A Multi-Objective Fuzzy Stopping

in a Stochastic and Fuzzy Environment

Y. YOSHIDA^{§}, M. YASUDA^{†}, J. NAKAGAMI^{†} and M. KURANO^{‡}

§Faculty of Economics and Business Administration, The University of Kitakyushu, Kitakyushu 802-8577, Japan. ^{†}Faculty of Science, ^{‡}Faculty of Education, Chiba University, Chiba 263-8522, Japan.

Abstract: In a stochastic and fuzzy environment, a multi-objective fuzzy stopping problem is discussed. The randomness and fuzziness are evaluated by probabilistic expectations and linear ranking functions respectively. Pareto optimal fuzzy stopping times are given under the assumption of regularity for stopping rules, by using λ-optimal stopping times.

Keyword: Multi-objective optimal stopping; Fuzzy stochastic systems; Fuzzy stopping; Pareto optimal.

1. Introduction

This paper presents a multi-objective fuzzy stopping model of ‘fuzzy stochastic systems’ in cooperation with sequences of ‘fuzzy random variables’. The fuzzy random variable, which is a fuzzy-number-valued extension of classical random variables, was studied by Puri and Ralescu [8] and has been discussed by many authors. It is one of the successful hybrid notions of randomness and fuzziness. On the other hand, stopping problems for a sequence of real-valued random variables had a long history and were studied extensively. Their applications are well-known in various fields (Chow et al. [2], Shiryayev [10]) and especially in the finance theory, recently. The optimal fuzzy stopping for fuzzy random variables is discussed by Yoshida et al. [13], and dynamic fuzzy systems without randomness are studied by Yoshida [14, 15, 16]. This paper analyzes a multiobjective stopping model for fuzzy stochastic systems, by extending the results of the classical stochastic systems (Aubin [1], Ohtsubo [7]).

We also discuss the optimization by ‘fuzzy’ stopping times. Fuzzy stopping times are introduced for dynamic fuzzy systems by Kurano et al. [6] and they are discussed by Yoshida et al. [12], and this paper applies the notion of fuzzy stopping times in a stochastic and fuzzy environment. In this paper, we evaluate the randomness and fuzziness regarding the stopped fuzzy stochastic systems by probabilistic expectations and linear ranking functions respectively. And we give Pareto optimal fuzzy stopping times for the multi-objective model, by introducing the notion of λ-optimal stopping times.

In Section 2, the notations and definitions of fuzzy random variables are given. In Section 3, fuzzy stopping times are introduced. We formulate a multi-objective optimal stopping problem for fuzzy stochastic systems by fuzzy stopping times. In Section 4, we give Pareto optimal fuzzy stopping times for the problem under the assumption of regularity for stopping rules.

2. Fuzzy random variables

Some mathematical notations of fuzzy random variables are given in this section. Let (Ω, M, P) be a probability space, where M is a σ-field and P is a non-atomic probability measure. Let ℝ be the set of all real numbers, let B denote the Borel σ-field of ℝ and let I denote the set of all bounded closed sub-intervals of ℝ. A fuzzy number is denoted by its membership function ã : ℝ ↦ [0, 1] which is normal, upper-semicontinuous, fuzzy convex and has a compact support. Refer to Zadeh [17] for the theory of fuzzy sets. R denotes the set of all fuzzy numbers. The α-cut of a fuzzy number ã(∈ R) is given by

ã_{α} := {x ∈ ℝ | ã(x) ≥ α} (α ∈ (0, 1]) and ã_{0} := cl{x ∈ ℝ | ã(x) > 0},

where cl denotes the closure of an interval. In this paper, we write the closed intervals by

[ã]_{α} := [[ã]_{α}^{−}, [ã]_{α}^{+}] for α ∈ [0, 1].

