Mean Square Numerical Methods for Initial Value Random Differential Equations ()
In this paper, the random Euler and random Runge-Kutta of the second order methods are used in solving random differential initial value problems of first order. The conditions of the mean square convergence of the numerical solutions are studied. The statistical properties of the numerical solutions are computed through numerical case studies.
1. Introduction
Random differential equations (RDE) are defined as differential equations involving random inputs. In recent years, increasing interest in the numerical solution of (RDE) has led to the progressive development of several numerical methods. This paper is interested in studying the following random differential initial value problem (RIVP) of the form:
(1.1)
Randomness may exist in the initial value or in the differential operator or both. In [1,2], the authors discussed the general order conditions and a global convergence proof is given for stochastic Runge-Kutta methods applied to stochastic ordinary differential equations (SODEs) of Stratonovich type. In [3,4], the authors discussed the random Euler method and the conditions for the mean square convergence of this problem. In [5], the authors considered a very simple adaptive algorithm based on controlling only the drift component of a time step. Platen, E. [6] discussed discrete time strong and weak approximation methods that are suitable for different applications. Other numerical methods are discussed in [7-12].
In this paper the random Euler and random RungeKutta of the second order methods are used to obtain an approximate solution for equation (1.1). This paper is organized as follows. In Section 2, some important preliminaries are discussed. In Section 3, the existence and uniqueness of the solution of random differential initial value problem is discussed and the convergence of random Euler and random Runge-Kutta of the second order methods is discussed. In Section 4, the statistical properties for the exact and numerical solutions are studied. Section 5 presents the solution of some numerical examples of first order random differential equations using random Euler and random Runge-Kutta of the second order methods showing the convergence of the numerical solutions to the exact ones (if possible). The general conclusions are presented in the end section.
2. Preliminaries
2.1. Mean Square Calculus [13]
Definition1. Let us consider the properties of a class of real r.v.’s 
whose second moments, 
are finite. In this case, they are called“second order random variables”, (2.r.v’s).
Definition 2. The linear vector space of second order random variables with inner product, norm and distance, is called an 
-space.
A s.p. 
is called a “second order stochastic process” (2.s.p) if for
, the r.v’s 
are elements of 
-space.
A second order s.p. 
is characterized by
,
.
2.1.1. The Convergence in Mean Square
A sequence of r.v’s 
converges in mean square (m.s) to a random variable 
if 
i.e. 
or
where lim is the limit in mean square sense.
2.1.2. Mean-Square Differentiability
The random process 
is mean-square differentiable at t if 
exists, and is denoted by
3. Random Initial Value Problem (RIVP)
3.1. Existence and Uniqueness
Let us have the random initial value problem
(3.1)
where 
is second order random process. This equation is equivalent to integral equation
(3.2)
Theorem (3.1.1)
If we have the random initial value problem (3.1) and suppose the right-hand side function 
is continuous and satisfies a mean square (m.s) Lipschitz condition in its second argument:
(3.3)
where C is a constant or
(3.4)
where c(t) is a continuous function {because in every finite interval c(t) ≤ constant}.
then the solution of equation (3.1) exists and is unique.
The proof
The existence can be proved by using successive approximations. Let
(3.5)
and for n 
1
(3.6)
For n 
1 we obtain:
where
(3.7)
For n > 1 we obtain:
(3.8)
Successively, we can obtain the following:
Hence
(3.9)
Since:
is convergent for finite t,
(3.10)
hence we can have the following
(3.11)
Accordingly,
Hence:
This yield 
Then 
exists. i.e.
(3.12)
Since 
is the general solution of equation (3.6) and 
is the general solution of equation (3.2).
To prove the uniqueness of the solution, let 
is a solution of the initial-value problem (3.1), or, which is the same, of the integral equation (3.2), and 
is the solution of
(3.13)
to prove the uniqueness of the solution we want to prove that
(3.14)
By subtraction (3.2) and the corresponding integral equation for 
Since 
then:
(3.15)
(3.16)
i.e;
(3.17)
where
.
(3.18)
From equation (3.17) we have:
(3.19)
Note that: at 
we obtain 
then: 
From (3.19) 
must satisfy the following condition:
(3.20)
which is in contradiction with being an independent free constant, hence the only solution of the integral equation (3.17) is
(3.21)
Hence 
i.e., the solution of equation (3.1) exists and is unique.
3.2. The Convergence of Euler Scheme for Random Differential Equations in (m.s.) Sense
Let us have the random differential equation
(3.22)
where X0 is a random variable and the unknown 
as well as the right-hand side f (X,t) are stochastic processes defined on the same probability space.
