Our main results and their proofs are presented in this section. We start off with the second moment in Section 3.1, two cases of the θ are discussed. Then the same structure is used for Section 3.2 to investigate the small moments.
3.1 The second moment
First we discuss the situation for , which has the linear growth condition on both drift and diffusion coefficients. Second we relax the constraint on the drift coefficient when .
The boundedness of the underlying SDE is well known. We state the following theorem and refer the readers to Chapter 5 of [18] for the proof.
Theorem 3.1 Assume that f and g satisfy the local Lipschitz condition. Assume that there exists a negative constant μ and positive constants σ, , such that for any ,
(3.1)
and
(3.2)
If
then the underlying solution of SDE (2.1) is asymptotically bounded in the second moment
(3.4)
Now we consider reproducing this boundedness property by the STM.
3.1.1
Theorem 3.2 Let (3.1), (3.2) and (3.3) hold; furthermore, if f satisfies the linear growth condition
(3.5)
where κ and are positive, then for , the STM solution (2.2) satisfies
Moreover, let the stepsize , then
(3.6)
Proof Due to (3.1), (3.2), (3.3) and (3.5), we obtain
where . Taking expectation on both sides, noting that , yields
(3.7)
where
and
Then, for , we have . From (3.7), we deduce
Let , then assertion (3.6) holds. □
This theorem shows that the STM can reproduce the upper bound of true solution (3.4) for the case of . The result of the EM boundedness, Theorem 5.2 in [19], is reproduced perfectly as a special case.
3.1.2
We try to release the constraint on the drift coefficient when and reproduce the boundedness property in STM as well. To show the theorem of this case, we first present the following lemma.
Lemma 3.3 Let conditions (3.1) and (3.3) hold, then for any with , we have the inequality
where .
Proof
□
For the theorem below, we denote
Theorem 3.4 Let (3.1), (3.2) and (3.3) hold. If , then for any STM (2.2) satisfies
Moreover, let the stepsize , then
(3.8)
Especially, when , thus
(3.9)
Proof Using Lemma 3.3 with , we have
By conditions (3.1)-(3.3) and Lemma 3.3, we have
where , , . Noting that
when , thus , we have easily; when , thus , we still have .
Let , we have
Let , assertion (3.8) and the special case hold. □
Without the linear growth condition on the drift coefficient, this theorem shows that the STM can still reproduce the boundedness property of true solution (3.4). The result of the BEM boundedness, Theorem 5.4 in [19], is recovered perfectly as a special case when .
3.2 The small moment
In this section, we discuss the asymptotic boundedness of STM in the p th moment for small p. First we discuss the situation for , which has a linear growth condition on both drift and diffusion coefficients. Second we release the constraint on the drift coefficient when .
3.2.1
We begin by imposing the linear growth condition on both drift and diffusion coefficients of SDE (2.1):
(3.10)
where κ and a are positive constants. We first present the theorem on the asymptotic boundedness in small moment of the solution of (2.1).
Theorem 3.5 Let (3.10) hold. If there exists a positive constant D such that for any ,
(3.11)
where λ is a positive constant and is a polynomial of with degree i, then there exists such that for all the solution of (2.1) obeys
(3.12)
where C is a positive constant dependent on κ, a, p, D, but independent of .
Following the same technique as the one used in Theorem 5.2 in [18], by choosing the Lyapunov function , it is straightforward to prove this theorem. So we omit it here. Now we give the result for the STM solution.
Theorem 3.6 Let (3.10) and (3.11) hold, and . Then, for any , there exists a pair of constants and such that for and , the STM solution (2.2) satisfies
(3.13)
where is a constant dependent on κ, a, p and D, but independent of and Δt.
Especially, when ,
Proof From (2.2), for we have
For the constant D in (3.11), we have
where
For any we have
Clearly , recalling the fundamental inequality
(3.14)
we have
Hence the conditional expectation
(3.15)
Since is independent of , we have , . By (3.10) we can get
(3.16)
Similarly, we can show that
(3.17)
and
(3.18)
where is a positive constant dependent on κ, and is a positive constant dependent on a. and may change from line to line. Now consider the following fraction:
(3.19)
For , it is obvious that the fraction has an upper bound. Substituting (3.16), (3.17) and (3.18) into (3.15), then using (3.10), (3.11) and the argument for (3.19), we have that
where is a positive constant dependent on κ and p, is a positive constant dependent on κ, a, p and D, and both of them may change from line to line. Taking expectations on both sides, we obtain
(3.20)
For any , by choosing sufficiently small such that and sufficiently small , for and , we have
(3.21)
where is a constant dependent on θ, κ and p. Then further reducing gives that for ,
Using these three inequalities together with (3.21), we have from (3.20) that
(3.22)
Since for any ,
then by further reducing such that for any , we obtain
Together with (3.22), we arrive at
Due to , we have . Then, by iteration and letting , we have
□
The theorem shows that the STM can reproduce the boundedness property of true solution (3.4). The result of the EM boundedness, Theorem 3.2 in [19], is recovered as a special case when .
3.2.2
In this part, we consider the case . One may notice from the next theorem that in this case the parameter θ exists in the conditions, therefore the boundedness of the underlying equation may not be fully reproduced under the same conditions. However, as we stated in Section 1 that the asymptotic moment boundedness of the numerical as a stand-alone result is a key component in the study of numerical stationary distribution. Thus we still keep the next theorem and the problem that if one could construct some θ independent sufficient conditions for this case remains open.
Theorem 3.7 Assume that the drift coefficient satisfies (3.1) and the diffusion coefficient satisfies (3.10) if the following holds for some positive constant λ:
where and D is some positive constant larger than . Then
where and is a positive constant dependent on κ, a, p and D.
Proof We start off with
where , . Then we have
Using (3.14), we have
(3.23)
where
Similar to the proof of Theorem 3.6, we compute that
and
Substituting these three estimates into (3.23), we get
where is a positive constant dependent on κ and p, is a positive constant dependent on κ, a, p and D, and both of them may change from line to line. Taking expectations on both sides, we obtain
For any , by choosing sufficiently small such that , then choose sufficiently small for and . For any and any , we have
Then, by iteration and letting , we have
Then
The proof is complete. □