STUDY ON GEOMETRIC BROWNIAN MOTION WITH APPLICATIONS

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G ^LOBAL J ^{OURNAL OF} E ^NGINEERING S ^{CIENCE AND} R ^ESEARCHES

STUDY ON GEOMETRIC BROWNIAN MOTION WITH APPLICATIONS

Entisar Alrasheed sidahmed^*1and Budoor Mohammed Abdelati²

*1

University of Bahri- College of Applied and Industrial Sciences-Department of Mathematics

2

University of Nile Valiy- College of Education – Department of Mathematics

ABSTRACT

The purpose of this paper is to give a detail study on geometric Brownian motion with some applications.

Keywords- Geometric Brownian motion, Ito lemma, delays geometric Brownian motion, geometric Brownian with jump.

I. INTRODUCTION

A random differential equation is a differential equation in which one or more of the terms is a random process resulting in a solution which is itself a random process. The existence and uniqueness of solution for the random differential equation is found in[1]. A geometric Brownian motion (GBM for briefly) is an important example of random processes satisfying a random differential equation and play great role in mathematical finance as it use for modeling values of investments or stock prices in the Black–Scholes model, GBM is widely used model of stock price behavior ,see[2]. The economist prefers geometric Brownian motion model for market prices as it is everywhere positive with probability 1.

Let



be the set of all possible outcomes of any random experiment , throughout this study we will denote by

X

_t

to the continuous time random process defined on the filtered probability space

  , ^F ,   ^F

t t_T

, ^P 

where,

F

is the σ-algebra of events,

  F

_t _t__T denotes the information generated by the process

X

_t over time interval

[ T 0 , ]

and P is the probability measure . Furthermore, the process

X

_t is adapted to the filtration

  F

_t _t__T.

Definition: A random variable

X

have the lognormal distribution (with parameters



and



) if log

( X )

is normally distributed (log

( X ) ~

N(



,



²)).

Consider the complete filtered probability space

  , F ,   F

_t__o

, P 

that satisfy the usual conditions.

Definition: On the time interval

[ 0 ,  ]

, a real-valued random process

W

_t

 W ( t ,  )

is Brownian motion (wiener process) if it satisfies the following properties:

0 )

1 W

₀



2) For

0  s  t  T , W

_t

 W

_s has normal distribution

N ( 0 , t  s )

with mean zero and variance

t  s

3) For

0  s   t   s  t  T , W

_t

 W

_s independent of

W

_t_

 W

_s_

1-dimensional Itô formula:

Let

X

_t be 1- dimensional Itô processes given by

d X

_t

 u ( t ) d t  v ( t ) d B

_t

Let

g ( x , t )  C

²

 [ 0 ,  )  R 

, then

Y

_t

 g  t , X

_t



is again Itô processes and

   

²



₂

^,   

²

2 1 ,

,

t t

t t t

t

d X

x X t X g

x d X t t g

t d X t Y g

d 

 



 



 

Where

 d X

_t



²

  d X

_t

  . d X

_t



is computed according to the following rules

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  , 0 , ( 1 ) .

, 0 .

.

. d t d B d t d B d t d B d B d B

²

d t d B d B i j

t

d 

_t



_t



_t _t



_t



_i _j

  

(1) also known as Ito multiplication table .For the proofsee [1]

Ito formula is fundamental theorem for evaluating Ito integral.

Multi-dimensional Itô process:

Let

B ( t ,  )   B

₁

( t ,  ) , B

₂

( t ,  ) ,  B

_m

( t ,  ) 

, denote to m- dimensional Brownian motion , for each

U

_i

( t ,  ) , 1  i  n , V

_i_j

( t ,  ) , 1  j  m

, the multi-dimensional Itô process defined as :

) ( )

( t d B t V

t d U X

d

_t



_i



_i _i

i i

t

U V

X , ,

and

B

_t defined in matrix and vector form as:

 





 







 







 









 





 







 





 







) (

, ,

,

) (

3 2 1

2 1

2 22

1 2

1 12

1 1

3 2 1

t B

B v

v v

v

v v

v V

u u u u

U

t X

X

m t

m n n

n

k m

j i

n i

n t

 











Theorem: Multi-dimensional Itô formula:

Consider the multi-dimensional Ito process

d X

_t

 U

_i

d t  V

_i

( t ) d B

_i

( t )

, where the vector



n



i

U U U U

U 

₁

,

₂

,

₃



and the matrix

V

ij

  V

₁₁

, V

₁₂

,  , V

₁d



have

L

₂ components and

B

_t is the vector of d independent Brownian motions.

Let

g  t , X

_t



be twice continuously differentiable function from

R

^dinto

R

then

Y

_t

 g  t , X

_t



is also an Ito process and

     

j i j

i t d k

j i i i

t d k

i t

k

d X d X

x x

X t X g

x d X t t g

t d X t Y g

d  

 



 



   



, 2

, 1

,

²

1 , 1

Where

d X

_i

d X

_jis computed by using the Ito multiplication rule in (1)

Definition: A geometric Brownian motion is a continuous-time random process , and we say that the random process

S

_t fellow the geometric Brownian motion if it satisfy the following random differential equation

) 2

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 variation around the trend and

W

_t is a Wiener process or Brownian motion.

