Download formulation of the optimal stopping problem

[Sidahmed* 4(1): January, 2017]
ISSN 2349-4506
Impact Factor: 2.785
Global Journal of Engineering Science and Research Management
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


 Ft t T
(the filtration) which satisfy
the usual conditions and hypotheses where T is the time interval needed for the experiment, stochastic process
X t which adapted to the filtration  Ft t T , 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
process St fellow the geometric Brownian motion if it satisfy the following stochastic differential equation :
d S t   S t d t   S t d Wt
(1)
 represents the constant drift or trend of the process ,  represents the amount of random variation
around the trend and Wt is a Wiener process or Brownian motion. If we denote by S 0  S 0 , to the initial value
where
F0  measurable, then by using Ito lemma the geometric Brownian motion (1) has general solution
which is
of
the form:

2
S t  S 0 exp    
2



 t   Wt 



(2)
FORMULATION OF THE OPTIMAL STOPPING PROBLEM
For the mathematical formulation of the problem of this type we will consider the filtered probability space
 , f , Ft t  T , p , and the standard Brownian motion B  Bt ; 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 ( the
R n which must satisfy the following condition:
b) g is continuous .
a) g   0 ,    R n
reward function ) on
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F   ft
Denote by
such that
  * , X

E e

 to the natural filtration of X t and let   be a real constant
g  X    exist and well define for every F- stopping time  . Define the value function
 of the optimal stopping problem with reward function g and discount rate 
  , X   E  e
X   sup E  e
g  X 
*

x

 
x



3
x R n
Where the supremum is taken over all stopping times  and
probability law of the process
as:

Xt , t  0 .
Then the optimal stopping problem is to find the value function
E x denotes the expectation with respect to the
  * , X
 as well as an optimal stopping time
 for which the supremum is attained if such time exist .
n
Definition :A measurable function f : R  0 ,  is called supermeanvalued with
n
respect to X t if for all stopping times  and  x R .

f ( x)  E x  f  X t  
4
f : R n  0 , 
f  x is also lower semiconscious (l.s.c) , then f is called l.s.c super harmonic. and if
is l.s.c , then by using Fatou lemma for any sequence  k of stopping times such that  k  0 a.s we get
If

 
  
5
f ( x)  E x lim f X tk  lim E x f X tk
k 
k 
By combining (4 ) and ( 5) we see that for such sequence
f ( x)  lim E
k 
x
 f X  
k
f  x
formula
6
, x
Denote by A to the characteristic operator of
Dynkin’s
 k , if f  x is l.s.c super harmonic then,
 
X t . If f  C 2 R n then by
is
super
Af 0
Definition : Let h be a real measurable function on
function and f ( x )  h we say that
X t ) . The function h  x  such that
harmonic
w
.r
.
t
Xt
7
iff
R , if f  x is super harmonic (super mean valued )
n
f  x is super harmonic (super mean valued ) majorant of h (w.r. to
hx   inf f ( x) ; x  R n
8
f
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 then
Let g  0 and let

h  f then h is called the least superharmonic majorants of h.
f  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

g ( x)  g  ( x)
,
9
 x  Rn
Definition :A lower semi continuous function
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f : R n  0 ,  is called excessive (w.r.t X t ) if
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Global Journal of Engineering Science and Research Management
f ( x)  E x  f  X s 
10
, , s  0, x  Rn
Theorem : ( Existence for optimal stopping) see [1]

Let
a
g  denote the optimal reward and g the least super harmonic majorants of
continuous
reward
function
g0
,
then

11
i ) g  ( x)  g ( x)
ii ) for   0 define the domain D as:





D   x ; g ( x)  g ( x)   




Suppose that the function g is bounded , then the optimal time
domain .the domain .
12

is the first time that the process exit from the
D is close to being optimal ,in the sense that :
  
g  ( x)  E x g X  
13
 2 ,  x
iii)
For arbitrary continuous g  0 Let
iv)


D   x ; g ( x)  g  ( x) 


14
be the continuation region
N 1, 2 , 3 define g N  g  N , DN   x ; g N ( x)  gˆ N ( x )
For
D N  D N 1 , D N  D  g
1
 0 , N  , D  N D N

g  ( x)  lim E x g  X  N
N 
D  
v)
In particular ,if
vi)
uniformly integrable w.r.t .
Let the value function
g  ( x)  E x
And
  D
a.s .
, If
N  


a.s.
N N .
and
D
Then
x
Q for all N then
15
 
Q x and the family g X  N

N
is
Qx.
g  (the stopping function ) be
 g X  

D
16
is the optimal stopping time.
Theorem: (Uniqueness for optimal stopping) see [1]
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Define the domain D such that:


D   x ; g ( x)  g  ( x)   R n


17
Suppose there is exist an optimal stopping time
problem
      x ,   for the
3  for all x. then
  D
18
for all x  D
,
and
g  ( x)  E x
Hence
 g X  
D
19
, for all x  R n
 D is 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
 
(i )   C  V   C V
(ii )   g on V and   g on  V
Define the domain

(iii ) E y 

T

D   x  V ;  ( x)  g ( x ). and suppose Yt spend zero time on  V a . s , i . e

  D  Yt  d t   0 ,  y  V
0

And suppose that
(i v )  D is a Lipschitz surface , i.e  D is locally graph of a function
h : R n1  R such that there exist k   with
h ( x)  h ( y)  K x  y
,
x, y
Also suppose that:
(v)   C 2  V \  D and the second order derivatives of  are locally bounded near
D
(v i ) L   f  0 on V \  D
(v i i ) L   f  0 on D
(v iii )  D  inf t  o ; yt  D   , a . s. R y  y  V
and
( i x ) the family is uniformly integrable w.r.t. R y ,  y  V , then
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


f  y d t  g  y   ;



 ( y )   ( y )  sup E y 
 T
t
t
20
y V
0
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    and
time
  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 time t on the open market which described by Geometric
-
Brownian motion .
There is fixed sale tax or transaction cost a  0 and discounting factor
 t
 Xt
a
- The discounted net of the sale is given by e
In this case the optimal stopping problem is to find the stopping time

E( s x ) e
 
 X
a

0

 which maximizes
 (maximizes the expected profit), if so then  will be the right time to sell the stocks,
see [1].
REFERENCE
1.
2.
3.
4.
Oksendal, Bernt K. Stochastic Differential Equations: An Introduction with Applications, Springer,
(2002), ISBN 3-540-63720-6.
Zhijun Yang . Geometric Brownian motion Model in Financial Market.
Thomas G. Kurtz: Lectures on Random Analysis. University of Wisconsin- Madison , WI 53706-1388
September (2001).
Hull, John .Options, Futures, and other Derivatives (2009), (7 ed.).
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