

 a R( A , A )  0 for all f   .From the inner i 1

2 i

product, we get the following property: n

i 1

f ( B)   ai R( Ai , B)    ai R( Ai , ),R( B, )   f , R( B, )

(2)

Then for all f   , B  A , | f ( B) | |  f , R( B, ) |   f , f  R( B, ), R( B, ) . The inequality being merely the Schwartz inequality for semi-inner products, which holds as long as  f , f   0 . Thus, if 2

 f , f   0 then (2) implies that f  0 for all B  A . Consequently, (1) defines a proper inner product on  , and so we thus obtain a norm:

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Reproducing Kernel Hilbert Spaceof A Set Indexed Brownian Motion

f  f, f 

For { f n }n 1   we have:

| f n ( B)  f m ( B) |2 |  f n  f m , R( B, ) |2  f n ( B)  f m ( B)

R( B, )  2

 f n ( B)  f m ( B) R( B, B) 2



Thusitfollows that if { f n }n 1 is Cauchy in



then it is a pointwise Cauchy. The closure of  denoted

by H . Since T is separable and R continuous then H is also separable.Since H is a separable Hilbert space, it must have a countable orthonormal basis. Define H ( X ) , the so-called “linear part” of the L2 ( A ) space of the set indexed stochastic process X , as the closure in L2 ( A ) of

 n   ai X Ai , Ai  A, ai  , n  1 ,  i 1  2 thinned out by identifying all elements indistinguishable in L ( A ) (In other word, elements x , y for which

E[( x  y ) 2 ]  0 ). This contains all distinguishable random variables, with finite variance, obtainable as linear combination of values of the process. There is a linear, one-one mapping T :   L ( A) defined by: 2

 n  n Τ ( f )  T   ai R ( Ai , )    ai X Ai  i 1  i 1 Note that T is clearly norm preserving and so extends to all H with range equal to all of H ( X ) . (In other words, there is exists Tr such that Tr |  T , Dom(Tr)  H and Im(Tr)  H ( X ) ). Since H is separable, we now know that H ( X ) isalso separable. We can use this to build an orthonormal basis for H ( X ) . If {n }n 1 is an orthonormal basis for H then {en }n 1 an orthonormal basis for H ( X ) when 



en  T(n ) . X is a set indexed Brownian motion therefore E[en ]  0 for all n and 

X A   ei E[ X Aei ] almost surely, i 1



where the series converges in L ( A ) . Easy to see that {en }n 1 an orthonormal sequences of centered Gaussian variables. Since T was an isometry, it follows that

E[ X Aei ]   R( A, ), i   i ( A) for all i



Then

X A   eii ( A) almost surely. i 1

RKHS and the KL representation of a set indexed Brownian motion on A ([0,1] ) and on A = A( Ls) Now, we willpresent two special cases of a RKHS and the KL representation of a set indexed Brownian motion, when: i.

T  [0,1]d and A([0,1]d )  {[0, x] : x  [0,1]d } (see Examples after Definition 1).

ii.

T  [0,1]d and A = A( Ls) (see Examples afterDefinition 1).

First case: T  [0,1] and A = A([0,1] )  {[0, x] : x  [0,1] } d

Let X  { X A : A  A}

set

indexed

Brownian

motion

when

T  [0,1]d

and

A = A([0,1] )  {[0, x] : x  [0,1] } . d

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Reproducing Kernel Hilbert Spaceof A Set Indexed Brownian Motion Let A  [0, a1 ]  [0, a2 ]  ...  [0, ad ]  A, B  [0, b1 ]  [0, b2 ]  ...  [0, bd ]  A then

R( A, B)  E[ X A X B ]  min{a1 , b1}min{a2 , b2 }  min{ad , bd } Values in set  are: n

i 1

f ( A)   ai R( Ai , A)    i min{a1 (i ), a1}min{a2 (i ), a2 }  min{ad (i), ad } n

j 1

i 1

g ( A)    j R( B j , A)    j min{b1 ( j ), a1}min{b2 ( j ), a2}  min{bd ( j ), ad } When

Ai  [0, a1 (i)]  [0, a2 (i)]  ...  [0, ad (i)]  A and B j  [0, b1 ( j )]  [0, b2 ( j )]  ...  [0, bd ( j )]  A The inner product on  is: n

 f , g     i  j R( Ai , B j )  i 1 j 1

   i  j min{a1 (i), b1 ( j )}min{a2 (i), b2 ( j )}  min{ad (i), bd ( j )} i 1 j 1

but 1

min{ak (i), bk ( j )}   I[0,ak (i )] ( x) I[0,bk ( j )] ( x)dx for k  1, 2,..., d 0

1 , x  D ). 0 , x  D

When I D is the indicator function of D  A ( I D ( x )   Therefore, we can rewrite the above as follows: n

 f , g   i  j  I[0,a1 (i )] ( x1 ) I[0,b1 ( j )] ( x1 )dx1   I[0,ad (i )] ( xd ) I[0,bd ( j )] ( xd )dxd  i 1 j 1

  i  j i 1 j 1



I[0,a1 (i )] ( x1 ) I[0,b1 ( j )] ( x1 )  I[0, ad ( i )] ( xd ) I[0,bd ( j )] ( xd )dx1  dxd 

  [ I

[0,1]d i 1





[0,1]d



[0, a1 ( i )]