A map X̃ : Ω ↦ R is called a fuzzy random variable if

{(ω, x) ∈ Ω × ℝ | X̃(ω)(x) ≥ α} ∈ M × B for all α ∈ [0, 1].  (2.1)

The condition (2.1) is also written as

{(ω, x) ∈ Ω × ℝ | x ∈ [X̃(ω)]_{α}} ∈ M × B for all α ∈ [0, 1],  (2.2)

where [X̃(ω)]_{α} = [[X̃(ω)]_{α}^{−}, [X̃(ω)]_{α}^{+}] := {x ∈ ℝ | X̃(ω)(x) ≥ α} is the α-cut of the fuzzy number X̃(ω) for ω ∈ Ω. We can find some equivalent conditions ([9]), however, in this paper, we adopt a simple equivalent condition in the following lemma.

Lemma 2.1 (Wang and Zhang [11, Theorems 2.1 and 2.2]). For a map X̃ : Ω ↦ R, the following (i) and (ii) are equivalent:

(i) X̃ is a fuzzy random variable.

(ii) The maps ω ↦ [X̃(ω)]_{α}^{−} and ω ↦ [X̃(ω)]_{α}^{+} are measurable for all α ∈ [0, 1].

Now we introduce expectations of fuzzy random variables for the description of stopping models for fuzzy stochastic systems. A fuzzy random variable X̃ is called integrably bounded if ω ↦ [X̃(ω)]_{α}^{−} and ω ↦ [X̃(ω)]_{α}^{+} are integrable for all α ∈ [0, 1]. Let X̃ be an integrably bounded fuzzy random variable. The expectation E(X̃) of the fuzzy random variable X̃ is defined by a fuzzy number ([5, Lemma 3]):

E(X̃)(x) := sup_{α∈[0,1]} min {α, 1_{[E(X̃)]_{α}}(x)} for x ∈ ℝ,  (2.3)

where 1_{D} is the classical indicator function of a set D and we put closed intervals

[E(X̃)]_{α} := [∫_{Ω}[X̃(ω)]_{α}^{−} dP (ω), ∫_{Ω}[X̃(ω)]_{α}^{+} dP (ω)] , α ∈ [0, 1].

3. Fuzzy stopping in a multi-objective model

Let k be a positive integer. In this section, we formulate a multi-objective optimal ‘fuzzy’ stopping problem for k fuzzy stochastic systems. Let {1, 2, ⋯, k} denote the set of k objects which are described by fuzzy stochastic systems with the time space ℕ := {0, 1, 2, ⋯}. For an object i = 1, 2, ⋯, k, let {X̃_{n}^{i}}_{n=0}^{∞} be a sequence of fuzzy random variables such that, for n = 0, 1, 2, ⋯,

E( max_{1≤i≤k} sup_{n≥0}[X̃_{n}^{i}(·)]_{0}^{+}]) < ∞ and E( min_{1≤i≤k}[X̃_{n}^{i}(·)]_{0}^{−}]) > −∞

where the interval [[X̃_{n}^{i}(ω)]_{0}^{−}, [X̃_{n}^{i}(ω)]_{0}^{+}] is the 0-cut of the fuzzy number X̃_{n}^{i}(ω). Let M_{n}, n = 0, 1, 2, ⋯ denotes the smallest σ-field on Ω generated by all random variables [X̃_{n}^{i}(ω)]_{α}^{−} and [X̃_{n}^{i}(ω)]_{α}^{+} (i = 1, 2, ⋯, k; m = 0, 1, 2, ⋯, n; α ∈ [0, 1]), and M_{∞} denotes the smallest σ-field containing ⋃_{n=0}^{∞} M_{n}. Then we call ({X̃_{n}^{i}}_{n=0}^{∞}, {M_{n}}_{n=0}^{∞}) the fuzzy stochastic system for an object i. A map τ : Ω ↦ ℕ ∪ {∞} is called a stopping time if it satisfies

{ω | τ(ω) = n} ∈ M_{n} for all n = 0, 1, 2, · · · .  (3.1)

Then we have the following lemma which is trivial from the definitions.

Lemma 3.1. Let i = 1, 2, ⋯, k be an object and let τ be a finite stopping time. We define

X̃_{τ}^{i} (ω) := X̃_{n}^{i}(ω), ω ∈ {ω | τ(ω) = n} for n = 0, 1, 2, · · · .  (3.2)

Then, X̃_{τ}^{i} is a fuzzy random variable.

