This study is focused on the approximate solution for the class of stochastic delay differential equations. The techniques applied involve the use of Caratheodory and Euler Maruyama procedures which approximated to stochastic delay differential equations. Based on the Caratheodory approximate procedure, it was proved that stochastic delay differential equations have unique solution and established that the Caratheodory approximate solution converges to the unique solution of stochastic delay differential equations under the Cauchy sequence and initial condition. This Caratheodory approximate procedure and Euler method both converge at the same rate. This is achieved by replacing the present state with past state. The existence and uniqueness of an approximate solution of the stochastic delay differential equation were shown and the approximate solution to the unique solution was also shown.
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.
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).
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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 [
Consider the stochastic delay equation
d x ( t ) = α ( x ( t ) , x ( t − r ) ) d t + β ( x ( t ) , x ( t − r ) ) d B ( t ) ,
for t 0 ≤ t ≤ T with initial value
x ( t 0 ) = x 0 ∈ L 1 2 (1)
where α ( x ( t ) ) is called the drift coefficient and β ( x t ) is the diffusion coefficient.
β ( t ) 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
x ( t ) = x 0 + ∫ 0 t α ( x ( s ) , x ( t − r ) ) d s + ∫ 0 t β ( x ( s ) , x ( t − r ) ) d B ( s ) (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 = { β t , t ≥ 0 } .
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.
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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.
Caratheodory’s approximate solutions, is said to be an approximate solution, if for every integer n ≥ 1 , with x q ( t ) = x 0 , for on t 0 ≤ t ≤ T , and
x q ( t ) = x 0 + ∫ t 0 t α ( x q ( s − 1 q ) , x q ( s − t ( s ) − 1 q ) ) d s + ∫ t 0 t β ( x q ( s − 1 q ) , x q ( s − τ ( s ) − 1 q ) ) d b ( t ) (3)
For all q ∈ ℕ .
We compute x n ( t ) step by step on the intervals [ t 0 , t 0 + 1 q ] , [ t 0 + 1 q , t 0 2 q ] ; for t 0 ≤ t ≤ t 0 1 q , x n ( t ) can be compute by
Step I
x q ( t ) = x 0 + ∫ t 0 t ∀ ( x q ( s − 1 q ) , x q ( s − t ( s ) − 1 q ) ) d s + ∫ t 0 1 q t β ( x q ( s − 1 q ) , x q ( s − τ ( s ) − 1 q ) ) d b ( t ) (4)
for n − 1 and q = 1 .
Step II
For [ t 0 + 2 q < t ]
x q ( t ) = x q ( t 0 + 2 q ) + ∫ t 0 + 2 q t ∀ ( x q ( s − 1 q ) , x q ( s − t ( s ) − 2 q ) ) + ∫ t 0 + 1 q t β ( x q ( s − 1 q ) , x q ( s − τ ( s ) − 2 q ) ) d b ( t ) (5)
Euler-Maruyama approximate solution is said to be an approximate solution of equation if for every integer n − 1 , x n ( t 0 ) = x 0 , for
t 0 + k − 1 n < t ≤ ( t 0 + k n ) ∧ T , k = 1 , 2 , ⋯
and
x n ( t ) = x 0 + ∫ t 0 t ∀ ( x n ( s ) , s ) d s + ∫ t 0 t β ( x n ( s ) , s ) d b ( s ) (6)
For t 0 + k − 1 n , we compute x n ( t ) = x 0 as
x n ( t ) = x n ( t 0 + k − 1 n ) + ∫ t 0 + ( k − 1 n ) t x n ( t 0 + k − 1 n , s ) d s + ∫ t 0 + k − 1 n t β ( x n ( t 0 + k − 1 n , s ) ) d B ( s )
For t 0 + k n
x n ( t ) = x n ( t 0 + k n ) + ∫ t 0 + k n t α ( x n ( t 0 + k n ) , s ) d s + β ( x n ( t 0 + k n ) , s ) d B ( s ) (7)
Thus, Caratheodory’s and Euler Maruyama approximation procedures converge to stochastic delay deferential equation.
1) B 0 = 0 ,
2) Path of Weiner process is continuous functions of t ∈ [ 0 , 0 ] ,
3) Increments of Weiner process on non-overlapping intervals are independent i.e. for ( s 1 , t 1 ) ∩ ( s 2 , t 2 ) = ∅ , the random variables W t 2 − W s 2 , W t 1 − W s 1 are independent,
4) Expectation E W t = 0 ,
5) For any t 1 ⋯ t n , the random vector ( W t 1 ⋯ W t n ) is Gaussian.
