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A C T A U N I V E R S I T A T I S L O D Z I E N S I S

FO LIA O E C O N O M IC A 206, 2007

G rażyn a Trzpiot*

M ULTIV A LU ED M ARKOV P R O C E SSE S

Abstract. M ultivalued random variables and stochastic processes can be use in integral geom etry, m athem atical econom ics or stochastic optim ization. U sin g the m ethods o f selection operators we can give the selection characterization o f identically distributed multivalued random variables. In this paper the regular selections and M arkov selections for multivalued stochastic processes will be studied.

Key words: m utivalued random variable, mutivalued stochastic processes.

1. IN T R O D U C T IO N

We present a concept of selection operators for m ultivalued random variables. F o r multivalued stochastic processes the some clue problem is the question of existing the vector-valued selection processes. In this paper we continue our work on properties o f m ultivalued random variables and stochastic process ( T r z p i o t 1999, 2000, 2004). First sections contain basic definition, next we remain characterizations of identically distributed m ul­ tivalued random variables and the selection problem o f multivalued random variables converging in distribution. Finally, we study the selection problem for multivalued M arkov processes.

2. M U L T IV A L U E D R A N D O M VA RIA BLE

Given a probability m easure space ( 0 , A , ц) random variable in classical definition is a m apping from Q to R. M ultivalued random variable is a m apping from Ü to all closed subset of X .

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We have a real Banach space X with m etric d. F o r any nonem pty and closed sets А, В с: X we define the H ausdorff distance h(A, В) o f A and B.

Definition 1. The exccss for two nonem pty and closed sets be defined by e(A, B) = sup d(x, B), where d(x, B) = inf||x — y||,

x e A y e B

the Hausdorff distance o f A and В is given by h(A, В ) = m ax {e(A, B), e(B, A)}, the norm || A || of set A we get as

M il = K ^ , {0}) = sup II x II. x e A

The set of all nonem pty and closed subsets o f X is a m etric space with the H ausdorff distance. The set o f all nonem pty and com pact subsets of A" is a complete, separable m etric space with the m etric h.

Definition 2. A multivalued function cp.Q -* 2X with nonem pty and closed values, is said to be (weakly) measurable, if cp satisfies the following equi­ valent conditions:

a) <p~\C) = { cu eß ■-(pico) n G ф 0} e A for every G open subset of X , b) d(x, is m easurable in со for every x e X ,

c) there exists a sequence {fn} of measurable functions f n. Q - * X such that cp(a>) = c l { f n(co)} for all а ) е й .

Definition 3. A m easurable multivalued function <p: £2 -* 2X with nonempty and closed values is called a multivalued random variable.

A multivalued function cp is called strongly m easurable, if there exist a sequence {ęoj o f simple functions (measurable functions having a finite num ber of values in 2X), such that h(ipn(co), <p(co)) -> 0 a.e.

Since set o f all nonem pty and com pact (or convex and com pact) subsets o f AT is a complete separable metric space with the m etric h, so m ultifunction ę \ Q ~ * 2х is m easurable, if and only if is strongly m easurable. This is equivalent to the Borel m easurability of <p.

Let -Kí-X-) 1 denote all nonem pty and closed subsets o f X . As the cr-field on K ( X ) , we get the er-field generated by cp~\G) = {соe &: <p(co) n G Ф 0},

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for every open subset G of X . The smallest ст-algebra containing these <p~[(G) we de noted by Acp

1. Two m ultifunctions <p and у/ are independent, if Acp and Ay/ are independent.

2. Tw o m ultifunctions (p and \y are identically distributed, if ц(ср~1(С)) — for all closed C c i X .

Definition 4. We say that a sequence of m ultivalued random variables (pn'.Q~* 2K(X) is independent, if so is {<?„} considered as m easurable functions from (Я, А, ц) to д а о , G).

Definition 5. Tw o multivalued random variables tp, if/ : £2 -> 2ВД are iden­ tically distributed, if cp(co) = y/(£2) a.e.

Particularly for ę n with com pact values independence (identical dist­ ributedness) of {ęn} coincides with that considered as Borel m easurable functions to all nonem pty, compact subsets o f X .