Now, for an object i, we consider the estimation of the fuzzy stochastic system stopped at a finite stopping time τ, by the evaluation of the fuzzy random variable X̃_{τ}^{i}. Let a map g : I ↦ ℝ satisfy the following three conditions (L.i) – (L.iii):

(L.i) g([a, b] + [c, d]) = g([a, b]) + g([c, d]) for [a, b], [c, d] ∈ I.

(L.ii) g(λ[a, b]) = λg([a, b]) for [a, b] ∈ I, λ ≥ 0.

(L.iii) a ≤ g([a, b]) ≤ b for [a, b] ∈ I.

This function is called a ‘linear ranking function’ which is used for the evaluation of fuzzy numbers (Fortemps and Roubens [4]). In the conditions, (L.i) and (L.ii) means linearity and (L.iii) means the regularity about the estimation of α-cuts of fuzzy numbers. Then, g preserves the order of intervals corresponding to the ‘fuzzy max order’

g([a, b]) ≤ g([c, d])

for [a, b], [c, d] ∈ I such that [a, b] ⪯ [c, d], which means that a ≤ c and b ≤ d. Then, the following lemma can be checked easily.

Lemma 3.2. For a map g : I ↦ ℝ, the following statements (i) – (iii) are equivalent:

(i) g is a linear ranking function.

(ii) g satisfies

g(λ_{1}[0, 1] + λ_{2}) = λ_{1}g([0, 1]) + λ_{2} for λ_{1} ≥ 0, λ_{2} ∈ ℝ.

(iii) g satisfies

g([a, b]) = a(1 − k) + bk for [a, b] ∈ I,

where k := g([0, 1]) ∈ [0, 1].

From Lemma 3.2(iii), there exists p, q (p, q ≥ 0, p + q = 1) such that

g([a, b]) = pa + qb  (3.3)

for an interval [a, b] ∈ I. Also we note that if we define

g_{(p,q)}(ũ) := ∫_{0}^{1}(pũ_{α}^{−} + qũ_{α}^{−}) dα,  (3.4)

then ũ ⪯ ṽ in the fuzzy max order implies g_{(p,q)}(ũ) ≤ g_{(p,q)}(ṽ). However the reverse does not hold in general. From (3.2), for ω ∈ Ω, the α-cut of the fuzzy number X̃_{τ}^{i} (ω) must be a closed interval [X̃_{τ}^{i} (ω)]_{α}. Therefore, from the definition (2.3), the expectation is given by the closed interval

E([X̃_{τ}^{i} (·)]_{α}).  (3.5)

Using the above linear ranking function g, we put

g(E([X̃_{τ}^{i} (·)]_{α})).  (3.6)

Therefore, the evaluation of the fuzzy random variable X̃_{τ}^{i} is represented by the following integral:

∫_{0}^{1} g(E([X̃_{τ}^{i} (·)]_{α})) dα.  (3.7)

Lemma 3.3. For an object i = 1, 2, ⋯, k and a finite stopping time τ, it holds that

∫_{0}^{1} g(E([X̃_{τ}^{i} (·)]_{α})) dα = ∫_{0}^{1} E(g([X̃_{τ}^{i} (·)]_{α})) dα = E (∫_{0}^{1} g([X̃_{τ}^{i} (·)]_{α}) dα) .

(3.8)

Proof. The properties (3.3) and (3.4) of g imply immediately, for each α, g(E([X̃_{τ}^{i} (·)]_{α})) = E(g([X̃_{τ}^{i} (·)]_{α})). Therefore

∫_{0}^{1} g(E([X̃_{τ}^{i} (·)]_{α})) dα = ∫_{0}^{1} E(g([X̃_{τ}^{i} (·)]_{α})) dα.

Also, by Fubini’s theorem, we have

∫_{0}^{1} E(g([X̃_{τ}^{i} (·)]_{α})) dα = E (∫_{0}^{1} g([X̃_{τ}^{i} (·)]_{α}) dα) .