We now discuss Caratheodory’s approximate solutions to stochastic delay differential equations and show that the solution x n ( t ) of delay equation approximates the solution x ( t ) of the original equation. The idea is to replace the resent state x ( t ) with the past x ( t − 1 q ) to obtain the equation.
d x ( t ) = α ( x ( t ) ) ( x − t ) d t + β ( x ( t ) ) d b ( t ) t ∈ [ t 0 , T ] with initial condition.
Now replace the present state x ( t ) by its past x ( t − 1 q ) , we have
d x n ( t ) = x ( x n ( t − 1 n ) , t ) d t + β ( x n ( t − 1 n ) , t ) d B ( t ) , for t ∈ [ t 0 , T ] (8)
also replace the past x ( t − τ ( t ) ) with x ( t − τ ( t ) − 1 n )
d x n ( t ) = α ( x n ( t − 1 n ) , x n ( t − T ( t ) − 1 n ) ) + β ( x n ( t − 1 n ) , x n ( t − τ ( t ) − 1 n ) ) d B ( t ) (9)
Now Rd-valued stochastic process x ( t ) on the interval t 0 ≤ t ≤ T is called a solution of the equation if the following properties are satisfied;
1) It is continuous and { x ( t ) } t 0 ≤ t ≤ T is F t -adapted,
2) { α ( x ( t ) , t ) } ∈ L 2 ( [ t 0 , T ] , R d ) and { β ( x ( t ) , t 2 ) } ∈ L 2 ( [ t 0 , T ] , R d × m ) ,
3) x ( t 0 ) = x 0 and, for every t 0 ≤ t ≤ T and x ( t ) = x 0 + ∫ t 0 t α ( x ( s ) , s ) d s + ∫ t 0 t β ( x ( s ) , s ) d B ( s ) .
A solution x ( t ) is said to be unique if any other solution x ( t ) is similar (indistinguishable) from it, that is p { x ( t ) = x ( t ) for all t 0 ≤ t ≤ T } = 1 .
Now we consider the stochastic differential delay Equation (1) with initial details x ( t 0 ) = α 0 . We define the Caratheodory approximation as follows. Let n ≥ 1 define x n ( t ) = x 0 , for every integer n ≥ 1 on t 0 − 1 ≤ t ≤ t 0 , and x n ( t ) = x 0 + ∫ t 0 t α ( x n ( s − 1 q ) , x n ( s − t ) ) d s + ∫ t 0 t β ( x n ( s − t ( s ) − 1 q ) , s ) d s , for t 0 ≤ t ≤ T .
We compute x n ( t ) step by step on the intervals [ t 0 , t 0 + 1 m ] , ( t 0 + 1 q , t 0 + 2 m ] . For t 0 ≤ t ≤ t 0 + 1 m , x n ( t ) can be constructed by x n ( t ) = x 0 + ∫ t 0 t α ( x 0 , s ) d s + ∫ t 0 t β ( x 0 , s ) d B ( s ) as follows,
Step I
For [ t 0 + 1 q ] , we have
x q ( t ) = x q ( t 0 + 1 m ) + ∫ t + 1 q t α ( x q ( s − 1 q ) , x q ( s − t ) ) d s + ∫ t 0 + 1 q t β ( x q ( s − τ ( s ) − 1 q ) , s ) d B ( s ) (10)
Step II
For [ t 0 + 1 q ] , we have
x q ( t ) = x q ( t 0 + 2 q ) + ∫ t + 1 q t α ( x q ( s − 1 q ) , x q ( s − t ) ) d s + ∫ t 0 + 1 q t β ( x q ( s − t ( s ) − 2 q ) , s ) d B ( s ) (11)
Suppose that | f ( x , t ) | 2 ∨ | g ( x , t ) | 2 ≤ k ( 1 + k ) , for all n ≥ 1 and t 0 ≤ s < t ≤ T with t − s ≤ 1 .
E | x n ( t ) − x n ( s ) | 2 ≤ C 2 ( t − s ) ,
where C 2 = 4 k ( 1 + C 1 ) and
c 1 = c 1 + 3 E | x q I 2 | e 3 k ( T − t 0 ) ( T − t 0 + 1 ) .