Definition 6. A selection of the m easurable m ultifunction <p: Q - » 2X is a m easurable function f : Q - * X , such th at f((o)etp(a>) for all coe£2.

Let Lľ(£2,A), for 1 o o , denote the X - valued, I f - space. We introduce the multivalued I f space.

Definition 7. The m ultivalued space Lp[i2, K(A")], for 1 < p < oo denote the space o f all m easurable multivalued functions <p:£2 - » 2K{X\ such that II (p II = II p(-)|| is in U .

Then U[Q, K(X)] becomes a complete m etric space with the m etric H p given by

H p(<p, w) = {5aK<P(a>), ЧУ(а>)УйцУ1р for 1 < p < со, Я «(P. V) = esssup h((p(co), ч/(со)),

(o eQ

where <p and i// are considered to be identical, if <p(a>) = y/(cu) a.e.

We can define similarly other U space for set o f different subsets of X (convex and closed, weakly com pact or com pact). We denote by LP[Q K (X)] the space o f all strongly m easurable functions in LP[ Q ,K ( X )]. Then all this space is complete metric space with the m etric H p.

Definition 8. The m ean E(<z>), for a m ultivalued random variables cp:Q -> 2K(X) is given as the integral \ a<pdn of <p defined by

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E(<p) = jo

4

>dp = { J o / d ^ : / e S{<p)}, where

S(<p) = { f eV[£2, X \ :f(co)e<p(co) a.e.}.

The m ean E(^>) exist, if S(<p) is nonempty. M ultifunction (p is an integ- rable, if II (p(co)\\ is an integrable. If <p have an integral, then E(<p) is compact. If Ц is atomless, then E(<p) is convex. If (p have an integral2 and E(ę?) is nonem pty, then coE(ę?) = E(cop) (со - denote convex hull o f the set). Now we present some properties o f m ean of m ultivalued random variables.

Let cp,if/:£2 -+ 2K(jr) be two multivalued random variables with nonempty S((p) and then:

1) cl E(<p и у ) = cl(E(ę>) + E(i//)), where (cp u yj){(o) = с\((р{оз) + w(co)), 2) cl E (co ę) — со E(ę>), where (со <р)(оз) = со (p(co), the closed convex hull, 3) /j(c/E (9>), clE(v,)) = # ,( ? ,

v')-Lemat 1 (A u m a n 1965) Let ip:i2 -* 2K(-X) and 1 < p ^ oo. If Sp(<P) = [ f e U[£2, X] :/(&») e <p(co) a.e.}.

then exists a sequence { fn} contained in Sp((p) such th at <p(ca) = c[{fn(co)} for all ( o eQ .

Lemat 2 (A u m a n 1965). Let <p, y/:£2-* 2K<-X) and l < p < o o . If Sp(y/) = Sp(y/) Ф 0 then (p(y/) = y/(co) a.e.

3. M U L T IV A L U E D S T O C H A S T IC P R O C E S S

Let T denote the set o f positive integers or nonnegative real numbers. Definition 9. Multivalued stochastic process is a family of multivalued random variables indexed by T{<p„ t e T } .

Supposing th at P are the certain properties of stochastic processes. Definition 10. A vector valued stochastic process [ f t, t e T } will be called a P selection o f {(pn, n > 1}, if {/„ t e T } has the properties P and f te<pt, a.e. for each t e T .

Let {At, t e T } be an increasing family o f sub-a-algebras of A.

A m ultivalued stochastic process {<p„ t e T } is said to be integrable, if for each t e T is integrable bounded (respectively, A, m easurable).

2 The multivalued integral was introduced by R. J. A u m a n (1965). For detailed arguments concerning the measurability and integration o f m ultifuction w e refer to (C a s t a i n g, V a - l a d i e r 1977; D e b r e u 1967; R o c k e f e l l a r 1976).

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Definition 11. Let X be a separable Banach space. The m ap Г : K ( X ) -* X is called a selection operator, if Г ( А ) е А , for all A e K ( X ) .

