[Sidahmed* 4(1): January, 2017] ISSN 2349-4506
Impact Factor: 2.785
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THE FORMULATION AND METHOD OF SOLUTION OF OPTIMAL STOPPING
PROBLEM
Entisar Alrasheed sidahmed*
*
University of Bahri- College of Applied and Industrial Sciences-department of mathematics - Sudan
University of Bahri- College of Computer Science- Sudan
DOI: 10.5281/zenodo.264067
KEYWORDS
: Stopping time, optimal stopping problem, Variational inequality.
ABSTRACT
This paper is concerned with the formulation and method of solution of the optimal stopping problem, the existence and uniqueness of the optimal topping time are also present.
INTRODUCTION
The theory of optimal stopping usually concerned with the problem that choosing a specific time to take a particular action, theses type of problems known as optimal stopping problems ,see[1]. An optimal stopping problems have well known applications in stochastic analysis, control theory and finance. The financial applications of the optimal stopping problem is used to maximize the expected reward function or minimize an expected cost over all stopping times. An example of the applications of optimal stopping problem in finance is the model of the stock price that described by geometric Brownian motion, see [2].
Throughout this paper we will consider the following stochastic setting:
The probability space
,
f
,
p
, increasing sequence of sigma algebra
F
t tT (the filtration) which satisfy the usual conditions and hypotheses where T is the time interval needed for the experiment, stochastic processt
X
which adapted to the filtration
F
t tT , the stopping time
and the optimal stopping time
,see [3]. Definition: A geometric Brownian motion is a continuous-time stochastic process , and we say that the stochastic processS
t fellow the geometric Brownian motion if it satisfy the following stochastic differential equation :)
1
(
t t t
t
S
d
t
S
d
W
S
d
where
represents the constant drift or trend of the process ,
represents the amount of random variationaround the trend and
W
t is a Wiener process or Brownian motion. If we denote byS
0
S
0
, to the initial value which isF
0
measurable, then by using Ito lemma the geometric Brownian motion (1) has general solution of the form:)
2
(
2
exp
2
0
tt
S
t
W
S
FORMULATION OF THE OPTIMAL STOPPING PROBLEM
For the mathematical formulation of the problem of this type we will consider the filtered probability space
,
f
,
F
t tT,
p
, and the standard Brownian motionB
B
t;
t
0
, suppose that the state of the system of this type is described by Geometric Brownian motion , and let g be a given Boreal function ( thereward function ) on
R
n which must satisfy the following condition:
nR
g
[Sidahmed* 4(1): January, 2017] ISSN 2349-4506
Impact Factor: 2.785
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lobal
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ournal of
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ngineering
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cience and
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esearch
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anagement
Denote byF
f
t to the natural filtration ofX
t and let
be a real constantsuch that
E
e
g
X
exist and well define for every F- stopping time
. Define the value function
*,
X
of the optimal stopping problem with reward function g and discount rate
as:
*,
sup
n
3
R x x
x
X
g
e
E
X
e
E
X
Where the supremum is taken over all stopping times
andE
x denotes the expectation with respect to the probability law of the processX
t,
t
0
.Then the optimal stopping problem is to find the value function
*,
X
as well as an optimal stopping time
for which the supremum is attained if such time exist .Definition :A measurable function
f
:
R
n
0
,
is called supermeanvalued with respect toX
t if for all stopping times
and
x
R
n.
f
(
x
)
E
x
f
X
t
4
If
f
x
is also lower semiconscious (l.s.c) , thenf
is called l.s.c super harmonic. and iff
:
R
n
0
,
is l.s.c , then by using Fatou lemma for any sequence
k of stopping times such that
k
0
a.s we get
(
)
lim
lim
5
k
k t
x k t k
x
X
f
E
X
f
E
x
f
By combining (4 ) and ( 5) we see that for such sequence
k, iff
x
is l.s.c super harmonic then,
,
6
lim
)
(
x
E
f
X
x
f
k
x
k
Denote by
A
to the characteristic operator ofX
t. Iff
C
2
R
n then byDynkin’s formula
f
x
is super harmonic w .r . tX
t iff
7
0
f
A
Definition : Let h be a real measurable function on
R
n, iff
x
is super harmonic (super mean valued ) function andf
(
x
)
h
we say thatf
x
is super harmonic (super mean valued ) majorant of h (w.r. tot
X
) . The functionh
x
such that
inf
(
)
;
n
8
f
f
x
x
R
x
h
Where the inf being taken over all super mean valued majorants of h , is called the least super mean valued
majorant of h . Suppose now there exists a function
h
such that
h
is super harmonic majorants of h and If f is any other super harmonic majorants of h thenh
f
then
h
is called the least superharmonic majorants of h.Let
g
0
and letf
x
be a supermeanvalued majorant of g. then if
is an stopping time
f
(
x
)
E
x
f
X
E
x
g
X
,f
(
x
)
sup
E
x
g
X
g
(
x
)
.