These complete the proof of this lemma. □

In the following definition, we modify fuzzy stopping times introduced by Kurano et al. [6] in order to apply them to fuzzy random variables.

Definition 3.1. A map τ̃ : ℕ × Ω ↦ [0, 1] is called a fuzzy stopping time if it satisfies the following (i) – (iii):

(i) For each n = 0, 1, 2, ⋯, the map ω ↦ τ̃(n, ω) is M_{n}-measurable.

(ii) For almost all ω ∈ Ω, the map n ↦ τ̃(n, ω) is non-increasing.

(iii) For almost all ω ∈ Ω, there exists an integer m such that τ̃(n, ω) = 0 for all n ≥ m.

Regarding the grade of membership of fuzzy stopping times, ‘τ̃(n, ω) = 0’ means ‘to stop at time n’ and ‘τ̃(n, ω) = 1’ means ‘to continue at time n’ respectively. And the intermediate value ‘0 < τ̃(n, ω) < 1’ is a notion of ‘fuzzy stopping’. It is easy to check the following lemma regarding construction of fuzzy stopping times ([6]).

Lemma 3.4.

(i) Let τ̃ be a fuzzy stopping time. Define a map τ̃_{α} : Ω ↦ ℕ by

τ̃_{α}(ω) := inf{n | τ̃(n, ω) < α}, ω ∈ Ω for α ∈ (0, 1],  (3.9)

where the infimum of the empty set is understood to be +∞. Then, we have:

(a) {ω | τ̃_{α}(ω) ≤ n} ∈ M_{n} for n = 0, 1, 2, ⋯;

(b) τ̃_{α}(ω) ≤ τ̃_{α′}(ω) a.a. ω ∈ Ω if α ≥ α′;

(c) lim_{α′↑α} τ̃_{α′}(ω) = τ̃_{α}(ω) a.a. ω ∈ Ω if α > 0;

(d) τ̃_{0}(ω) := lim_{α↓0} τ̃_{α}(ω) < ∞ a.a. ω ∈ Ω.

(ii) Let {τ̃_{α}}_{α∈[0,1]} be maps τ̃_{α} : Ω ↦ ℕ satisfying the above (a) (b) and (d). Define a map τ̃ : ℕ × Ω ↦ [0, 1] by

τ̃(n, ω) := sup_{α∈[0,1]} min{α, 1_{{ω|τ̃_{α}(ω)>n}}(ω)} for n = 0, 1, 2, ⋯ and ω ∈ Ω.

(3.10)

Then τ̃ is a fuzzy stopping time.

Fuzzy stopping times are always finite from Definition 3.1(iii). Now, by using Lemma 3.4 and the linear ranking function g, we consider the estimation of the fuzzy stochastic system stopped at a ‘fuzzy’ stopping time τ̃ regarding the i-th object. Let i = 1, 2, ⋯, k be an object and let τ̃ be a fuzzy stopping time. stopping time τ̃. From Lemma 3.1, we have [X̃i ; τ̃_{α}(ω)]_{α} := [X̃_{n}^{i}(ω)]_{α} for ω ∈ {ω | τ̃_{α}(ω) = n}, where τ̃_{α}(ω) are ‘classical’ stopping times given by (3.9). By Lemma 3.3, we define a random variable

Gi ; τ̃(ω) := ∫_{0}^{1} g([X̃i ; τ̃_{α}(ω)]_{α}) dα, ω ∈ Ω.  (3.11)

Note that (3.11) is well-defined since the function α ↦ g([X̃i ; τ̃_{α}(ω)]_{α}) is leftcontinuous on (0, 1]. Therefore the expectation E(Gi ; τ̃) is the evaluation (3.7) of the fuzzy random variable X̃τ̃. By Fubini’s theorem, we have

E(Gi ; τ̃) := E (∫_{0}^{1} g([X̃i ; τ̃_{α}(·)]_{α}) dα) = ∫_{0}^{1} E(g([X̃i ; τ̃_{α}(·)]_{α})) dα  (3.12)

for fuzzy stopping times τ̃. Then, Pareto optimal solutions for the multiobjective stopping model are characterized as follows.

