数理解析研究所講究録 1207 巻 2001 年 157-164

157
On a Fuzzy Extension of Stopping Times
北九州市立大学経済学部
千葉大学理学部
千葉大学理学部

吉田祐治

安田正實
中神潤一
蔵野正美

千葉大学教育学部(Yuji YOSHIDA)(Masami YASUDA)(Jun-ichi NAKAGAMI)(Masami KURANO)

0. Introduction
We have studied fuzzy stopping stopping problems in dynamic fuzzy systems and fuzzy
stochastic systems in our papers ([7, 10, 11]). However, in our papers, the problems aredefined by -cuts and the formulations are not given in direct forms. In this talk, we
discuss the definition of fuzzy stopping times and formulation of anew fuzzy stopping
model from the viewpoint of Zadeh’s extension principle.
$\alpha$1. Fuzzy random variables
Fuzzy random variables were first studied by Puri and Ralescu [8] as afuzzy-valued
extension of real random variables, and they have been studied by many authors. In
the rest of this section, we give some mathematical notations regarding fuzzy randomvariables.
Let
be aprobability space, whereis a -field andis anon-atomic
probability measure.
denotes the set of all real numbers, anddenotes the Borel-field of R. Let
be the set of all non-empty bounded closed intervals, and let be
the Hausdorff metric on. Afuzzy number is denoted by its membership function:
which is normal, upper-semicontinuous, fuzzy convex and has acompact
support (Kurano et al. [6]). Refer to Zadeh [12] regarding fuzzy set theory. 7% denotes theset of all fuzzy numbers. In this paper, we identify fuzzy numbers with its corresponding
membership functions. The -cut of afuzzy number
is given by$(\Omega, \mathcal{M}, P)$$P$$\mathcal{M}$$\mathrm{c}\mathrm{r}$$B$$\mathbb{R}$$\mathrm{C}(\mathbb{R})$$\mathrm{c}\mathrm{r}$$\delta$$\mathrm{C}(\mathbb{R})$$\tilde{a}$$\mathbb{R}\vdasharrow[0,1]$$\alpha$$\tilde{a}(\in \mathcal{R})$$\tilde{a}_{\alpha}:=\{x\in \mathbb{R}|\tilde{a}(x)\underline{>}\alpha\}(\alpha\in(0,1])$and$\tilde{a}_{0}:=\mathrm{c}1\{x\in \mathbb{R}|\mathrm{a}(\mathrm{x})>0\}$,where cl denotes the closure of an interval. In this paper, we write the closed intervals by$\tilde{a}_{\alpha}:=[\tilde{a}_{\alpha}^{-},\tilde{a}_{\alpha}^{+}]$

for $\alpha\in[0,1]$ .

Hence we introduce apartial order , so called the fuzzy $\max$ order, on fuzzy numbers:Let
be fuzzy numbers. Then$[succeq]$$\mathcal{R}([2])$$\tilde{a},\tilde{b}\in \mathcal{R}$$\tilde{a}[succeq]\tilde{b}$means that

and$\tilde{a}_{\alpha}^{-}\geq\tilde{b}_{\alpha}^{-}$

Afuzzy-number-valued mapping$\tilde{X}$:$\Omega$}$arrow \mathcal{R}$$\tilde{a}_{\alpha}^{+}\geq\tilde{b}_{\alpha}^{+}$

for all $\alpha\in[0,1]$ .

is called afuzzy random variable ([8]) if$\{(\omega, x)\in\Omega\cross \mathbb{R}|\tilde{X}(\omega)(x)\geq\alpha\}\in \mathcal{M}\cross B$

for all $\alpha\in[0,1]$ .

(1.1

158
where $X(r.z)$ is afuzzy number in and its a-cut is written as
for w @ . Therefore, the condition (1.1) is also written as{ xC$\ovalbox{\tt\small REJECT} \mathrm{q}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}.((.>)\ovalbox{\tt\small REJECT}$$[X_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}(\mathrm{J}), \ovalbox{\tt\small REJECT} \mathrm{y}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}(\mathrm{J})]\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT} El$$\mathrm{R}|^{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT} \mathrm{X}^{\ovalbox{\tt\small REJECT}}(\mathrm{c}\cdot\ovalbox{\tt\small REJECT})(x)\ovalbox{\tt\small REJECT} a\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$

for all $\alpha\in[0,1]$ .

