A Comparative Survey of an Approximate Solution Method for Stochastic Delay Differential Equations ()
1. Introduction/Background of the Study
Stochastic differential equations have many applications in science and engineering. One of the important applications is the stochastic representation of solution to ordinary differential equations. A lot of monographs, conference papers and journal articles have been written by various authors on the Caratheodory’s, Euler-Maruyama approximate solution to stochastic delay differential equation which is usually represented by means of ordinary differential equation. This paper therefore aims at obtaining the error between the approximate solution and the accurate solution. It would also investigate the effect of Caratheodory’s scheme to the stochastic delay differential equation. To achieve this, the work will be among others;
1) Find the error (difference) between approximation solution and accurate solution of stochastic delay differential equation.
2) Establish the difference between approximate solutions and the unique solution to stochastic delay differential, under non uniform Lipschits condition and non linear growth condition.
3) Compare Caratheodory approximation and Euler-Manyama approximate method of convergence.
4) Construct to show that approximate solution converges to a unique solution.
2. Related Literature
Stochastic differential equations have many applications in science and engineering. One of the important applications is the stochastic representation of solution to ordinary differential equations which includes stochastic approximate solution. An approximate solution is one of the essential concepts in the study of stochastic differential equations. A solution is said to be approximated where there is difficulty in finding exact solution or analytical solution. The approximate procedure is known as Caratheodory’s approximate procedure. Approximate equations are defined on partitions of time interval and their coefficients are Taylor approximations of the coefficients of the equation. The Euler-Maruyama method was developed as one of the powerful numerical methods for the stochastic differential delay equations with Markovian switching (SDDEWMS).
[1] considered some class of control system governed by the neutral stochastic functional differential equations ith unbounded delay and studied the approximation controllability of the systems. [2] established the difference between an approximate solution and an accurate/exact solution for a stochastic differential delay equation where the approximate solution as were called Caratheodory’s, was constructed from successive approximation. [2] obtained the difference between approximate solution and accurate solution for the stochastic differential equations where the approximate solution is called Caratheodory’s approximate solution, which has been constructed by successive approximation. [2] obtained the difference between approximate solution and accurate solution for the stochastic differential equations where the approximate solution is called Caratheodory’s approximate solution which has been constructed by successive approximations. [3] presented result on an analytic approximate method for the class of stochastic differential equations with coefficients that do not satisfy the Lipschitz and linear growth conditions but behaved like a polynomial. Furthermore, equations from this class have unique solutions with bounded moments and their coefficients satisfy polynomial conditions. [4] considered the stochastic differential equation and defined the Caratheodory’s approximate solution of stochastic differential delay equation.
[5] discussed Caratheodory’s and Euler-Maruyama’s approximation solutions to stochastic differential delay equation.
To make the theory more understandable, we impose the non-uniform Lipschitz condition and non-linear growth condition. The Euler method discretisation has an optimal strong convergence rate and [5] established Caratheodory’s and Euler approximate solutions to stochastic differential delay equation.
Consider the stochastic delay equation
,
for
with initial value
(1)
where
is called the drift coefficient and
is the diffusion coefficient.
is a Brownian noise that defines the randomness of the physical system and it is often called the white noise.
The Brownian noise is the simplest intrinsic noise term that adequately model Browian motion. The integral form of (1) is
(2)
The first integral in (1) is a voltera integral equation, and second integral is an Ito stochastic integral with respect to the Brownian motion
.
However the Lipschitz condition only guarantees the existence and uniqueness of the solution, then the equation can be solved implicitly. Therefore we often seek the approximate solution rather than the accurate solution.
[5] discussed Caratheodory’s and Euler-Maruyama’s approximation solutions to stochastic differential delay equation. To make the theory more understandable, we impose the non-uniform Lipschitz condition and non-linear growth condition. [6] considered the class of semi linear stochastic evolution equation with delays and proved that the Caratheodory’s approximate solution converges to the solution of stochastic delay evolution equations. [7] obtained the estimate on difference between the Caratheodory approximate solution
and the unique solution
to the stochastic differential delay equation, and he obtained the estimate on difference between the Caratheodory approximate solution
and the unique solution
to the stochastic differential delay equation. [8] discussed the Caratheodory approximate solution for the class of doubly perturbed stochastic differential equation. [9] showed that stochastic differential equations with jumps and non-lipschitz coefficients have pair wise unique strong solutions by the Euler approximation method. [10] developed the approximate analytical solution of fractional delay differential equations of the initial value linear and nonlinear boundary problems. [11] considered the Caratheodory’s approximate solution of stochastic functional differential Equation (SFDEs) and obtained the existence theorem for stochastic functional differential equations. Thus, Caratheodory’s approximation procedure, extended the Caratheodory’s approximate scheme to the case of stochastic differential scheme to the case of stochastic differential delay equations.
[12] studied Euler-Maruyama method for the stochastic differential delay equations with Markovian switching (SDDEWMS). Approximate equations are defined on partitions of time interval and their coefficients are Taylor approximations of the coefficients of the equation. The Euler Maruyama method was developed as one of the powerful numerical methods for the stochastic differential delay equations with Markovian switching (SDDEWMS). [12] worked on an approximate analytical method and introduced new variational iteration method.