Definition 3.2. A fuzzy stopping time τ̃^{∗} is called Pareto optimal if there exists no fuzzy stopping time τ̃ such that

E(Gi ; τ̃) ≥ E(G_{τ̃^{∗}}^{i}) for all objects i = 1, 2, ⋯, k

and

E(Gi ; τ̃) > E(G_{τ̃^{∗}}^{i}) for some object i = 1, 2, ⋯, k.

4. Pareto optimal fuzzy stopping times

In this section, we give Pareto optimal solutions for the problem in Section 3. We introduce the following λ-optimal stopping times in order to obtain Pareto optimal stopping times. Real numbers {λ^{i}}_{i=1}^{k} are called weights of objects if they satisfy

∑_{i=1}^{k} λ^{i} = 1 and λ^{i} ≥ 0 (i = 1, 2, ⋯, k).  (4.1)

For a set of weights λ := {λ^{i}}_{i=1}^{k}, we define a fuzzy stochastic system {X̃_{n}^{λ}}_{n=0}^{∞}, which is {M_{n}}_{n=0}^{∞}-adapted, by

X̃_{n}^{λ}(ω)(x) := sup_{α∈[0,1]} min{α, 1_{[X̃_{n}^{λ}(ω)]_{α}}(x)}, ω ∈ Ω, x ∈ ℝ,

where the α-cuts [X̃_{n}^{λ}(ω)]_{α} are closed intervals given by

[X̃_{n}^{λ}(ω)]_{α} = ∑_{i=1}^{k} λ^{i}[X̃_{n}^{i}(ω)]_{α}

= ∑_{i=1}^{k} λ^{i} [[X̃_{n}^{i}(ω)]_{α}^{−}, [X̃_{n}^{i}(ω)]_{α}^{+}]

= [∑_{i=1}^{k} λ^{i}[X̃_{n}^{i}(ω)]_{α}^{−}, ∑_{i=1}^{k} λ^{i}[X̃_{n}^{i}(ω)]_{α}^{+}] , ω ∈ Ω.

For fuzzy stopping times τ̃, in the same way as (3.11) we define a random variable

Gλ ; τ̃(ω) := ∫_{0}^{1} g([X̃λ ; τ̃_{α}(ω)]_{α}) dα for ω ∈ Ω.

Similarly to the proof of Lemma 3.3, we can easily check that its expectation is reduced to the weighted sum of the expectations for objects:

E(Gλ ; τ̃) = ∑_{i=1}^{k} λ^{i}E(Gi ; τ̃).  (4.2)

Now we give the definition of λ-optimal stopping times as follows.

Definition 4.1. Let λ := {λ^{i}}_{i=1}^{k} be a set of weights for objects. Then a fuzzy stopping time τ̃^{∗} is called λ-optimal if

E(G_{τ̃^{∗}}^{λ}) ≥ E(Gλ ; τ̃)

for all fuzzy stopping times τ̃.

Theorem 4.1. Let λ := {λ^{i}}_{i=1}^{k} be a set of weights for objects such that

∑_{i=1}^{k} λ^{i} = 1 and λ^{i} > 0 (i = 1, 2, ⋯, k).  (4.3)

Then a λ-optimal fuzzy stopping time τ̃^{∗} is Pareto optimal.

Proof. Let τ̃^{∗} be a finite λ-optimal fuzzy stopping time. If τ̃^{∗} is not Pareto optimal, then there exists a fuzzy stopping time τ̃ such that

E(Gi ; τ̃) ≥ E(G_{τ̃^{∗}}^{i}) for all objects i = 1, 2, ⋯, k

and

E(Gi ; τ̃) > E(G_{τ̃^{∗}}^{i}) for some object i = 1, 2, ⋯, k.