$\{(\omega, x)\in\Omega\cross \mathbb{R}|x\in\tilde{X}_{\alpha}(\omega)\}\in \mathcal{M}\cross B$(1.2)

The following lemma gives asimple characterization of fuzzy random variables.
Lemma 1.1 (Wang and Zhang [9, Theorems 2.1 and 2.2]). For amapping
the following (i) and (ii) are equivalent:(i)$\tilde{X}$$\tilde{X}$:$\Omega\vdash+\mathcal{R}$,is afuzzy random variable.

(ii) The mappings$\omega$$\succ*\tilde{X}_{\alpha}^{-}(\omega)$

and$\vdash+\tilde{X}_{\alpha}^{+}(\omega)$$\omega$are measurable for all $\alpha\in[0,1]$ .

From [6, Lemma 3] and Lemma 1.1, we obtain the following lemma regarding the
a-cuts
.
$X\sim\alpha(\omega)=[\tilde{X}_{\alpha}^{-}(\omega),\tilde{X}_{\alpha}^{+}(\omega)]=\{x\in \mathbb{R}|\tilde{X}(\omega)(x)\geq\alpha\}$Lemma 1.2.
(i) Let$\omega$

be afuzzy random variable. Then the, have the following properties $(a)-(c)$ :$\tilde{X}$$\in\Omega$,(a)$\bigcap_{\alpha’<\alpha}\tilde{X}_{\alpha’}(\omega)=\tilde{X}_{\alpha}(\omega)$(b)$\mathrm{c}1(\bigcup_{\alpha>0}\tilde{X}_{\alpha}(\omega))=\tilde{X}_{0}(\omega)$(c) The mappings$\omega$$|arrow\tilde{X}_{\alpha}^{-}(\omega)$$\in\Omega$$\omega$

and

-cuts

$X\sim\alpha(\omega)=[\tilde{X}_{\alpha}^{-}(\omega),\tilde{X}_{\alpha}^{+}(\omega)]$$\omega$.

$\vdash+\tilde{X}_{\alpha}^{+}(\omega)$

are measurable for $\alpha\in[0,1]$ .

(ii) Suppose that afamily of interval-valued mappings$(\alpha\in[0,1])$ satisfies the following conditions $(a)-(c)$ :$X_{\alpha}=[X_{\alpha}^{-}, X_{\alpha}^{+}]$,(a)$\bigcap_{\alpha’<\alpha}X_{\alpha’}(\omega)=X_{\alpha}(\omega)$(b)$\mathrm{c}1(\bigcup_{\alpha>0}X_{\alpha}(\omega))=X_{0}(\omega)$(c) The mappings$\omega$$\omega$,$\vdash*X_{\alpha}^{-}(\omega)$$\in\Omega$$\omega$:$\Omega-+\mathrm{C}(\mathrm{R})$if $\alpha>0$ .

$\in\Omega$and,; $f\alpha>0$ .

$\omega\in\Omega$,$\alpha$.

$\omega|arrow X_{\alpha}^{+}(\omega)$

are measurable for $\alpha\in[0,1]$ .

Then, amembership function$\tilde{X}(\omega)(x):=\sup_{\alpha\in[0,1]}\min\{\alpha, 1_{X_{\alpha}(\omega)}(x)\}$gives afuzzy random variable$\tilde{X}$and,$\tilde{X}_{\alpha}(\omega)=X_{\alpha}(\omega)$$\omega\in\Omega$for$\omega$,$x\in \mathbb{R}$$\in\Omega$

,

(1.3)

and $\alpha\in[0,1]$ .

