International scientific conference of young researchers
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- II INTERNATIONAL SCIENTIFIC CONFERENCE OF YOUNG RESEARCHERS Baku Engineering University 30
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II INTERNATIONAL SCIENTIFIC CONFERENCE OF YOUNG RESEARCHERS Baku Engineering University 30 27-28 April 2018, Baku, Azerbaijan STOCHASTIC DIFFERANTIAL EQUATIONS AND THEIR APPLICATIONS Shams ANNAGHİLİ Bakı Engineering University sems.ennagili18@gmail.com AZERBAIJAN ABSTRACT There is a strong relations between ordinary differential equations and probability processes. In particular, stochastic differential equations and their applications to different real life problems will be considered.
Let us consider the probability space, that consists of three parts
, , . Here,
represents the set of all possible outcomes of the random experiment, -is the − field representing the set of all events, and
- the assignment of probabilities to the events. To understand stochastic differential equations it is helpful to begin with an example of deterministic differential equation. An ordinary differential equation ) ,
) (
t f dt t dx
dt x t f t dx ) , ( ) (
with initial conditions 0 ) 0 ( x x can be written in integral form t ds s x s f x t x 0 0 ) ( , ( ) ( , where
) , , ( ) ( 0 0
x t x t x is the solution with initial conditions 0 0 ) ( x t x . An example is given as ) ( ) ( ) ( t x t a dt t dx , 0 ) 0 ( x x
(1) Sometimes we meet case when, due to some unexpected randomness we can no longer assume that the initial condition 0
to be deterministic constant. If it happens, we may assume 0
to be a random variable ) (
X , where
-represents the result of experiment When we deal with (1), and suppose that ) (t a is not a deterministic parameter but rather a stochastic parameter, we get a stochastic differential equation. If we assume that ) (
( ) ( ) (
t h t f t a ) ( ) ( ) ( ) ( ) ( ) ( t t X t h t X t f dt t dX
(2) If in (2) we introduce dt t t dW ) ( ) ( where
) (t dW denotes differential form of the Brownian motion, we obtain: ) ( ) ( ) ( ) ( ) ( ) ( t dW t X t h dt t X t f t dX
(3) In general, a stochastic differential equation is given as ) , ( )) , ( , ( ) , ( , ( ) , ( t dW t X t g dt t X t f t dX (4)
where
-indicates that ) ,
t X X is a random variable and possesses the initial condition 0 ) , 0 (
X . For stochastic differential equations we often obtain a more realistic mathematical model of the situation. 1)
The simple population growth model: ) ( ) (
N t a dt dN , 0 ) 0 ( N N -constant, where ) (t N -is the size of population at time t , and ) (t a - is the relative rate of growth at time t. In some cases, it is possible that ) (t a is unknown, but Yüklə 36,69 Mb. Dostları ilə paylaş:
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