Then from (4.2) we have

E(Gλ ; τ̃) = ∑_{i=1}^{k} λ^{i}E(Gi ; τ̃) > ∑_{i=1}^{k} λ^{i}E(G_{τ̃^{∗}}^{i}) = E(G_{τ̃^{∗}}^{λ}).

This contradicts the λ-optimality of τ̃^{∗}, and so we obtain this theorem. □

Finally, in order to construct λ-optimal fuzzy stopping times, we introduce the following (λ, α)-optimal fuzzy stopping times.

Definition 4.2. Let λ := {λ^{i}}_{i=1}^{k} be a set of weights for objects and let α ∈ [0, 1]. A fuzzy stopping time τ̃^{∗} is called (λ, α)-optimal if

E(g([X̃_{τ̃_{α}^{∗}}^{λ}(·)]_{α})) ≥ E(g([X̃λ ; τ̃_{α}(·)]_{α}))

for all fuzzy stopping times τ̃.

In order to characterize (λ, α)-optimal stopping times, we let real random variables

γ_{n,α}^{λ} := ess sup_{τ: stopping times, τ≥n} E(g([X̃_{τ}^{λ}(·)]_{α})|M_{n}) for n = 0, 1, 2, ⋯, (4.4)

where the definition of the essential supremum is referred to [2, Chapter 1-6]. Define a stopping time σ_{α}^{λ} : Ω ↦ ℕ by

σ_{α}^{λ}(ω) := inf {n | g([X̃_{n}^{λ}(ω)]_{α}) = γ_{n,α}^{λ}(ω)}  (4.5)

for ω ∈ Ω and α ∈ [0, 1], where the infimum of the empty set is understood to be +∞. Then the following lemma can be checked easily by Chow et al. [2, Theorem 4.1].

Lemma 4.1. Let λ := {λ^{i}}_{i=1}^{k} be a set of weights for objects. Suppose

P (σ_{α}^{λ} < ∞) = 1 for all α ∈ [0, 1].  (4.6)

Then, for α ∈ [0, 1], the following (i) and (ii) hold:

(i) γ_{n,α}^{λ}(ω) = max{g([X̃_{n}^{λ}(ω)]_{α}), E(γ_{n+1,α}^{λ} | M_{n})(ω)} a.a. ω ∈ Ω for n = 0, 1, 2, ⋯;

(ii) σ_{α}^{λ} is (λ, α)-optimal and E(γ_{0,α}^{λ}) = E(g([X̃_{σ_{α}^{λ}}^{λ}(·)]_{α})).

In order to construct an optimal fuzzy stopping time from the (λ, α)-optimal stopping times {σ_{α}^{λ}}_{α∈[0,1]}, we need the following regularity condition.

Assumption A (Regularity). The map α ↦ σ_{α}^{λ}(ω) is non-increasing for almost all ω ∈ Ω.

Under Assumption A, we can define a map σ̃^{λ} : ℕ × Ω ↦ [0, 1] by

σ̃^{λ}(n, ω) := sup_{α∈[0,1]} min{α, 1_{{ω|σ_{α}^{λ}(ω)>n}}(ω)} for n = 0, 1, 2, ⋯ and ω ∈ Ω.

(4.7)

Put the α-cut (3.9) of σ̃^{λ}(n, ω) by σ̃_{α}^{λ}(ω). Then σ̃_{α}^{λ}(ω) and σ_{α}^{λ}(ω) may not equal only at most countable many α ∈ (0, 1], so we obtain the following result.

Theorem 4.2. Let λ := {λ^{i}}_{i=1}^{k} be a set of weights for objects satisfying (4.3). Suppose (4.6) and Assumption A hold. Then σ̃^{λ} is a λ-optimal fuzzy stopping time and it is also Pareto optimal.

Proof. From Assumption A and Lemma 3.4(ii), σ̃^{λ} is a fuzzy stopping time and we obtain

E(Gλ ; τ̃) ≤ ∫_{0}^{1} sup_{τ} E (g([X̃_{τ}^{λ}(·)]_{α}) ) dα

for all fuzzy stopping times τ̃ by Lemma 4.1. And also the last equals

∫_{0}^{1} E(γ_{0,α}^{λ}) dα = ∫_{0}^{1} E(g([X̃_{σ_{α}^{λ}}^{λ}(·)]_{α})) dα.