II. SOLUTION OF THE GEOMETRIC BROWNIAN MOTION

Denote by

S

₀

 S   0

, to the initial value which is

F

₀



measurable, then under Ito interpretation, equation (2) has the analytic solution:

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) 3 2 (

exp

²

0

 



 

  



 



 



_t

t

S t W

S   

Which is log-normally distributed random variable with expected value

E   S

t

 S

₀

e

^^t and variance

  S

_t

 S

₀²

e

² ^t

( e

²^t

 1 )

Var

^ ^ .

Geometric Brownian motion is that it is Markovian since it satisfies the Markov chain property, thus the future states

 ^t ^h 

S 

depends only on the future increment of the Brownian motion,see [4].

Definition: A random process

X

_t is said to be

F

_t



martingale if the following conditions are satisfied:

1-

X

_t is adapted to the filtration

  F

_t _t__T

2-

^ t ^ 0 , E   X

t

^ ^

3- For every

s

and

t

such that

0  s  t

,

E  X

t

F

s

  X

s

Condition (3) means that the information in the σ-algebra

F

_sis quiet enough to determine the value of

X

_s, It is easy to prove that geometric Brownian motion is martingale.

III. MULTIDIMENSIONAL GEOMETRIC BROWNIAN MOTION

The GBM (2) can be extending to multidimensional case as fellow:

) 4 ( ,

3 , 2 , 1 , ) ( )

( )

(

_,

1

n i

t W d t

S t d t S t

S

d

ⁿ _i _j _j

j i i

i

    







Where

S

_i

( t ) , 

_i

, , 

_i_,_j and

W

_j

(t )

defined in vector and matrix notation as fellow:

 







 









 







 









 





 







 







 









m t t t t

j

k m m

m

k k

j i

m i

m t t t t

i

W W W W

t W

S S S S

t S

 











2 2 1

2 1

2 22

1 2

1 12

1 1

. 3 2 1

2 2 1

) ( , )

(





If

S

_i

( 0 )

is the initial value, then by using Itô’s lemma the general solution of (4) is:

 





 



   



 

 

  



) 2 (

exp 1 ) 0 ( ) (

1 ,

1 2

,

t W t

S t

S

ⁿ _i

j i j

n

j i j

i i

i

  

Càdlàg function: A function

f : E  M

is called a càdlàg function if for every

t  E

the function

f (t )

is right-continuous with left limits.

If the sample path of the random process

X

_t is right continuous with left limit, we set _s

t s t

t s

X





 

lim

, and





 X

_t

X

_t

X

_t ,

 X

_t represent the size of jump of the process

X

_t at time t , if there is no jump then

 X

is zero,see[6].

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Definition: A jump process is type of random process that has discrete movements called jumps, the very useful jump process is the standard Poisson process

N

_t with jumps of size +1 only, that is

  N

_t

 N

_t

 N

_t _

 1 

, and whose paths are constant in between two jumps. If

N

_t jumps occur in

 ⁰ ^, ^t 

,then for some



_t and for each

s  0

and

t  0 , N

_s__t

 N

_s ^∼Poisson with mean



_t . Definition: The process

) 5 ( ,

1

R t Z

Y

^t

N k

k

t

  



Is called a compound Poisson process.

Random Integrals with Jumps: Given the random process

X

_twe let the random integral of

X

_twith respect

Y

_t

defined by

 



T N

i

k t t

t

i

k

Z

X Y

d X

0 1

In particular the compound Poisson process

Y

_t in (5) admits the random integral representation

) 6 (

0

0 N s

t

Y Z d N

Y ^ ^ 

_s

Ito formula with jump: consider the case of a standard Poisson process

N

_t with intensity



_t and

N

_s_ be the left limit of the Poisson process at time s, then

     

   

s

N k

s s

s N

k

s s

N k t

N d N

f N

f

N d N

f N

f

k f k f f

N f

t t

t



 

  



















1 1

1

1 1 )

0 (

For the compound Poisson process

Y

_t

     

   

s

N k

s s

s N

k

s N

k

T k

T t

N d Y

f Y

f

N d Y f Z

Y f

Y f Z

Y f f

Y f

t t

s t

k k



 

  

 



















1 1

1

) 0 (

And for the process

) 7

t

(

t

i

u d t v d W d N

Y

d    

Where



is the fraction rate.

And if

f ( t )  C

²

 [ 0 ,  )  R 

, the process

Y

_t

 f ( t )

is again Ito process and

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         

   

  ⁽ ⁸ ⁾

2 1

0

0 0

2

0 0

s s

s t

s s t

i

N d Y f Y f

s d Y f v W

d Y f u W

d Y f u Y

f Y f





 

 

 





The Ito formula for jump processes used the following extension Ito multiplication table:

.

d t d B

_t

d N

_t

t

d

⁰ ⁰ ⁰

B

t

d

⁰

^d ^t

⁰

N

t

d

⁰ ⁰

d N

_t

see[7]

IV. GEOMETRIC BROWNIAN MOTION WITH JUMPS

Generally the random differential equations with jump component has great

importance in different areas of science. The problem of this kind of processes with jumps is considered in[1], [3].