If | f ( x , t ) | 2 ∨ | g ( x , t ) | 2 ≤ k ( 1 + k α I 2 ) for all n ≥ 1 ,
sup t 0 ≤ t ≤ T E | x n ( t ) | 2 ≤ C 1 = ( 1 + 3 E | x 0 | 2 ) e 3 k ( t − t 0 ) ( T − t 0 + 1 ) for all t 0 ≤ t ≤ T
Proof
Let n ≥ 1 and if x n ( t ) with condition such that | f ( x , t ) | 2 ∨ | g ( x , t ) | 2 ≤ k ( 1 + | x | 2 ) .
Then { x n ( t ) } t 0 ≤ t ≤ T ∈ M 2 ( [ t 0 , T ] , R 2 ) .
By the Caratheodory approximate solution we have
| x n ( t ) | 2 ≤ 3 | x 0 | 2 + 3 | ∫ t 0 t f ( x n ( s − 1 m ) , x n ( s − r ) ) d s | 2 + 3 | ∫ t 0 t g ( x n ( s − 1 m ) , x n ( s − r ) ) d B ( s ) | 2
Using the Holder inequality, we have
E | x n ( t ) | 2 ≤ 3 E | x 0 | 2 + 3 ( t − t 0 ) E ∫ t 0 t | f ( x n ( s − 1 m ) , x n ( s − r ) ) | 2 d s + 3 E ∫ t 0 t | g ( x m ( s − 1 m ) , x m ( s − r ) ) | 2 d s ≤ 3 E | x 0 | 2 + 3 k ( T − t 0 + 1 ) ∫ t 0 t [ 1 + E | x n ( s − 1 m ) | 2 ] d s ≤ 3 E | x 0 | 2 + 3 k ( T − t 0 + 1 ) ∫ t 0 t [ 1 + sup t 0 ≤ t ≤ T E | x n ( r ) | 2 ] d s
for all t 0 ≤ t ≤ T .
Then
1 + sup t 0 ≤ t ≤ T E | x n ( r ) | 2 ≤ 1 + 3 E | x 0 | 2 + 3 k ( T − t 0 + 1 ) ∫ t 0 t [ 1 + sup t 0 ≤ t ≤ T E | x n ( r ) 2 | ] d s
By the Gronwall inequality
⇒ 1 + sup t 0 ≤ r ≤ t E | x n ( r ) | 2 ≤ ( 1 + 3 E | x 0 | 2 ) e 3 k ( t − t 0 ) ( T − t 0 + 1 )
for all t 0 ≤ t ≤ T .
Suppose f ( x , t ) and g ( x , t ) are continuous. Let x 0 be a bounded Rd-valued, F-measurable random variable. Suppose further that there exists a continuous increasing concave function C : R + → R + such that
∫ 0 t d u c ( u ) = ∞ (12)
for all x , y ∈ R d , t 0 ≤ t ≤ T .
| F ( x , t ) − f ¯ ( y , t ) | 2 ∨ | g ( x , t ) − g ¯ ( g , t ) | 2 ≤ c ( | x − y | 2 ) (13)
Then d x ( t ) = f ( x ( t ) , t ) d t + g ( x ( t ) , t ) d B ( t ) , has a unique solution x ( t ) and Caratheodory approximate solutions x n ( t ) converges to x ( t ) . In the sense of E ( sup t 0 ≤ t ≤ T | x m ( t ) − x ( t ) | 2 ) = 0 .
Proof
We divide the proof into two parts
1) Uniqueness
Let x ( t ) and x ^ ( t ) be two solution of d x ( t ) = f ( x ( t ) , t ) d t + g ( x ( t ) , t ) d B ( t ) .
From equation, we show that
E ( sup t 0 ≤ t ≤ T | x ( r ) − x ^ ( r ) | 2 ) ≤ 2 ( T − t 0 + 4 ) ∫ t 0 t E k ( | x ( s ) − x ^ ( s ) | 2 ) d s (14)
Since k ( ∗ ) is concave, by Jensen inequality we have
E k ( | x ( s ) − x ^ ( s ) | 2 ) ≤ k ( E | x ( s ) − x ^ ( s ) | 2 ) ≤ k [ E ( sup t 0 ≤ t ≤ T | x ( r ) − x ^ ( r ) | 2 ) ]
For ε > 0 ,
E ( sup t 0 ≤ t ≤ T | x ( r ) − x ^ ( r ) | 2 ) ≤ ε + 2 ( T − t 0 + 4 ) ∫ t 0 t k [ E ( sup t 0 ≤ t ≤ T | x ( r ) − x ^ ( r ) | 2 ) ] d s
for all t 0 ≤ t ≤ T .