Definitions [6,7]
• Let g: 
is an m.s. bounded function and let h > 0 then The “m.s. modulus of continuity of g” is the function
• The function g is said to be m.s uniformly continuous in T if:
Note that:
(The limit depends on h because g is defined at every t so we can write
)
In the problem (3.22), we find that the convergence of this problem depends on the right hand side (i.e. 
then we want to apply the previous definition on 
hence:
Let 
be defined on 
where S is bounded set in 
Then we say that f is “randomly bounded uniformly continuous” in S, if
(note that
)
3.2.1. Random Mean Value Theorem for Stochastic Processes
The aim of this section is to establish a relationship between the increment 
of a 2-s.p. and its m.s. derivative 
for some 
lying in the interval 
for
. This result will be used in the next section to prove the convergence of the random Euler method.
Lemma (3.3.2) [6,7]
Let 
is a 2-s.p., m.s. continuous on interval 
. Then, there exists 
such that
(3.25)
The proof
Since 
is m.s. continuous, the integral process
is well defined and the correlation function
is well defined, is a deterministic continuous function on T × T.
For each fixed r, the function 
is continuous and by the classic mean value theorem for integrals, it follows that:
(3.26)
Note that by definition of 
expression (3.26) can be written in the form
Since 
We must prove that for the value 
satisfying (3.26) one get:
(3.27)
The proof of (3.27)
As
(3.28)
and since:
then by substituting in (3.28)
(3.29)
And since:
then by substituting in (3.28) we have:
i.e. 
we obtain
Theorem (3.3.1) [6,7]
Let 
be a m.s. differentiable 2-s.p. in 
and m.s. continuous in
. Then, there exists 
such that
,
The proof
The result is a direct consequence of Lemma (3.3.2) applied to the 2-s.p. 
And the integral formula
(3.30)
The proof of (3.30)
Let 
be a m.s. differentiable on T and let the ordinary function 
be continuous on 
whose partial derivative 
exist If 
(3.31)
Then
(3.32)
Let 
in Equations (3.31) and (3.32) we have the useful result that:
If 
is m.s. Riemann integrable on T then:
Then we have:
3.2.2. The Convergence of Random Euler Scheme
In this section we are interested in the mean square convergence, in the fixed station sense, of the random Euler method defined by
(3.33)
where 
and 
are 2-r.v.’s , 
, 
and f:
, 
satisfies the following conditions:
C1: 
is randomly bounded uniformly continuousC2: 
satisfies the m.s. Lipschitz condition
where
(3.34)
Note that under hypothesis C1 and C2, we are interested in the m.s. convergence to zero of the error
(3.35)
where 
is the theoretical solution 2-s.p. of the problem (3.22),
.
Taking into account (3.22), and Theorem (3.3.1), one getsSince from (3.22) we have at 
then
Note 
and we can use 
instead of 
and from Theorem (3.3.1) at 
then we have:
then
Note that we deal with the interval 
and hence 
was the starting in the problem (3.22) and here 
is the starting and since Euler method deal with solution depend on previous solution and if we have 
instead of 
then we can use 
instead of
.
Then the final form of the problem (3.22) is
, for some
(3.36)
Now we have the solution of problem (3.22) is 
At 
then 
and the solution of Euler method (3.33) is 
Then we can define the error
By (3.33) and (3.36) it follows that
This implies
Hence
(3.37)
Since:
(3.38)
Since the theoretical solution 
is m.s. bounded in
, 
and Under hypothesis C1, C2 We obtain
• 
• 
(*)
Since 
is Lipschitz constant (from C2) and from Theorem (3.3.1) we have 
and note that the two points are 
and 
in (*) then we have:
Since 
and 
• 
Then by substituting in (3.38) we have
(3.39)
Then by substituting in (3.37) we have
Since:
is geometrical sequence.
Then:
Then we get
Taking into account that 
where
.
(3.40)
Note that:
The term: 
as
(
as
)
(3.41)
And the second term:
we have:
(3.42)
The first limit in (3.42) equal zero and:
The computation of 
as follows:
Let 
then by tacking the logarithm of the two sides we have:
By using the (L’Hospital’s Rule):
(3.43)
Then 
which implies that
hence
By substituting in (3.42):
(3.44)
By substituting from (3.44) and (3.42) in (3.40) hence 
i.e., 
converge in m.s to zero as
hence
.
3.3. The Convergence of Runge-Kutta of Second Order Scheme for Random Differential Equations in Mean Square Sense
In this section we are interested in the mean square convergence, in the fixed station sense, of the random Runge-Kutta of second order method defined by
(3.45)
where 
and 
are 2-r.v.’s, 
, 
and f:
, 
satisfies the following conditions:
C1: 
is randomly bounded uniformly continuousC2: 
satisfies the m.s. Lipschitz condition
where
(3.46)
Note that under hypothesis C1 and C2, we are interested in the m.s. convergence to zero of the error
(3.47)
where 
is the theoretical solution 2-s.p. of the problem (3.22),
.
