Riesz Space and Fuzzy Upcrossing Theorems Wen-Chi Kuo1 , Coenraad C.A. Labuschagne1 and Bruce A. Watson2 1

2

School of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, South Africa [email protected], [email protected] Centre for Applicable Analysis and Number Theory and School of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, South Africa [email protected]

In our earlier paper, Discrete-time stochastic processes on Riesz spaces, we introduced the concepts of conditional expectations, martingales and stopping times on Dedekind complete Riesz space with weak order units. Here we give a construction of stopping times from sequences in a Riesz space and are consequently able to prove a Riesz space upcrossing theorem which is applicable to fuzzy processes.

1 Introduction The concepts of conditional expectations, martingales and stopping times on Dedekind complete Riesz space with weak order units were introduced in [3]. The concepts of generalized stochastic processes have also been studied by [1, 2, 5]. Here we construct stopping times from sequences in a Riesz space and as a consequence are able to prove a Riesz space upcrossing theorem. We conclude the paper by applying our result to fuzzy processes. Let E be a Riesz space with weak order unit. A positive order continuous linear projection, T , on E, with T e a weak order unit of E for each weak order unit e of E and R(T ) a Dedekind complete Riesz subspace of E is called a conditional expectation on E. A filtration on E is a family of conditional expectations, (Ti )i∈N , on E with Ti Tj = Tj Ti = Ti for all j ≥ i. If (Ti ) is a filtration on E then there is a weak order unit e ∈ E that is invariant under Ti for all i ∈ N. To see this, let u be any weak order unit and e = T1 u. Then e is a weak order unit and Tj e = Tj T1 u = T1 u = e. The pair (fi , Ti )i∈N is a (sub, super) martingale on the Riesz space E with weak order unit, if (Ti ) is a filtration on E, fi ∈ R(Ti ) for all i ∈ N and fi (≤, ≥) = Ti fj , for all i ≤ j. Let (Ti ) be a filtration on a Riesz space E with weak order unit. We define a stopping time P to be an increasing family of positive order continuous

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linear projections (Pi ) with R(Pi ) a Dedekind complete Riesz subspace of E, and having P0 = 0, Pi ≤ I, for all i, and Tj Pi = Pi Tj , for all i ≤ j. We say that the stopping time P is bounded if there exists N so that Pn = I for all n ≥ N. Further details about the theory resulting from these definitions and their consequences can be found in [3].

2 Preliminaries Let T be a conditional expectation on a Riesz space, E, with weak order unit and denote by E + = {f ∈ E | f ≥ 0} the positive cone of E. If f ∈ E + then the band, Bf , generated by f is given by Bf = {g ∈ E | |g| ∧ nf ↑n |g|} and the band projection, Pf , onto Bf is given by Pf g = sup g ∧ nf,

for all

n

g ∈ E+,

and Pf is then defined to be the unique extension of Pf to E. In particular Pf g = Pf g + − Pf g − . Through out this section, we consider positive elements first and then use linearity and positivity of the maps to extend the results to general elements of E, see [7]. Theorem 1. ([4]) Let T be a conditional expectation on a Dedekind complete Riesz space, E, with weak order unit and let P be the band projection of E onto the band, B, in E generated by 0 ≤ g ∈ R(T ). Then T P = P T .

3 Construction of Stopping Times We now inductively apply the above reasoning to build stopping times. Theorem 2. Let (Ti ) be a filtration on a Dedekind complete Riesz space, E, with weak order unit and let (fi ) be an increasing sequence in E with fi ∈ R(Ti ) for all i ∈ N. Then Pf := (Pf + ) is a stopping time adapted to the i filtration (Ti ), where Pf + denotes the projection onto the band Bf + . i

i

Proof. From Theorem 1, Ti Pf + = Pf + Ti , for j ≥ i, and as i

i

fj+

fi+

i

i

fi+

∈ R(Tj ), we

have Tj Pf + = Pf + Tj . Also, ≥ and thus Bf + ⊂ Bf + from which it i i i j follows that Pf + Pf + = Pf + = Pf + Pf + . j