Then we have
9
,
)
(
)
(
x
g
x
x
R
ng
[Sidahmed* 4(1): January, 2017] ISSN 2349-4506
Impact Factor: 2.785
G
lobal
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ournal of
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cience and
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esearch
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anagement
,
,
0
,
10
)
(
x
E
xf
X
ss
x
R
nf
Theorem : ( Existence for optimal stopping) see [1]
Let
g
denote the optimal reward and
g
the least super harmonic majorants ofa continuous reward function
g
0
, then
11
)
(
)
(
)
g
x
g
x
i
)
ii
for
0
define the domain D as:
12
)
(
)
(
;
x
g
x
g
x
D
Suppose that the function g is bounded , then the optimal time
is the first time that the process exit from the domain .the domain .D
is close to being optimal ,in the sense that :
2
,
13
)
(
x
E
g
X
x
g
x
iii) For arbitrary continuous
g
0
Letiv)
;
(
)
(
)
14
x
g
x
g
x
D
be the continuation region
For
N
1
,
2
,
3
defineg
N
g
N
,
D
N
x
;
g
N(
x
)
g
ˆ
N(
x
)
and
N
ND . Then
NN N
N
N
D
D
D
g
N
D
D
D
1,
10
,
,
, If
N
a.s.Q
x for all N then
15
lim
)
(
x NN
E
g
X
x
g
v) In particular ,if
D
a.s .Q
x and the family
N N
X
g
isvi) uniformly integrable w.r.t .
Q
x.Let the value function
g
(the stopping function ) be
16
)
(
D
X
g
E
x
g
x And
D is the optimal stopping time.[Sidahmed* 4(1): January, 2017] ISSN 2349-4506
Impact Factor: 2.785
G
lobal
J
ournal of
E
ngineering
S
cience and
R
esearch
M
anagement
Define the domain D such that:
17
)
(
)
(
;
g
x
g
x
R
nx
D
Suppose there is exist an optimal stopping time
x
,
for theproblem
3
for all x. then
18
,
for
all
x
D
D
and
,
19
)
(
x
E
xg
X
for
all
x
R
ng
D
Hence
Dis an optimal stopping time for the problem (3), for the proof see [1]Theorem : ( Variational inequality for optimal stopping) see [1]
Suppose that we can find a function
:
V
R
such that the following conditions are hold
V
C
V
C
i
)
(
V
on
g
and
V
on
g
ii
)
(
Define the domain
D
x
V
;
(
x
)
g
(
x
)
.
and supposeY
t spend zero time on
V
a
.
s
,
i
.
e
Y
d
t
y
V
E
iii
T
t D
y
0
,
)
(
0
And suppose that
D
v
i
)
(
is a Lipschitz surface , i.e
D
is locally graph of a functionR
R
h
:
n1
such that there existk
withh
(
x
)
h
(
y
)
K
x
y
,
x
,
y
Also suppose that:
V
D
C
v
)
\
(
2 and the second order derivatives of
are locally bounded near
D
D
V
on
f
L
i
v
)
0
\
(
D
on
f
L
i
i
v
)
0
(
t
o
y
D
a
s
R
y
V
iii
v
)
D
inf
;
t
,
.
.
y
(
and)
[Sidahmed* 4(1): January, 2017] ISSN 2349-4506
Impact Factor: 2.785
G
lobal
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ournal of
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S
cience and
R
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;
20
sup
)
(
)
(
0
V
y
y
g
t
d
y
f
E
y
y
t ty T
And
21
D
Which is an optimal stopping time for this problem .
METHOD OF SOLUTION OF THE OPTIMAL STOPPING PROBLEM
We can solve the optimal stopping problem by using the variational inequality theorem which state that if we can find a function
which satisfy the conditions in the above theorem , then we can find a solution
andtime
which is an optimal stopping time for the problem.Application: Now we want give an example to illustrate the formulation of an optimal stopping problem in selling assets.
For the formulation we will consider the following assumptions
-
X
t is the price of a person’s assets at timet
on the open market which described by Geometric Brownian motion .- There is fixed sale tax or transaction cost
a
0
and discounting factor
0
- The discounted net of the sale is given by
e
t Xt a In this case the optimal stopping problem is to find the stopping time
which maximizes
X a
x s
e
E
( ) (maximizes the expected profit), if so then
will be the right time to sell the stocks, see [1].REFERENCE
1. Oksendal, Bernt K. Stochastic Differential Equations: An Introduction with Applications, Springer, (2002), ISBN 3-540-63720-6.
2. Zhijun Yang . Geometric Brownian motion Model in Financial Market.
3. Thomas G. Kurtz: Lectures on Random Analysis. University of Wisconsin- Madison , WI 53706-1388 September (2001).