Next we need to introduce expectations of fuzzy random variables in order to describe
an optimal stopping model in the next section. Afuzzy random variable
is called
integrably bounded if bothandare integrable for all $\alpha\in[0,1]$ .
$\tilde{X}$$\omega$$\mapsto+\tilde{X}_{\alpha}^{-}(\omega)$$\omega$$\vdash+\tilde{X}_{\alpha}^{+}(\omega)$

159
Let X be an integrably bounded fuzzy random variable. By classical expectation, we give
closed intervals,$E( \tilde{X})_{\alpha}:=[\int_{\Omega}\tilde{X}_{\alpha}^{-}(\omega)\mathrm{d}P(\omega),$ $\int_{\Omega}\tilde{X}_{\alpha}^{+}(\omega)\mathrm{d}P(\omega)]$$\alpha\in[0,1]$

.

(1.4)

Since the mappingis left-continuous by the monotone convergence theorem,from [6, Lemma 3] the expectationof the fuzzy random variable is defined by afuzzy number
,
.
(1.5)
$\alpha\vdasharrow E(\tilde{X})_{\alpha}$$E(\tilde{X})$$\tilde{X}$$E( \tilde{X})(x):=\sup\min\{\alpha, 1_{E(\overline{X})_{\alpha}}(x)\}$$x\in \mathbb{R}$$\alpha\in[0,1]$

2. Afuzzy stopping model
In this section, we discuss anew fuzzy stopping model for aprocess of fuzzy randomvariables. Let afuzzy stochastic process
be aprocess of integrably boundedfuzzy random variables such that the mapping
is continuous on
almostsurely and, whereis the right-end of the 0-cut of the fuzzy
number.
denotes afamily of the smallest -fields on generated by all
random variables
andsuch thatis right continuous,anddenotes the smallest -field containing. Amapping :issaid to be astopping time if$\{\tilde{X}_{t}\}_{t\geq 0}$$[0, \infty)$$t\vdasharrow\tilde{X}_{t}$$E( \sup_{t\geq 0}\tilde{X}_{t,0}^{+})<\infty$$\tilde{X}_{t}(\omega)$$\tilde{X}_{t,0}^{+}(\omega)$$\{\mathcal{M}_{t}\}_{t\geq 0}$$\Omega$$\mathrm{c}\mathrm{r}$$\tilde{X}_{s,\alpha}^{-}$$\tilde{X}_{s,\alpha}^{+}(s\leq t;\alpha\in[0,1])$$t\vdash+\mathcal{M}_{t}$$\mathcal{M}_{\infty}$$\bigcup_{t\geq 0}\mathcal{M}_{t}$$\mathrm{c}\mathrm{r}$$\{\omega|\tau(\omega)\leq t\}\in \mathcal{M}_{t}$$\tau$$\Omega\mapsto*[0, \infty]$

for all $t\geq 0$ .

(2.1)

Now we introduce new fuzzy stopping times by mappings : $[0, \infty)\cross\Omega\vdash+[0,1]$ such thatare fuzzy numbers for each C3 . The -cut of the fuzzy number
is givenby$\tilde{\tau}$$\tilde{\tau}(\cdot\backslash ,\omega)$$\Omega$$\omega$$\alpha$$\tilde{\tau}_{\alpha}(\omega):=\{t\geq 0|\tilde{\tau}(t,\omega)\geq\alpha\}(\alpha\in(0,1])$and$\tilde{\tau}(\cdot,\omega)$To’(w)$:=\mathrm{c}1\{t\geq 0|\tilde{\tau}(t,\omega)>0\}$, (2.2)where cl denotes the closure of an interval. Then we write the -cut(2.2) as aboundedclosed interval
for $\alpha\in[0,1]$ .
(2.3)$\alpha$$\tilde{\tau}_{\alpha}(\omega):=[\tilde{\tau}_{\alpha}^{-}(\omega),\tilde{\tau}_{\alpha}^{+}(\omega)]$

Definition 1. (A new fuzzy stopping time). Amapping :afuzzy stopping time if it satisfies the following (i) and (ii):$\tilde{\tau}$

(i)

(ii)

$\tilde{\tau}(\cdot, \omega)$$\tilde{\tau}_{\alpha}^{-}$

andis afuzzy number for each$\tilde{\tau}_{\alpha}^{+}$$\omega\in\Omega$$[0, \infty)\cross\Omega\vdasharrow[0,1]$.

are stopping times for each $\alpha\in[0,1]$ , where$\tilde{\tau}_{\alpha}^{\pm}$is defined by (2.3).