( x1 ) I[0,ad (i )] ( xd )]  j [ I[0,b1 ( j )] ( x1 )  I[0,bd ( j )] ( xd )]dx1  dxd  j 1



f I ( x1 ,..., xn )g I ( x1 ,..., xn )dx1  dxd 

[0,1]d

f I g I dS

[0,1]d

When n

f I ( x1 , x2 ,..., xd )    i [ I[0,a1 (i )] ( x1 ) I[0,ad (i )] ( xd ) i 1

g I ( x1 , x2 ,..., xd )    j [ I[0,b1 ( j )] ( x1 ) I[0,bd ( j )] ( xd ) j 1

Finally,  f , g  



f g dS . d

[0,1]

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Reproducing Kernel Hilbert Spaceof A Set Indexed Brownian Motion We

define

 I  { f : A   : f ()   f I dS , 



f ( A)   f , R( A, ) 



( f I )2 dS   }

then

R( A, )   I and

[0,1]d

f I ( x1,..., xd ) I[0,a1 ] ( x1 )  I[0,ad ] ( xd )dx1  dxd then  I   . The mapping

[0,1]d

  n T :   L2 ( A) is Τ ( f )  T   f I g I dS   i X Ai and from that we get in the same way   i 1 

X A   ei E[ X Aei ] almost surely. i 1

Karhunen-Loève expansion of a set indexed Brownian motion: Let 1 , 2 ,..., d and  1 , 2 ,..., d be a eigenvalues and normalized eigenfunctions of operator

 : L2 ( A)  L2 ( A)



 ( ( A))   ( ( x1 , x2 ,..., xd )) 

R( A, S ) ( S )dS 

[0,1]d





R ( A, S )   min{s1 , x1}min{s2 , x2 }  min{sd , xd } ( s1 , s2 ,..., sd )ds1ds2  dsd

[0,1]d

When A  [0, x1 ]  [0, x2 ]  ...  [0, xd ]  A , S  [0, s1 ]  [0, s2 ]  ...  [0, sd ]  A. That is

i and  i solve the integral equation



 ( x1 , x2 ,..., xd ) 

min{s1 , x1}min{s2 , x2}  min{sd , xd } ( s1, s2 ,..., sd )ds1ds2  dsd

[0,1]d

And



1 , i  j 0 , i  j

 i ( s1 , s2 ,..., sd ) j ( s1 , s2 ,..., sd )ds1ds2  dsd  

[0,1]d

Suppose that, there exist a { i }id1 such that

 ( x1 , x2 ,..., xd )   1 ( x1 ) 2 ( x2 )   d ( xd ) for all

( x1 , x2 ,..., xd )  [0,1]d then 1

 1 ( x1 )   d ( xd )    1 ( s1 ) min{s1 , x1}ds1    d ( sd ) min{sd , xd }dsd 0

d Denote    . If for all i ,  i ( xi ) 

(3)

  (s ) min{s , x }ds i

then we get the required on (3).It is clearthat,

 i ( xi )    i ( si ) min{si , xi }dsi   si dsi  xi   i (si )dsi for all i . Differentiating both sides with 0

respect to xi generates:

 i '( xi )    i ( si )dsi ,  i ''( xi )   i ( xi ) for all i , togetherwith boundary xi

condition  i (0)  0 . The solutions of this pair of differential equations are given by:

 i ,n ( xi )  2 sin( 12 (2n  1) xi ) ,  n   (2 n 21) 

Then

 n ( x1 , x2 ,..., xd )   1,n ( x1 ) 2,n ( x2 )   d ,n ( xd )   2 sin( 12 (2n  1) xi ) , n   (2 n 21)  d

i 1

The Karhunel-Loève expansion of X is obtained by setting

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i   i i , we get:

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Reproducing Kernel Hilbert Spaceof A Set Indexed Brownian Motion 

X A   n en n ( A) n 1

In other words,



n 1

i 1

X [0, x1 ][0, x2 ]...[0, xd ]  ( 2 2 )d  (2 n11)d en  sin( 12 (2n  1) xi )

(4)

Second case: T  [0,1] and A = A( Ls) d

Let x  ( x1 , x2 ,..., xd )  [0,1]d , wedenote [0, x]  [0, x1 ]  [0, x2 ]  ...  [0, xd ] . There existsan increasing sequence

finite

sub-classes

Am ([0,1]d )  {A1m ,..., Akmm }  A closed

under

intersection:

A ([0,1] )  {[0, x]: xi  2m , 0  ki  2 , i  1, 2,..., d}  .From the Definition 1 we derive that: m

A A( Ls) ,

gm ([0, x])  [0, ( 2km1 ,..., 2kmd )]  [0, 2km1 ]  [0, 2km2 ]  ... [0, 2kmd ] , g m ()   and for all

A  m g m ( A) . Let A A( Ls) then from (4),

X gm ([0, x1 ][0, x2 ]...[0, xd ])  X [0, k1 ][0, k2 ]...[0, kd ]  ( 2m



2 2 d



)

 n 1

en  sin( 12 (2n  1)

1 (2 n 1)d

i 1

ki 2m

)

The process X is continuous and then from Definition 1 we derive: 

lim X gm ([0, x1 ][0, x2 ]...[0, xd ])  lim( 2 2 ) d  (2 n11)d en  sin( 12 (2n  1)

m

X A  X m gm ( A)  X 

n 1

kd k1 k2 m g m ([0, m ][0, m ]...[0, m ]) 2 2 2

i 1



ki 2m

)

 lim( 2 2 )d  (2 n11)d en  sin( 12 (2n  1) m

n 1

i 1

ki 2m

)

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