Let G ( r ) = ∫ 1 r d u c ( u ) on r > 0 ,
And let 1 G ( ∗ ) be the inverse function of G ( ∗ )
From Equation (1) we have
lim e → 0 G ( ε ) = ϖ and D O M ( 1 G ) = ( − ϖ , G ( ϖ ) ) .
for ε > 0 ,
E ( sup t 0 ≤ r ≤ T | x ( r ) − x ^ ( r ) | 2 ) ≤ 1 g [ G ( ε ) + 2 ( T − t 0 + 4 ) ( T − t 0 ) ]
Letting ε > 0 , we have
E ( sup t 0 ≤ r ≤ T | x ( r ) − x ^ ( r ) | 2 ) = 0 (15)
We have x ( t ) = x ^ ( t ) for all t 0 ≤ t ≤ T .
2) Convergence
To show the existence then convergence,
Let x ( t 0 ) = x 0 , and n = 1 , 2 , 3 , ⋯ , we define a sequence
x n ( t ) = x 0 + ∫ t 0 t α ( x n − 1 ( s ) , s ) d s + ∫ t 0 t ( x n − 1 ( s ) , s ) d B ( s ) (16)
for t ∈ [ t 0 , T ] ; since x 0 ( t 0 ) ∈ M 2 ( [ t 0 , T ] , R d ) .
By induction, we have that
x n ( t 0 ) ∈ M 2 ( [ t 0 , T ] , R d ) ,
such that
E | x n ( t ) | 2 ≤ C 1 + 3 k ( T + 1 ) ∫ t 0 t E | x n − 1 ( s ) | d s ( s )
where C 1 = 3 E | x 0 | 2 + 3 k T ( T + 1 ) ; that is for any k ≥ 1 ,
max 1 ≤ n ≤ k E | x n ( t ) | 2 ≤ C 1 + 3 k ( T + 1 ) ∫ t 0 t max 1 ≤ n ≤ k E | x n − 1 ( s ) | 2 d s ≤ C 1 + 3 k ( T + 1 ) ∫ t 0 t ( E | x n | 2 d s ( s ) max 1 ≤ n ≤ k E | x n ( s ) | 2 d s ) ≤ C 2 + 3 k ( T + 1 ) ∫ t 0 t max 1 ≤ n ≤ k E | x n ( s ) | 2 d s .
where C 2 = C 1 + 3 k T ( T + 1 ) E | x 0 | 2 .
By the Gronwall inequality, we have
max 1 ≤ n ≤ k E | x n ( t ) | 2 ≤ C 2 e 3 k T ( T + 1 )
Since k is constant, we have
E | x n ( t ) | 2 ≤ C 2 e 3 k T ( T + 1 ) , for all t 0 ≤ t ≤ T and n ≥ 1
Since
| x 1 ( t ) − x ( t 0 ) | 2 = | x 1 ( t ) − x 0 | 2 ≤ 2 | ∫ t 0 t α ( x 0 , s ) d s 2 | = 2 | ∫ t 0 t β ( x 0 , s ) d β ( s ) 2 |
Taking the expectation we have
E | x 1 ( t ) − x ( t 0 ) | 2 ≤ 2 k ( t − t 0 ) 2 ( 1 + E | x 0 | 2 ) + 2 k ( t − t 0 ) ( 1 + E | x 0 | 2 ) ≤ C (17)
where C = 2 k c ( t − t 0 + 1 ) ( t − t 0 ) ( 1 + E | x 0 | 2 ) for n ≥ 0
E | x n + 1 ( t ) − x n ( t ) | 2 ≤ C [ M ( t − t 0 ) ] n ! , for t 0 ≤ t ≤ T , (18)
where M = 2 k ¯ ( T − t 0 + 1 ) .