Remark. We cannot replace Definition 1(ii) with the following condition$(\mathrm{i}\mathrm{i}’)$

The mapping$\omega\vdash\Rightarrow\tilde{\tau}(t,\omega)$is$\mathcal{M}_{t}$is called-measurable for $t\geq 0$$(\mathrm{i}\mathrm{i}’)$:160
since it holds that (ii)but the reverse does not hold from$\Rightarrow(\mathrm{i}\mathrm{i}’)$.

$\{\omega|\tilde{\tau}(t,\omega)\geq\alpha\}=\{\omega|\tilde{\tau}_{\alpha}^{-}(\omega)\leq t\}\cap\{\omega|t\leq\tilde{\tau}_{\alpha}^{+}(\omega)\}$Then the following lemma is trivial.
Lemma 2.1.
(i) Let be afuzzy stopping time. Define amapping(2.3). Tien, we have:$\tilde{\tau}$(a)$\tilde{\tau}_{\alpha}^{-}$

and$\tilde{\tau}_{\alpha}^{+}$,$\bigcap_{\alpha’<\alpha}\tilde{\tau}_{\alpha’}(\omega)=\tilde{\tau}_{\alpha}(\omega)$(c)$\mathrm{c}1(\bigcup_{\alpha>0}\tilde{\tau}_{\alpha}(\omega))=\tilde{\tau}_{0}(\omega)$$-(\mathrm{c})$:by (2.2) and$\Omega\vdash+[0, \infty)$

are stopping times for each $\alpha\in[0,1]$ ;(b)(ii) Let$\tilde{\tau}_{\alpha}$$\omega\in\Omega$,

if $\alpha>0$ ;$\in\Omega$$\omega$.

be intervai-valued mappings. Define amapping :$\{\tilde{\tau}_{\alpha}\}_{\alpha\in[0,1]}$$\tilde{\tau}_{\alpha}$$[0, \infty)\cross\Omega\vdasharrow[0,1]$$\tilde{\tau}$$\tilde{\tau}(t,\omega):=\sup\min\{\alpha$,:$\Omega\mapsto*\mathrm{C}(\mathbb{R})$satisfying the above (a)by$1_{\{\{v1\dot{t}\in\overline{\tau}_{\mathrm{Q}}(\omega)\rangle(\omega)\}}$

for $t\geq 0$ anci$\omega$$\in\Omega$

.

(2.4)

$\alpha\in[0,1]$

Then$\tilde{\tau}$

is afuzzy stopping time.

We can define the stochastic process stopped at afuzzy stopping time$\tilde{X}_{\overline{\tau}}(\omega)(x)=\sup_{t\geq 0}\min\{\tilde{X}_{t}(\omega)(x),\tilde{\tau}(t,\omega)\}$The reason is as follows. Fix$\omega$$\in\Omega$and$\tilde{X}.(\omega)(x)$$x\in \mathbb{R}$$\tilde{\tau}$

.

by
(2.5)

. By considering amapping: $[0, \infty)\vdash t[0,1]$(2.6), the fuzzy random variable (2.5) gives atype 2
is afuzzy set on the time space
1 of (2.6), and therefore (2.5) means the stochastic process stopped at afuzzy
extension
stopping time .
almost surely, by taking the a-cut
is continuous on
Since the mappingin (2.5) we obtain abounded closed interval$[0, \infty)$$\tilde{\tau}$$t\vdash+\tilde{X}_{t}$$\tilde{X}_{\overline{\tau},\alpha}(\omega)=$$\cup$$[0, \infty)$$\tilde{X}_{t,\alpha}(\omega)=[\min_{t\in\overline{\tau}_{\alpha}(\omega)}\tilde{X}_{t,\alpha}^{-}(\omega),$$t \in(\{v)\max_{\overline{\mathcal{T}}_{\mathrm{Q}}’}\tilde{X}_{t,\alpha}^{+}(\omega)]$

,

(2.2)

$t\in\tilde{\tau}_{q}(\omega)$

Fuzzy sets of type $m$ are defined inductively as follows.
(fuzzy sets of type $m$ ) are mappings:(fuzzy sets of type $m-1$ )This extension is also called Zadeh ’s support fuzzification ([2])$\mapsto*[0,1]$.