3. Methodology
We discuss the following types of approximate solutions, these are
1) Caratheodory approximate method;
2) Euler-Mmaruyama approximate method, together with properties of Brownian noise or wiener processes.
3.1. Caratheodory Approximate Method
Caratheodory’s approximate solutions, is said to be an approximate solution, if for every integer
, with
, for on
, and
(3)
For all
.
We compute
step by step on the intervals
,
; for
,
can be compute by
Step I
(4)
for
and
.
Step II
For
(5)
3.2. Euler-Maruyama’s Approximate Method
Euler-Maruyama approximate solution is said to be an approximate solution of equation if for every integer
,
, for
,
and
(6)
For
, we compute
as
For
(7)
Thus, Caratheodory’s and Euler Maruyama approximation procedures converge to stochastic delay deferential equation.
3.3. Properties of Brownian Noise or Weiner Process
1)
,
2) Path of Weiner process is continuous functions of
,
3) Increments of Weiner process on non-overlapping intervals are independent i.e. for
, the random variables
are independent,
4) Expectation
,
5) For any
, the random vector
is Gaussian.
4. Approximate Solutions
We now discuss Caratheodory’s approximate solutions to stochastic delay differential equations and show that the solution
of delay equation approximates the solution
of the original equation. The idea is to replace the resent state
with the past
to obtain the equation.
with initial condition.
Now replace the present state
by its past
, we have
, for
(8)
also replace the past
with
(9)
Now Rd-valued stochastic process
on the interval
is called a solution of the equation if the following properties are satisfied;
1) It is continuous and
is
-adapted,
2)
and
,
3)
and, for every
and
.
A solution
is said to be unique if any other solution
is similar (indistinguishable) from it, that is
.
Now we consider the stochastic differential delay Equation (1) with initial details
. We define the Caratheodory approximation as follows. Let
define
, for every integer
on
, and
, for
.
We compute
step by step on the intervals
,
. For
,
can be constructed by
as follows,
Step I
For
, we have
(10)
Step II
For
, we have
(11)
5. Theorems/Lemma
5.1. Lemma 1 [7]
Suppose that
, for all
and
with
.
,
where
and
.
5.2. Lemma 2 [7]
If
for all
,
for all
Proof
Let
and if
with condition such that
.
Then
.
By the Caratheodory approximate solution we have
Using the Holder inequality, we have
for all
.
Then
By the Gronwall inequality
for all
.
5.3. Theorem [7]
Suppose
and
are continuous. Let
be a bounded Rd-valued, F-measurable random variable. Suppose further that there exists a continuous increasing concave function
such that
(12)
for all
,
.
(13)
Then
, has a unique solution
and Caratheodory approximate solutions
converges to
. In the sense of
.
Proof
We divide the proof into two parts
1) Uniqueness
Let
and
be two solution of
.
From equation, we show that
(14)
Since
is concave, by Jensen inequality we have
For
,
for all
.
Let
on
,
And let
be the inverse function of
From Equation (1) we have
and
.
for
,
Letting
, we have
(15)
We have
for all
.
2) Convergence
To show the existence then convergence,
Let
, and
, we define a sequence
(16)
for
; since
.
By induction, we have that
,
such that
where
; that is for any
,
where
.
By the Gronwall inequality, we have
Since k is constant, we have
, for all
and
Since
Taking the expectation we have
(17)
where
for
, for
, (18)
where
.
By (17) we see that (18) holds when
We now show (18) for (n + 1)
(19)
so (18) holds for
. Hence, by induction, (18) holds for all
, by replacing “n” with
we have
(20)
By taking the expectation we have
Thus
(21)
Since
By the Borel Cantelli lemma yields that for almost all
, there exists a positive integer
such that
(22)
For n, Ζ n, with the probability 1, the partial sums
i.e.
(23)
Since
the uniformly convergent in E E [0, T], since
is continuous and
adapted and also a Cauchy sequence is L2, as
in L2, letting
in above equation, gives
for all
, for
.
Since
is concave and increasing, there exist a positive number “b” such that
, on
,
Let
, then
By the linear growth condition and
, we conclude that the approximate solution
converges to
in the sense of stochastic delay differential equation, i.e. Equation (1) hence the proof is completed.
6. Summary/Conclusions
The concept of approximations has been extended to stochastic delay differential equation. This was achieved by replacing the present state with past state through Caratheodory’s scheme. The Caratheodory’s and Euler-Maruyama approximation procedures were compared and the difference and error between them were obtained. The two procedures were shown to converge to stochastic delay differential equation.
The work has proved the existence and uniqueness of an approximate solution of the stochastic delay differential equation using Brownian white noise of the Ito-type. Relevant lemma and theorem associated with the approximate solution method are included. By linear growth condition it was shown that the approximate solution converges to the unique solution in the sense of stochastic delay equation.
Acknowledgements
We sincerely acknowledge the ingenuity of all the researchers who have worked in this area’ this paper “A comparative survey of an approximate solution method for stochastic delay differential equations” was done by Donatus Anonwa and Christian E Emenonye. The two did the work separately and met to combine their ideas. Donatus Anonwa prepared the final manuscript which was edited and later proof read by Christian Emenonye. We sincerely acknowledge the reviewers and the editorial team of this Journal for their competence in the discharge of their work.