1.2
Markov
times
measurable modification5
([72],
chap.
4,
p.
42).
Each real adapted process
with right (left) continuous trajectories is a progressively measurable process
([72],
chap.
4,
p.
43).
1.2
Markov
times
In the present section we shall define and discuss the properties of Markov
times, which play a decisive role in the theory of optimal stopping rules. The
discussion will deal only with the case of continuous time. The definitions
and results can be carried almost automatically to the case of discrete
time-
in which case as a rule they become simpler.
Let
(R,
.P)
be a measure space, let T
=
[0,
a),
and let
F
=
{9,),
t
E
T,
be a nondecreasing sequence of sub-o-algebras, i.e.,
4,
G
4,
G
9
for
s
l
t.
Definition
1.
The random variable
T
=
z(w) with values in
T
=
[0,
co]
is
said to be a Markov time with respect to the system
F
=
{F,},
t
E
T,~
if
for each t
E
T
{r.,
:
z(~c,)
l
t)
E
.F,.
Markov times can be interpreted as random variables independent of the
"future."
Definition
2.
The Markov time z
=
z(m) defined in a probability space
(Q,
.F,
P) is said to be a stopping time or ajnite Markov time if
P{z(w)
<
00)
=
1.
With each Markov time z
=
z(o) we may associate the aggregate
9,
of the sets
A
E
.F
for which
A
n
{o
:
z(o)
I
t}
E
,F,
for all
t
E
T.
It is easy
to verify that
9,
is a o-algebra.
An obvious interpretation of the o-algebra
.Fr
is the following.
By
9,
we
shall understand the totality of events related to some physical process
and observed before time t. (For example, let
9,
=
o{co:
t,,
s
I
t) be a
o-algebra generated by values
t,,
s
I
t,
of an observable process
X
=
{t,),
t
E
T). Then
9,
is the totality of events to be observed over the random
time
z.
The process
2
=
{[,(w)),
t
E
T,
is said to be a modificat~on of the process
X
=
{(,(w)),
t
E
T.
if, for each
t
E
T,
P(o: [,(w)
#
(,(o))
=
0.
'
The words "with respect to the system
F
=
{F,),
t
E
T"
will be omitted when ambiguity
is
impossible.
剩余227页未读，继续阅读
merz9b
- 粉丝: 1
- 资源: 4
我的内容管理
展开
最新资源
- 新型矿用本安直流稳压电源设计：双重保护电路
- 煤矿掘进工作面安全因素研究：结构方程模型
- 利用同位素位移探测原子内部新型力
- 钻锚机钻臂动力学仿真分析与优化
- 钻孔成像技术在巷道松动圈检测与支护设计中的应用
- 极化与非极化ep碰撞中J/ψ的Sivers与cos2φ效应：理论分析与COMPASS验证
- 新疆矿区1200m深孔钻探关键技术与实践
- 建筑行业事故预防：综合动态事故致因理论的应用
- 北斗卫星监测系统在电网塔形实时监控中的应用
- 煤层气羽状水平井数值模拟：交替隐式算法的应用
- 开放字符串T对偶与双空间坐标变换
- 煤矿瓦斯抽采半径测定新方法——瓦斯储量法
- 大倾角大采高工作面设备稳定与安全控制关键技术
- 超标违规背景下的热波动影响分析
- 中国煤矿选煤设计进展与挑战：历史、现状与未来发展
- 反演技术与RBF神经网络在移动机器人控制中的应用
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