161$\mathrm{w}^{\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}X_{\ovalbox{\tt\small REJECT}},.(w)}$$-\#((\ovalbox{\tt\small REJECT} \mathrm{J})(=)$

is the -cut of$\mathrm{c}\mathrm{r}$and$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}.,.(\mathrm{u})\ovalbox{\tt\small REJECT}$$[_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}.(\mathrm{u}),\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}\mathrm{j}.(\mathrm{u})]$is the -cut of$0$.

Lemma 2.2. Let
Proof.
andsurely.

$X_{\ovalbox{\tt\small REJECT}}(\mathrm{J})(\cdot)$$\tilde{X}_{\overline{\tau},\alpha}(\omega)$$\tilde{\tau}$be afuzzy stopping time. Thenis afuzzy random variable.

$\tilde{X}_{\tilde{\tau}}$is abounded closed interval and the mappings }are measurable since
is continuous on$\omega$$\omega\vdash+\max_{t\in\overline{\tau}_{\alpha}(\omega)}\tilde{X}_{t,\alpha}^{+}(\omega)$$arrow\min_{t\in\overline{\tau}_{\alpha}(\omega)}\tilde{X}_{t,\alpha}^{-}(\omega)$$t\vdash+\tilde{X}_{t}$$[0, \infty)$

almost$\square$Now we introduce an estimation of the fuzzy random variables
A $\in[0,1]$ . Put amappingby$\tilde{X}_{\overline{\tau}}$. Fix aparameter$g:\mathrm{C}(\mathbb{R})\vdasharrow \mathbb{R}$,$g([a, b]):=\lambda a+(1-\lambda)b$and give the ranking method of a fuzzy number$\int_{0}^{1}g(\tilde{a}_{\alpha})$$\tilde{a}$$[a, b]\in \mathrm{C}(\mathbb{R})$

,

(2.8)

by (see Campos and Munoz [1])da.

We note that$\int_{0}^{1}g(E(\tilde{X}_{\tilde{\tau}})_{\alpha})\mathrm{d}\alpha=\int_{0}^{1}E(g(\tilde{X}_{\tilde{\tau},\alpha}))$for fuzzy stopping times$\tilde{\tau}$

do

(2.9)

, and in this paper we discuss the following problem.

Problem 1. Find afuzzy stopping time$\tilde{\tau}^{*}$

such that(2.10)$\int_{0}^{1}E(g(\tilde{X}_{\tilde{\tau}^{*},\alpha}))\mathrm{d}\alpha\geq\int_{0}^{1}E(g(\tilde{X}_{\overline{\tau},\alpha}))\mathrm{d}\alpha$

for all fuzzy stopping times , where Ais called apessimistic-Optimistic index for decision
makers.
$\tilde{\tau}$

Remark. If$E(\tilde{X}_{\tilde{\tau}}*)[succeq] E(\tilde{X}_{\tau}$- $)$, then (2.10) holds, where$[succeq] \mathrm{i}\mathrm{s}$the fuzzy $\max$ order on 7%.