i

j

u t

Riesz Space and Fuzzy Upcrossing Theorems

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The following theorem, which is a direct consequence of Theorem 2, gives that if (fi ) a sequence in E with fi ∈ R(Ti ) for all i ∈ N, where (Ti ) is a filtration on E, then there exists a stopping time P adapted to (Ti ) with the property that g ∈ R(Pi ) if and only if g is in the band in E generated by f1+ ∨ ... ∨ fi+ . In applications this type of stopping time is of critical importance: for example, let (Ω, F, µ) be a probability space and E = L1 (Ω, F, µ). The stopping time P corresponds to the stopping time τ : Ω → N ∪ {∞}, defined by  ∞, fi (x) ≤ 0 for all i ∈ N , τ (x) = min{i | fi (x) > 0}, fj (x) > 0 for at least one j ∈ N which gives the least i for which fi (x) is positive. Corollary 1. Let (Ti ) be a filtration on a Dedekind complete Riesz space, E, with weak order unit and let (fi ) be a sequence in E with fi ∈ R(Ti ) for all i ∈ N. Then P := (Pgi ) is a stopping time adapted to the filtration (Ti ), where Wi gi = j=1 fj+ . We can now deduce a relation between orderings on the sequences (fi ) and the orderings of the stopping times they generate. Definition 1. Let F and G be stopping times adapted to the filtration (Ti ). We say G ≤ F if Fi ≤ Gi for all i ∈ N. Lemma 1. Let (Ti ) be a filtration on a Dedekind complete Riesz space, E, with weak order unit. Let (fi ) and (gi ) be increasing sequences in E with fi , gi ∈ R(Ti ) and 0 ≤ fi ≤ gi , for all i ∈ N. Then, as stopping times, G ≤ F where F = (Pfi ) and G = (Pgi ). Proof. Let q ∈ E + , then q ∧ nfi ≤ q ∧ ngi . Taking suprema over n we thus u t obtain that Pfi q ≤ Pgi q. Hence G ≤ F . To conclude this section with the Riesz space analogue of the classical result that if (fi , Fi ) is a sub (super) martingale, then ((fi − a)+(−) , Fi ) is a positive sub martingale for each a ∈ R. Theorem 3. Let (fi , Ti ) be a sub (super) martingale on the Riesz space E with weak order unit. Then ((fi − g)+(−) , Ti ) is a sub martingale on E for each g ∈ R(T1 ). Proof. As R(Ti ) is a Riesz space in its own right, it follows that (fi − g)+(−) ∈ R(Ti ), hence it remains only to verify the sub martingale inequality. From the assumption that (fi , Ti ) is a sub (super) martingale and that g ∈ R(T1 ) ⊂ R(Ti ) for all i, it follows that (fi − g, Ti ) is a sub (super) martingale and thus that for all i ≤ j, (fi − g) ≤ (≥) Ti (fj − g).

(1)

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But fj − g ≤ (fj − g)+ (respectively g − fj ≤ (fj − g)− ) and Ti is a positive map, hence Ti (fj −g) ≤ Ti ((fj −g)+ ) (respectively Ti (g −fj ) ≤ Ti ((fj −g)− )). Combining this with the fact that Ti ((fj − g)± ) ≥ 0 we have (Ti (fj − g))+(−) ≤ Ti ((fj − g)+(−) ). Together (1) and (2) give (fi − g)+(−) ≤ Ti ((fj − g)+(−) ).

(2) u t

4 Upcrossing Theorem In L1 (Ω, F, µ), if (fi , Fi ) is a martingale and a < b are real numbers, let Ui (a, b) denote the upcrossing to time i. This process has associated with it an upcrossing yield (b − a)Ui (a, b). This concept of upcrossing yield can be naturally extended to Riesz space stochastic processes as below. Definition 2. Let g ≤ f be elements of a Riesz space E and let (fi ) ⊂ E. We define the upcrossing yield of the sequence (fi ) across the order interval [g, f ] to be N X YN (g, f ) = QiN (f − g) i=1

where

S0m

=0=

Qm 0 ,

Sn1

=

Qm n =

PW n

n X j=2

Snm+1 =

n X j=2

i=1 (g−fi )

+

and

m m ) − Sj−2 PW ni=j (fi −f )+ (Sj−1 m PW ni=j (g−fi )+ (Qm j−1 − Qj−2 ).