By (17) we see that (18) holds when n = 0
We now show (18) for (n + 1)
| x n + 2 ( t ) − x n + 1 ( t ) | 2 ≤ 2 | ∫ t 0 t [ α ( x n + 1 ( s ) , s ) − α ( x n ( s ) , s ) ] d s | + 2 | ∫ t 0 t [ β ( x n + 1 ( s ) , s ) − β ( x n ( s ) , s ) ] d B ( s ) | 2 (19)
E | x n + 2 ( t ) − x n − 1 ( t ) | 2 ≤ 2 k ¯ ( t − t 0 + 1 ) E ∫ t 0 t | x n + 1 ( s ) − x n ( s ) | 2 d s ≤ M ∫ t 0 t E | x n + 1 ( s ) − x n ( s ) | 2 d s ≤ M ∫ t 0 t C [ M ( s − t 0 ) ] n n ! d s = C [ M ( t − t 0 ) ] n + 1 ( n + 1 ) !
so (18) holds for n + 1 . Hence, by induction, (18) holds for all n ≥ 0 , by replacing “n” with n − 1 we have
sup t 0 = t ≤ T | x n + 1 ( t ) − x n ( t ) | 2 ≤ 2 R ( T − t 0 ) ∫ t 0 T | x n ( s ) − x n − 1 ( s ) | 2 d s + 2 sup t 0 ≤ t ≤ T | ∫ t 0 T [ g ( x n ( s ) , s ) − g ( x n − 1 ( s ) , s ) ] d B ( s ) | 2 (20)
By taking the expectation we have
E ( sup t 0 = t ≤ T | x n + 1 ( t ) − x n ( t ) | 2 ) ≤ 2 k ^ ( T − t 0 + 4 ) ∫ t 0 T E | x n ( s ) − x n − 1 ( s ) | 2 d s ≤ 4 M ∫ t 0 T C [ M ( s − t 0 ) ] n − 1 ( n + 1 ) ! d s = 4 C [ M ( T − t 0 ) ] n n !
Thus
{ sup t 0 = t ≤ T | x n + 1 ( t ) − x n ( t ) | > 1 2 n } ≤ 4 C [ 4 M ( T − t 0 ) ] n n ! (21)
Since
∑ n = 0 ϖ 4 C [ 4 M ( T − t 0 ) ] n n ! < + ∞
By the Borel Cantelli lemma yields that for almost all w ∈ R , there exists a positive integer n 0 = n 0 ( ϖ ) such that
sup t 0 ≤ t ≤ T | x n + 1 ( t ) − x n ( t ) | ≤ 1 2 n (22)
For n, Ζ n, with the probability 1, the partial sums
x ( t 0 ) + ∑ i = 0 n − 1 [ x i + 1 ( t ) − x i ( t ) ] = x n ( t )
i.e.
x n ( t ) = x 0 + ∑ i = 0 n − 1 [ x i + 1 ( t ) − x i ( t ) ] (23)
Since x n ( t 0 ) = x 0 the uniformly convergent in E E [0, T], since x ( t ) is continuous and f t adapted and also a Cauchy sequence is L2, as x n ( t ) → x ( t ) in L2, letting n → ϖ in above equation, gives
E | x ( t ) | 2 ≤ C 2 e e k T ( T + 1 )
for all t 0 ≤ t ≤ T , for x ( t ) ∈ M 2 ( [ t 0 , T ] , R d ) .
Since ∈ ( u ) is concave and increasing, there exist a positive number “b” such that
C ( u ) ≤ b ( 1 + u ) , on u ≥ 0 ,
Let a = sup t 0 ≤ t ≤ T ( | f ( 0 , t ) | 2 ∨ | g ( 0 , t ) | 2 ) < ∞ , then
( | f ( x , t ) | 2 ∨ | g ( x , t ) | 2 ) ≤ 2 ( | f ( 0 , t ) | 2 ∨ | g ( 0 , t ) | 2 ) + 2 ( | f ( x , t ) − f ( 0 , t ) | 2 ∨ | g ( x , t ) − g ( 0 , t ) | 2 ) ≤ 2 a + 2 c ( | x | 2 ) ≤ 2 a + 2 b ( 1 + | x | 2 ) ≤ 2 ( a + b ) ( 1 + | x | 2 ) .
By the linear growth condition and c = 2 ( a + b ) , we conclude that the approximate solution x n ( t ) converges to x ( t ) in the sense of stochastic delay differential equation, i.e. Equation (1) hence the proof is completed.
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.
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.
The authors declare no conflicts of interest regarding the publication of this paper.
Emenonye, E.C. and Anonwa, D. (2023) A Comparative Survey of an Approximate Solution Method for Stochastic Delay Differential Equations. Applied Mathematics, 14, 196-207. https://doi.org/10.4236/am.2023.143012