3. An optimal fuzzy stopping time
In this section, we consider about the construction of an optimal fuzzy stopping time for
Problem 1. For afuzzy stopping time , from (2.7) and (2.8) we have$\tilde{\tau}$$g(\tilde{X}_{\tilde{\tau},\alpha}(\omega))=\lambda$$t \in[(\omega),\tilde{\tau}_{a}^{+}(\omega)]\min_{\tilde{\gamma^{-}}_{\mathrm{O}}’}\tilde{X}_{t,\alpha}^{-}(\omega)+(1-\lambda)$

Fix$\omega\in\Omega$$t \in[(\{v),\tilde{\tau}_{\alpha}^{+}((v)]\max_{\tilde{\tau}_{\mathrm{o}}^{-}’}\tilde{X}_{t,\alpha}^{+}(\omega),$, $\alpha\in[0,1]$ .
(3.1)$\alpha$$\in[0,1]$

for simplicity, and define$\mathrm{Y}_{\tau^{-}(\omega),\tau(\omega)}+(\omega):=\lambda\min_{+t\in[\tau^{-}(\omega),\tau(\omega)]}\tilde{X}_{t,\alpha}^{-}(\omega)+(1-\lambda)\max_{\tau^{-}t\in[((v),\tau(\omega)]}\tilde{X}_{t,\alpha}^{+}(\omega)+$for pairs of finite stopping times
following optimal stopping problem.

$(\tau^{-}, \tau^{+})$

satisfying$0\leq\tau^{-}\leq\tau^{+}$’$\omega\in\Omega$(3.2). Then we consider the162
Problem 2. Find apair of finite stopping times (r , r”) satisfying 0
maximizes the expected value.

r$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$r^{+}$

which$E(1\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}-,.+)$First we deal with an optimal stopping with respect tooptimal stopping with respect to$\tau^{+}$, and next we discuss an$\tau^{-}$

Problem 3. Fix aconstant time
expected value.

$s_{1}$. Find afinite stopping time$\tau_{1}$which maximizes the$E(\mathrm{Y}_{s_{1},\tau_{1}})$

Define Snell’s envelope for Problem 3by(3.3)$U_{s_{1\prime}s_{2}}:= \mathrm{e}\mathrm{s}\mathrm{s}\sup_{+\tau\geq s_{2}}E(\mathrm{Y}_{s_{1},\tau}+_{\mathrm{t}\cdot)(\cdot)1\mathcal{M}_{s_{2}})}$

for pairs of constant times$(s_{1}, s_{2})$satisfying$s_{2}\geq s_{1}\geq 0$. Define,$Q(s_{1}, \omega):=\inf\{s_{2}|s_{2}\geq s_{1}, U_{s_{1\prime}s_{2}}(\omega)=\mathrm{Y}_{s_{1\prime}s_{2}}(\omega)\}$$\omega$$\in\Omega$

,

(3.4)

where the infimum of the empty set is understood to be, is finite almost. Ifsurely, then it is an optimal stopping time for Problem 3.
Next we consider Problem 2. Define Snell’s envelope for Problem 2by$Q(s_{1}$$+\infty$$\cdot$

$)$

(3.5)

$L_{s_{1}}:= \mathrm{e}\mathrm{s}\mathrm{s}\sup_{\mathcal{T}^{-\geq s_{1}}}E(\mathrm{Y}_{\tau^{-},Q(\tau^{-}(\cdot),\cdot)}(\cdot)|\mathcal{M}_{s_{1}})$for constant times$s_{1}$. Put apair of stopping times

satisfying$(\sigma^{-}, \sigma^{+})$$\sigma^{-}(\omega):=\inf\{s_{1}|L_{s_{1}}(\omega)=E(\mathrm{Y}_{s_{1\prime}Q(s_{1\prime}\cdot)}(\cdot)|\mathcal{M}_{s_{1}})(\omega)\}$,$\omega$$0\leq\sigma^{-}\leq\sigma^{+}$$\in\Omega$

,

by
(3.6)

and
$\sigma^{+}(\omega):=Q(\sigma^{-}(\omega),\omega)$,$\omega$$\in\Omega$where the infimum of the empty set is understood to beIf
is finite almost surely, then we obtain,$+\infty$

(3.7)

.