1) Г is called a continuous selection operator (or m easurable operator), if Г is continuous with respect to topology on K ( X ) generated by the subbase { A e K ( X ) , a < d ( x , A ) < b } (a, b e R , x e l } , D enote Borel er-algebra o f this topology by á?. This is separable and completely m ertizable topology space ( K (X ), W).

2) Г is called a linear selection operator, if for any A, B e K ( X ) Г(а1А + a 2B) = a, Г(А ) + а2Г(В),

3) Г is called a Lipschitz selection operator, if there exists a constant k > 0 such th at for any A, B e K ( X )

\ \ Г ( А ) - П В ) \ \ < Ы ( А , В ) .

Theorem 1 (T r z p i o t 2004). Let X be a separable Banach space. Then there exists a sequence of m easurable selection operators {Гп, n ^ 1} such th at for each A e K ( X )

А = с1{Гп(А), n > 1}.

G. S a l i n e t t i and R. W e t s (1979) studied the distribution theory of multivalued random variables in finite-dimensional Banach spaces and they proved that:

• M ultivalued random variables and <p2 are identically distributed, if and only if the real-valued stochastic process {d(x, <p{), x e X ) and {d(x,<p2), x e X } have the same finite dimensional distribution.

• If a sequence o f multivalued random variables {<pn, n > 1} converges in distribution to <p, then there exist selections {fn, 1} of {tpn, n ^ 1} such th at { | | / J n > 1} converges in distribution to ||/ ||, where / is a vector valued random variables with f e t p a.e.

Theorem 2 ( T r z p i o t 2004). Let I be a finite-dimensional Banach space, and let and (p2 be two multivalued random variables. Then the following are equivalent:

1) <p{ and (p2 are identically distributed,

2) there exist selection sequences { f [n, n > 1} and {ft, n > 1} o f <pl and q>2, such that <pt(&) = c l n > 1}, i = 1,2,

3) the real-valued stochastic process {d(x,<pl), x e X } , and {d(x, <p2), x e l } have the same finite dimensional distribution.

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Given a complete probability space (Ü, A, h) and increasing family o f sub-cr-algebra o f A: {At, t e R + }. A multivalued stochastic process {(pt,

t e T } is said to be regular and right-continuous with respect to topology space (K ( X ), W), if it is adapted and for each a>eQ, has a left-hand limit and is right-continuous with respect to topology space (K ( X ), И0 for every i e R + .

Theorem 3 ( T r z p i o t 2004). Let AT be a separable Banach space and let {<pt, í e R + } c LP[Q, K C(X)] be a regular and right-continuous with respect to topology space (K ( X ), W). Then {ęt, i e R + } has a regular and right- -continuous selection.

Based on Theorem 3 ( T r z p i o t 2004) we can write the following theorem, which is a m ultivalued version o f Theorem 2 ( T r z p i o t 2004).

4. M U L T IV A L U E D M ARK O V P R O C E S S E S

Now we will study the selection problem for M arkov process. As in Definition 11, the topology on K ( X ) is generated by the subbase { A e K ( X ) , a < d ( x , A ) < b } (a, b e R , x e X ) } . Denote Borel er-algebra of this topology by B. This is separable and completely mertizable topology space (K (X ), W). So we start from Definition 12.

Definition 12. Multivalued stochastic process is called a M arkov process, if it is a M arkov process with the m easurable space (K ( X ), W) being its state-space.

D enote by D an index set and by a{<pd, d e D } the ст-algebra generated by the family o f m ultivalued random variables. D enote by ( Z N, YN) the countable product space o f m easurable space (Z, У).

Theorem 4. Let X be a separable Banach space and let {cpt, i e R + } be a m ultivalued M arkov process and A, — cr{<p„ s < t}. Then exist the family of vector valued stochastic process í e R +, n > 1} such that

(P'(co) = cl {/"(си), i e R + , 1}.

Additionally if we write x,(co) = ...f " ,...,...), then {x„ £ e R +} is a M arkov process with the state space (X N, &8N).

Proof. Let {Fn, 1} be a sequence o f m easurable selection operators on

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f"(co) = r n(<pt(co)). So we have a family o f adapted process which satisfying the first thesis.