m+1 for all i ∈ N and thus S m ≤ Qm ≤ S m+1 . We note that Sim ≥ Qm i ≥ Si The following results from [3] will be needed in order to prove a vector valued upcrossing theorem. Let (Ti )i∈N be a filtration and P a bounded stopping time. For each (fi ) with P fi ∈ R(Ti ) for all i ∈ N, Pthe stopped process (fP , TP ) defined by fP = i (Pi − Pi−1 )fi and TP f = i (Pi − Pi−1 )Ti f. Let P be a stopping  time and k ∈ N, we define P ∧ k to be the family of Pi , i < k . If P and S are stopping times compatible projections (P ∧ k)i = I, i ≥ k with the filtration (Ti )i∈N . We say S ≤ P if Pi ≤ Si for all i ∈ N and in this case for all i ≤ j then Pi Sj = Sj Pi = Pi and [Sj − Sj−1 ][Pi − Pi−1 ] = 0 for all j ≥ i + 1.

Riesz Space and Fuzzy Upcrossing Theorems

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Theorem 4. (Hunt’s Optional Stopping Theorem [3, Theorem 5.5]) Let E be a Riesz space with weak order unit, (fi , Ti )i∈N a (sub, super) martingale on E and S ≤ P bounded stopping times, then TS fP = (≥, ≤)fS . i.e. (fP , TP ) is a (sub, super) martingale over the family of all stopping times compatible with the filtration (Ti ). The following theorem provides a Riesz space analogue to the upcrossing theorem of probability spaces, see [6]. Theorem 5. (Upcrossing Theorem) Let (fi , Ti ) be a sub (super) martingale in a Riesz space E with weak order unit. Then for g, f ∈ R(T1 ) with g ≤ f we have T1 YN (g, f ) ≤ T1 (fN − g)+ (resp. T1 (fN − f )− ). Proof. In the proof we use the notation introduced in the definition of the upcrossing yield. Let N ∈ N be fixed through out the proof. Let hi = (fi −g)+ , then (hi , Ti ) is a sub martingale and as S n+1 ≥ Qn we have that S n+1 ∧ N ≥ Qn ∧ N . Thus from Theorem 4, hQn ∧N ≤ TQn ∧N (hS n+1 ∧N ). Consequently 0 ≤ TQn ∧N (hS n+1 ∧N − hQn ∧N ), and as T1 TQn ∧N = T1 we have that 0 ≤ T1 (hS n+1 ∧N − hQn ∧N ). Thus # "N X [hQn ∧N − hS n ∧N ] = T1 [hQN ∧N − hS 1 ∧N ] T1 n=1

+

N −1 X n=1

T1 [hQn ∧N − hS n+1 ∧N ] ≤ T1 [hQN ∧N − hS 1 ∧N ].

(3)

Direct calculation shows that N

(Q ∧ N )i =



I, 0,

i≥N i≤N −1

and hence the stopped process hQN ∧N = hN . Also hS 1 ∧N ≥ 0, thus (3) can be refined to "N # X [hQn ∧N − hS n ∧N ] ≤ T1 hN . (4) T1 n=1

From the definition of Sin and the facts that for fixed n, Qni is an increasing family of positive projections, n − Sin = P(fi+1 −g)− (Qni − Qni−1 ) Si+1 i h i X PW i+1 (fk −g)− − PW ik=j (fk −g)− (Qnj−1 − Qnj−2 ) + j=2

k=j

≤ P(fi+1 −g)− .

n Thus as hi+1 ≥ 0 and Si+1 ≥ Sin , we have

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Kuo, Labuschagne and Watson n 0 ≤ (Si+1 − Sin )hi+1 ≤ P(fi+1 −g)− hi+1

and the definition of hi+1 gives 0 = P(fi+1 −g)− (fi+1 − g)+ = P(fi+1 −g)− hi+1 . Hence n (Si+1 − Sin )hi+1 = 0.

(5)

Proceeding in a similar manner we obtain that 0 ≤ Qni+1 − Qni ≤ P(fi+1 −f )+ which when applied to (fi+1 − f )− gives 0 ≤ (Qni+1 − Qni )(fi+1 − f )− ≤ P(fi+1 −f )+ (fi+1 − f )− = 0 and thus proving that 0 = (Qni+1 − Qni )(fi+1 − f )− .