Geometric Brownian motion with Jumps play essential role in financial market when the stock prices, and prices of other assets show jumps which usually

caused by unpredictable events or sudden shift.

Consider that an asset price

S

_t with the percentage drift



and the percentage volatility



follow GBM, but each jump reduces the asset price by a fraction



, then

S

_t should satisfy a differential equation like

) 9 ) (

( ) ( ) ( )

( )

( t S t d t S t d W t S t d N t S

d      

Such equation is known as geometric Brownian motion with jump.

Through integration we come to random integral equation

) 10 ) (

0 ( ) ( ) 0 (

) 0 (

) 0 ( )

( t S u d N

_u

u W d u t S

u d u t S

S t

S         _ 

Where

^S   ⁰

is the initial value. Again under Itô's interpretation (8) , the general solution of (9) is:

) 11 ( )

1 2 (

exp 2

0 N t

W t t

t S

S       _ ^  





 





 





 





V. DELAY GEOMETRIC BROWNIAN MOTION

Delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.

Delay geometric Brownian motion (DGBMs for briefly) or geometric Brownian motion with delay is the solution of a random differential equation where the drift and diffusion coefficient depend linearly on the past of the solution.

Let

S (t )

be the price of risky asset at time

t

, and assume that

S (t )

is the geometric Brownian motions that is:

) 12 ) (

( ) ( ) ( )

( )

( t r S t d t V t S t d W t S

d  

where

r  0

is the risk-free interest rate ,

V (t )

represents the volatility and

W

_t is a Wiener process or Brownian motion

The case when the volatility

V (t )

can be represented as a function of

S ( t   )

, say

V  S ( t   ) 

, equation (12) becomes:

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 ( )  ( ) ( ) ( 13 ) )

( )

( t r S t d t V S t S t d W t S

d    

As the option price at time

t  

is clearly dependent on the underlying asset price

S ( t   )

.

In the other case if the volatility is a function of the past states

S ( t  

₁

)

,

S ( t  

₂

)

,

 , S ( t  

_n

)

, then

) (t

S

satisfy:

 ( ) , ( ) , , ( )  ( ) ( ) ( 14 )

) ( )

( t r S t d t V S t

₁

S t

₂

S t S t d W t

S

d         

_n

Such equations ( (13) and (14)) are delay geometric Brownian motionssee[5].

Applications: Geometric Brownian motion has many applications in real life, for example in mathematical finance and economics as we mentioned above the geometric Brownian motion is essential model for the stock prices.

As any other differential equation in describing natural phenomenon geometric Brownian motion model is used to study and describe the growth of populations as it consider the random influences ( noise element) on the growth process which are neglected in deterministic approach .

VI. GEOMETRIC BROWNIAN MOTION AS POPULATION GROWTH MODEL

Consider the simple (deterministic) population growth model

) 14 ( )

costant (

) 0 ( , ) ( ) ) (

(

N

0

N t N t t a

d t N

d  

where

N ( t )

is the size of population at time

t

and

a (t )

is the relative rate of growth at time

t

, for some random environmental effects

a ( t )  r ( t )  noise

, assume that

r ( t )

is nonrandom function and let the noise term be the white noise

W

_twhich we do not know exactly what it is behavior , but only it is probability distribution, then

) 15 ( )

( )

( t r t W

_t

a   

where



is constant , when we substitute

a (t )

in (15) in (14) we get

) 16 ( )

( )

(

_t

t

r N t d t N t d W

N

d   

Which is a geometric Brownian motion.

The geometric Brownian motion model (16 ) take into the consideration all the random environmental factors that effect on the population growth process so it can be regarded as more adequate models for the development for the population growth.

REFERENCE

1. Oksendal, Bernt K. Stochastic Differential Equations: An Introduction with Applications, Springer, (2002), ISBN 3-540-63720-6

2. Hull, John .Options, Futures, and other Derivatives (2009), (7 ed.).

3. Thomas G. Kurtz: Lectures on Random Analysis. University of Wisconsin- Madison , WI 53706-1388 September 2001.

4. Zhijun Yang . Geometric Brownian motion Model in Financial Market.

5. Xuerong Mao. Delay geometric Brownian motion in financial option valuation. Department of Statistics and Modeling Science-University of Strathclyde -Glasgow, G1 1XH.

6. Entisar Alrasheed Sidahmed. Applications of controlled model with jump in finance. International journal of engineering science and research technology IJESRT4(4), April 2015.

7. N. Privault. http://www.ntu.edu.sg/home/nprivault/indext.html. pp(365-394) April 25, 2013.

STUDY ON GEOMETRIC BROWNIAN MOTION WITH APPLICATIONS

G LOBAL J OURNAL OF E NGINEERING S CIENCE AND R ESEARCHES