T o p ro o f th at {xt, r e R +, n > 1} is a M arkov process it is sufficient to show th a t for each bounded m easurable function h : (X N, 38N) -> R, one has

E[A(xr) H J = E[/j(x,)|ct(x,)], t > s , t, s e R + ,

where E[/i|j4] denotes the conditional expectation with respect to A. Define H : ( K ( X ) , W ) - > R as follows

H ( A ) ш h ( r i( A ) , r 2( A ) ,Г „ (с ) ,...), A e K ( X ) . T hen H ( - ) is a bounded m easurable function on K ( X ) . F rom the M arkov property o f {tp,, r e R +} follows that

E [ H ( ^ ) |Ą J = ЦН(<р^ I a.e. t > s , t, s e R +, that means

E [ /j( x ,)|Ą J = E[/j(x()|ct(í»s)], a.e. t > s , t, s e R + , and because <р,(а>) = cl{f?(co), i e R +, « > 1 } we have

o-(í»s) = o ( f ns, n > 1) = a(x, s).

According Theorem 2 the statistical law o f <pt is completely determinated by that of {f,, í e R + , n ^ 1}, so we can call the Theorem 4 the discretization theorem for m ultivalued M arkov process.

If the m ultivalued M arkov process takes the values in a finite subset Z of positive integer, then the process {x„ i e R + , 1} presented on Theorem 4 m ay be regarded as a m odel o f interacting particle systems.

REFER EN C ES

A r t s l e i n Z., V i t a l e R. A . (1975), A strong law o f large numbers f o r random com pact sets, “A n nals o f Probability” , 3, 879-882.

A u m a n R. J. (1965), Integrals o f set-valued functions, “Journal o f M athem atical Analysis and A p p lication ” , 12, 1, 1-12.

B e r g e С. (1966), Espaces topologiques, D u n od , Paris.

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C a s t a i n g C., V a l a d i e r M . (1977), Convex Analysis and M easurable M ultifunctions, “ Lec­ tures N o tes o f M athem atics” , 580, Springer-Verlag, Berlin.

D e b r e u G . (1967), Integration o f correspondens, “Proceding 5th Berkeley Sym posium on M athem atics, Statistics and Probabilistics” , 1, 2, 351-372.

E n g e l k i n g R. (1975), Topologia ogólna, PW N, W arszawa. H a u s d o r f f F. (1957), S et Theory, Chelsea, N ew Jork.

H e s s C. (1991), Convergence o f Conditional Expectations f o r Unbonded Random Sets, Integrands, and Integral Functionals, “M athem atics o f O perations Research” , 16, 3, 627-649. R o c k e f e l l a r R. T. (1976), Integral functionals, normal integrands, mesurable selections,

“ Lectures N otes o f M athem atics” , 543, 157-207.

S a l i n e t t i , G. , W e t s R. (1979), On the convergence o f sequences o f convex S ets in Finite Dimensions, SIA M R eview, 21, 1.

S a p o r t a G. (1990), Prohabilités, analyse des données et statistiqu e, Edition Technique, Paris. T r z p i o t G . (1999), W ielowartościowe zmienne losowe tv badaniach ekonom icznych, A E K a­

towice.

T r z p i o t G. (2002), M ultivariate M ultivalued Random Variable, “A cta Universitalis Lodziensis”, F olia O econom ica, 162, 9-1 7 .

T r z p i o t G. (2006), M ultivalued Stochastic Processes, “A cta U niversitalis Lodziensis” , Folia O econom ica, 196, 93-102.

G rażyna Trzpiot

W IE L O W A R T O ŚC IO W E PR O C E SY M A R K O W A

W ielow artościow e zmienne losow e i w ielow artościow e procesy stochastyczne znajdują zastosow anie w geom etrii różniczkowej, w matematycznej ekonom ii oraz w zadaniach stocha­ stycznej optym alizacji. W ykorzystując operatory selekcyjne m ożliw a jest charakterystyka ciągu w ielow artościow ych zmiennych losow ych o takim samym rozkładzie. Przedmiotem badań zaprezentowanych w artykule są selektory wielow artościow ych procesów M arkowa.

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