Since σ̃_{α}^{λ}(ω) ≠ σ_{α}^{λ}(ω) holds only at most countable α ∈ (0, 1], we have

E (∫_{0}^{1} g([X̃_{σ_{α}^{λ}}^{λ}(·)]_{α}) dα) = E (∫_{0}^{1} g([X̃_{σ̃_{α}^{λ}}^{λ}(·)]_{α}) dα) .

Thus, by Fubini’s theorem, we can show

E(Gλ ; τ̃) ≤ ∫_{0}^{1} E(g([X̃_{σ_{α}^{λ}}^{λ}(·)]_{α})) dα
= E (∫_{0}^{1} g([X̃_{σ_{α}^{λ}}^{λ}(·)]_{α}) dα)
= E (∫_{0}^{1} g([X̃_{σ̃_{α}^{λ}}^{λ}(·)]_{α}) dα)
= ∫_{0}^{1} E(g([X̃_{σ̃_{α}^{λ}}^{λ}(·)]_{α})) dα
= E(G_{σ̃^{λ}}^{λ}).

Therefore σ̃^{λ} is λ-optimal. We also obtain Pareto optimality of σ̃^{λ} from Theorem 4.1. □

Acknowledgments

The author would like to thank the referees for valuable comments and suggestions.

References

[1] J.P.Aubin, Mathematical Methods of Game and Economic Theory (North-Holland, Amsterdam, 1979).

[2] Y.S.Chow, H.Robbins and D.Siegmund, The Theory of Optimal Stopping: Great Expectations (Houghton Mifflin Company, New York, 1971).

[3] J.C.Cox, S.A.Ross and M.Rubinstein, Option pricing: A simplified approach, J. Financial Economics 7 (1979) 229-263.

[4] P.Fortemps and M.Roubens, Ranking and defuzzification methods based on area compensation, Fuzzy Sets and Systems 82 (1996) 319-330.

[5] M.Kurano, M.Yasuda, J.Nakagami and Y.Yoshida, A limit theorem in some dynamic fuzzy systems, Fuzzy Sets and Systems 51 (1992) 83-88.

[6] M.Kurano, M.Yasuda, J.Nakagami and Y.Yoshida, An approach to stopping problems of a dynamic fuzzy system, preprint.

[7] Y.Ohtsubo, Multi-objective stopping problem for a monotone case, Mem. Fac. Sci. Kochi Univ. Ser. A 18 (1997) 99-104.

[8] M.L.Puri and D.A.Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986) 409-422.

[9] M.L.Puri and D.A.Ralescu, Convergence theorem for fuzzy martingales, J. Math. Anal. Appl. 160 (1991) 107-122.

[10] A.N.Shiryayev, Optimal Stopping Rules (Springer, New York, 1979).

[11] G.Wang and Y.Zhang, The theory of fuzzy stochastic processes, Fuzzy Sets and Systems 51 (1992) 161-178.

[12] Y.Yoshida, M.Yasuda, J.Nakagami and M.Kurano, A monotone fuzzy stopping time for dynamic fuzzy systems, Bull. Infor. Cyber. Res. Ass. Stat. Sci., Kyushu University 31 (1999) 91-99.

[13] Y.Yoshida, M.Yasuda, J.Nakagami and M.Kurano, Optimal stopping problems in a stochastic and fuzzy system, J. Math. Analy. and Appl. 246 (2000) 135-149.

[14] Y.Yoshida, Markov chains with a transition possibility measure and fuzzy dynamic programming, Fuzzy Sets and Systems 66 (1994) 39-57.

[15] Y.Yoshida, An optimal stopping problem in dynamic fuzzy systems with fuzzy rewards, Computers Math. Appl. 32 (1996) 17-28.

[16] Y.Yoshida, Duality in dynamic fuzzy systems, Fuzzy Sets and Systems 95 (1998) 53-65.

[17] L.A.Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.