$\sigma^{+}$$E(\mathrm{Y}_{\sigma^{-},\sigma}+)=E(\mathrm{Y}_{\sigma}-,Q(\sigma^{-}(\cdot),\cdot)(\cdot))\geq E(\mathrm{Y}_{\tau^{-},Q(\tau}-(\cdot),\cdot)(\cdot))\geq E(\mathrm{Y}_{\tau^{-},\tau}+)$for all pairs of finite stopping times
is optimal for Problem 2.

$(\tau^{-}, \tau^{+})$

satisfying$0\leq\tau^{-}\leq\tau^{+}$. Therefore(3.8)$(\sigma^{-}, \sigma^{+})$

Theorem 3.1. Let $\alpha\in[0,1]$ . Put apair of stopping timesby (3.6) and (3.7). If
is an optimal pair of stopping times for Problem
finite almost surely, then2.
$(\sigma^{-}, \sigma^{+})$$\sigma^{+}is$$(\sigma^{-}, \sigma^{+})$

Put
for $\alpha\in[0,1]$ . In order to construct an optimal fuzzy stopping time fromthe a-0ptimal stopping times, we need the following regularity condition$\sigma^{\pm}=\sigma_{\alpha}^{\pm}$$\{\sigma_{\alpha}^{\pm}\}_{\alpha\in[0,1]}$

163
Assumption A(Regularity). The mapping ex $+$is non-decreasing and left
continuous and the mapping a e
is non-increasing and left continuous for almost.
all$\mathrm{a}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}(\mathrm{u})$$\mathrm{a}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}(\mathrm{u})$$\langle\ovalbox{\tt\small REJECT} \mathrm{x}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\subset\ovalbox{\tt\small REJECT} El$

Under Assumption$\mathrm{A}$, we can define amapping$\tilde{\sigma}$:$[0, \infty)\cross\Omega\vdasharrow[0,1]$$\tilde{\sigma}(t,\omega):=\sup_{\alpha\in[0,1]}\min\{\alpha, 1+\{\omega|\sigma_{\overline{\alpha}}((v)\leq t\leq\sigma_{\alpha}(\omega)\}(\omega)\}$We also write its -cut of by$\alpha$$\tilde{\sigma}_{\alpha}(\omega)$

for $t\geq 0$ and(see (2.2)), then we haveby$\omega\in\Omega$

.

(3.9)

$\tilde{\sigma}_{\alpha}(\omega)=[\sigma_{\alpha}^{-}(\omega),\sigma_{\alpha}^{+}(\omega)]$

Theorem 3.2 (Optimal fuzzy stopping time). Suppose Assumption A holds. If$\infty)=1$ , thenis an optimal fuzzy stopping time for Problem 1.

.

$P(\sigma_{0}^{+}<$$\tilde{\sigma}$

Proof. From Assumption Aand Lemma 2.1,
and Theorem 3.1, we have$\sigma\sim$is afuzzy stopping time. By (3.1), (3.8)$E(g(\tilde{X}_{\tilde{\sigma},\alpha}))=E(\mathrm{Y}_{\sigma_{\overline{\alpha}},\sigma_{\alpha}}+)\geq E(\mathrm{Y}_{\tilde{\tau},\tilde{\tau}}+)=E(g(\tilde{X}_{\tilde{\tau},\alpha})\overline{\alpha}\alpha)$

for all fuzzy stopping times$\tilde{\tau}$

and all $\alpha\in[0,1]$ . Therefore,$\tilde{\sigma}$is optimal for Problem 1.

$\square$

The following result implies acomparison between the optimal values of the ‘classical’
stopping model and the ‘fuzzy’ stopping model (Problem 1). Then we find that the fuzzy
stopping model is more better than the classical one.
Corollary 3.1. It holds that, under the same assumptions as Theorem 3.2,$\int_{0}^{1}E(g(\tilde{X}_{\overline{\sigma},\alpha}))$$da \geq\int_{0}^{1}E(g(\tilde{X}_{\tau\alpha}.,))$

where is the optimal fuzzy stopping time and$of$ classical stopping times.
$\tilde{\sigma}$

Proof. Trivial.

$\tau^{*}is$da,(3.10)

an optimal stopping time in the class$\square$

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