The Convergence between Approximate Solutions and the Unique Solutions of SDEs

In this paper, we focus on Euler-Maruyama’s approximate solutions of SDEs. Thus, we will be able to show that the unique solution x(t )o f the stochastic differential equation converge to approximate solution.


Introduction
Nowadays, we know that not all the solutions of stochastic differential equations have the explicit expressions.Thus, we often attempt to seek for approximate solution rather than the accurate solution, such as Euler-Maruyama's approximate solutions.Of course, there exist amounts of conclusions between approximate solutions and the unique solutions of SDEs.Based on these facts, we will be able to show that the unique solution x(t) of the stochastic differential equation converge to Euler-Maruyama's approximate solution in this paper.
Firstly, we will give some preconditions.Let (Ω, F, P ) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions.Throughout this paper, unless otherwise specified, we let be both Borel measurable.We will consider the d-dimensional stochastic differential equation of Itô type with initial value x(t 0 ) = x 0 .Secondly, let us illustrate the definition of Euler-Maruyama's approximate solutions.They are defined as follows: For every integer n ≥ 1, define x n (t) = x 0 , and then for t ( Note that if define for t 0 ≤ t ≤ T .Then from the equations ( 2) and (3), we can obtain the expression of Euler-Maruyama's approximate solutions as follows: For the Euler-Maruyama's approximate solutions, the following results are well-known.Of course, we will list two important conditions before results for convenience: There exists two positive constants K, K such that Theorem 1.1 [2] Under the linear growth condition ( 6), for all n ≥ 1, we have where

Assume that the linear growth condition (6) holds and x(t) is the unique solution of the equation (1). Then there exists a positive constant
Based on the above works, the aims of this paper is to discuss that the unique solution x(t) of the stochastic differential equation converge to Euler-Maruyama's approximate solution.

The convergence between Euler-Maruyama's approximate solutions and the unique solution.
Now we can state our main result.
Let the linear growth condition (6) hold.Assume that the equation ( 1) has the unique solution (that means the pathwise uniqueness) x(t).Then the Euler-Maruyama's approximate solutions x n (t) converge to x(t) in the sense that

Some Lemmas
Before giving the proofs of main theorems, we need to show the following some lemmas based on Theorem 1.1.

Lemma 2.2
Under the linear growth condition ( 6), for all n ≥ 1 and p ≥ 2, sup where C is a positive constant related to K, p, T.
Proof Fix n ≥ 1 and p ≥ 2 arbitrarily.Note from equation ( 4) that for Using the Hölder inequality, the martingale inequality [Lemma 2.4] and the definition of Euler-Maruyama approximate solutions, we can then derive that E|x n (r)| p ds.

Lemma 2.3 Under the linear growth condition (6), for all n ≥ 1 and t
where C p is a positive constant related to K, p, T.
Hence, we can know that, where and

Proofs of main results
In this subsection, we start to prove our main theorems.
Note by the property (1) and Lemma 2.3 thatĒ|x n (t) − ȳn (t)| p = E|x n (t) − xn (t)| p → 0 as n → ∞,So we can see that both x0 (t) and z0 (t) are solutions of equation (1) with respect to the Brownian motion B0 (t) under the same initial condition.According to the uniqueness of solutions of stochastic differential equations, we get x0 (t) = z0 (t) which contradicts inequality (12).So the conclusion of this theorem is true.