(6)

As a consequence of (6), 0 ≤ (Qni+1 − Qni )(fi+1 − f )+ = (Qni+1 − Qni )(fi+1 − f ) from which it follows that (Qni+1 − Qni )(f − g) ≤ (Qni+1 − Qni )(fi+1 − g). This inequality along with 0 ≤ (Qni+1 − Qni )(fi+1 − g)− yields (Qni+1 − Qni )hi+1 ≥ (Qni+1 − Qni )(f − g).

(7)

Combining (5) and (7) yields n − Sin )]hi+1 ≥ (Qni+1 − Qni )(f − g). [(Qni+1 − Qni ) − (Si+1

Summing over i = 0, ..., N − 1 in (8) gives N X i=1

as

Qn0

=0=

S0n .

n )]hi ≥ QnN (f − g) [(Qni − Qni−1 ) − (Sin − Si−1

Adding n n − QnN )hN )]hN = (SN [(I − QnN ) − (I − SN

to both sides of the above equation we have n − QnN )hN + QnN (f − g), hQn ∧N − hS n ∧N ≥ (SN

which when summed over n = 1, ..., N yields

(8)

Riesz Space and Fuzzy Upcrossing Theorems N X

n=1

[h

Qn ∧N

−h

S n ∧N

]+

"

N X

n=1

#

n (QnN − SN ) hN ≥ YN (g, f ).

107

(9)

n Now as QnN = 0 for all n = 0, ..., N and as SN hN ≥ 0 we can deduce from (9) that N X

n=1

[hQn ∧N − hS n ∧N ] ≥ YN (g, f ).

(10)

Applying T1 to (10) and using (4) we have u t

T1 hN ≥ T1 YN (g, f ).

5 Application to Fuzzy Processes Let (Ω, A) be a measurable space, then a fuzzy random variable, [5], is a map f : Ω 7→ F (X) with {ω ∈ Ω | [f (ω)]α ∩ B 6= φ} ∈ A,

for all B open in X,

(11)

where X be a Banach space and F (X) = {ν | ν : X 7→ [0, 1], να closed for α ∈ [0, 1], να 6= φ for 0 < α ≤ 1} in which να = {x ∈ X | ν(x) ≥ α}.

(12)

This setting can be conveniently generalized by taking X to be an arbitrary topological space and, for convenience, replacing the interval [0, 1] by [0, ∞) (and when 1 is needed, considering the limit to infinity). Let F(X) = {ν | ν : X 7→ R} with pointwise addition, scalar multiplication and partial ordering making F(X) into a Riesz space. Let να be defined as in (12) for each ν ∈ F(X) and (Ω, A) be a measurable space, then a generalized fuzzy random variable is an element of the positive cone, [L(Ω, X)]+ , of the Dedekind complete Riesz space L(Ω, X), the fuzzy measurable maps. Here f ∈ L(Ω, X) if f : Ω 7→ F(X) and (11) holds. As L(Ω, X) is a Dedekind complete Riesz space with weak order unit, Theorem 5, is applicable. Restricting attention to the positive cone of L(Ω, X), we thus obtain a fuzzy upcrossing theorem.

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References 1. De Jonge, E. (1979). Conditional expectation and ordering. Annals of Probability 7, 179-183. 2. Goussoub, N. (1982). Riesz-space-valued measures and processes. Bull. Soc. Math. France 110, 233-257. 3. Kuo, W.-C., Labuschagne, C.C.A. and Watson, B.A. (2004). Discrete time stochastic processes on Riesz spaces. (Submitted). 4. Kuo, W.-C. (2004). Stochastic processes on Riesz spaces. M.Sc. Dissertation, School of Mathematics, University of the Witwatersrand, Johannesburg. 5. Li, S. and Ogura, Y. (1999). Convergence of set-valued and fuzzy-valued martingales. Fuzzy Sets and Systems 101, 453-461. 6. Stroock D.W. (1987). Lectures on Stochastic Analysis: Diffusion Theory. Cambridge Univeristy Press. 7. Zaanen, A.C. (1997). Introduction to Operator Theory in Riesz Space. SpringerVerlag, Heidelberg.