Taking into account (3.22), and Theorem (3.3.1), one getsSince from (3.22) we have at 
then
Note 
and we can use 
instead of 
And from Theorem (3.3.1) at 
then we obtain
Note that we deal with the interval 
and hence 
was the starting in the problem (3.22) and here 
is the starting and since Euler method deal with solution depend on previous solution and if we have 
instead of 
we can use 
instead of 
then the final form of the problem (3.22) is
, for some 
(3.48)
Now we have the solution of problem (3.22) is 
At 
then 
and the solution of Runge-Kutta of 2 order method (3.45) is 
Then we can define the error
By (3.45) and (3.48) it follows that
Then we obtain:
By taking the norm for the two sides:
(3.49)
Since:
(3.50)
Since the theoretical solution 
is m.s. bounded in
, 
and Under hypothesis C1, C2 We have
• 
• 
(*)
where 
Is Lipschitz constant (from C2) and:
From Theorem (3.3.1) we have
and note that the two points Are 
and 
in (*) then
where 
and 
• 
Then by substituting in (3.50) we have
(3.51)
And another term:
Since:
• 
• 
where
. Is Lipschitz constant (from C2) and:
From Theorem (3.3.1) we have
and note that the two points are 
and 
in (*) then we have:
where 
and 
And the last term:
Then by substituting in (3.49) we have
Then we have:
Since:
is geometrical sequence then we have:
Then we get:
Taking into account that 
where
(3.52)
Note that:
The term:
and the second term:
we have:
(3.53)
The first limit in (3.53) equals zero and:
The computation of
is as follows:
Let 
then by tacking the logarithm of the two sides we have:
By using the (L’Hospital’s Rule):
(3.54)
Then 
hence:
By substituting in (3.53):
(3.55)
By substituting from (3.55) and (3.53) in (3.51) then we obtain 
i.e. 
converges in m.s to zero as 
hence 
4. Some Results
Theorem 4.1
Let {Xn, n = 0, 1, ···}, {Yn, n = 0, 1, ···} be sequences of 2-r.v’s over the same probability space and let a and b be deterministic real numbers.
Suppose: 
and 
Then:
1) 
2) 
3) 
4) 
5) 
Definition 4.1 [13]. “The convergence in probability”
A sequence of r.v’s 
converges in probability to a random variable 
as 
if
Definition 4.2 [13]. “The convergence in distribution”
A sequence of r.v’s 
converge in distribution to a random variable 
as 
if
Lemma (4.1) [13]
The convergence in m.s implies convergence in probability
Lemma (4.2) [13]
The convergence in probability implies convergence in distribution
Theorem 4.2
If 
then PDF of 
PDF of 
i.e.; 
Proof Since we have shown that If 
then
i.e., if 
then 
Then we have: 
then
5. Numerical Examples
Example (5.1)
The differential equation with random term in it and random initial condition
K, D are independent Poisson random variables with joint PDF
1) The exact solution,
2) The numerical solution
Using the Random Euler Method:
at n = 1
at n = 2
at n = 3
at n = 4
and so on…
Then the general numerical solution is
i.e.,
.
This can be written in another form:
.
We can prove that:
1) 
Proof Since 
(if and only if) 
Then:
where 
i.e; 
We can verify theorem (4.1) as follows
2) 
Proof
Then:
i.e.;
.
3) 
Proof Since 
Then we have:
Then by taking the limit:
i.e.
.
4) 
Proof
i.e.,
.
5) 
Proof Since 
Let us define Z = D. Then the inverse transformation is:
, D = Z then we have D = Z and
Then:
Since 
then 
hence
.
For a numerical solution:
since 
Let 
then 
Then:
where 
Then by taking the limit we have
i.e.; 
B. Using the Random Runge-Kutta method:
at n = 0
At n = 1
At n = 2
Then the general solution is:
,
,
.
This can be written in another form:
.
We can prove that:
1) 
Proof
Since 
(if and only if) 
where 
i.e.;
.
Verification of Theorem (4.1):
2) 
Proof
i.e.;
.
3) 
Proof Since 
then:
i.e.;
.
4) 
Proof
i.e. 
5) 
Since 
Let us define Z = D. Then the inverse transformation is:
D = z then we have D = z and 
Since 
then 
this implies
.
Numericallysince 
Let 
then 
where 
i.e.;
.
Example (5.2)
Solve the problem
The exact solution
The numerical solution by the Euler method:
AT n = 1
At n = 2
At n = 3
And so on….
Then the general numerical solution:
.
It is clear that:
1) 
Since 
Verification of Theorem (4.1)
It is clear that:
2) 
3) 
Since 
4) 
5) 
Then 
which implies
Then 
which implies
Then: 
6. Conclusions
The initially valued first order random differential equations can be solved numerically using the random Euler and random Runge-Kutta methods in mean square sense. The existence and uniqueness of the solution have been proved. The convergence of the presented numerical techniques has been proven in mean square sense. The results of the paper have been illustrated through some examples.
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