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\(L^{p }\) (\(p>2\))-strong convergence of multiscale integration scheme for jump-diffusion systems
- Jiaping Wen1Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-019-1956-0
© The Author(s) 2019
- Received: 14 November 2018
- Accepted: 13 January 2019
- Published: 22 January 2019
Abstract
In this paper we shall prove the \(L^{p}\) (\(p>2\))-strong convergence of multiscale integration scheme for stochastic jump-diffusion systems with two-time-scale, which gives a numerical method for effective dynamical systems.
Keywords
- Fast-slow SDEs with jumps
- \(L^{p}\) (\(p>2\))-strong convergence
- Multiscale integration scheme
MSC
- 60H10
- 34F05
1 Introduction
Multiscale jump-diffusion stochastic differential equations arise in many applications and have already been studied widely. What is usually of interest for this kind of system (1) is the time evolution of the slow variable \(x_{t}^{\epsilon }\). Thus a simplified equation, which is independent of the fast variable and possesses the essential features of the system, is highly desirable. On the one hand, while averaging principle [1–6] plays an important role in the research of slow component by getting a reduced equation (2), the difficulty of obtaining the effective equation (2) lies in the fact that the coefficient \(\bar{a}(\cdot )\) is given via expectation with respect to measure \(\mu ^{x}(dy)\), which is usually difficult or impossible to obtain analytically, especially when the dimension m is large. On the other hand, even if we get the reduced equation, the equation cannot be solved explicitly. Therefore, the construction of the efficient computational methods is of great importance. Furthermore, the idea of multiscale integration schemes (cf. [7]) overcomes these difficulties exactly, which solves \(\bar{x}_{t}\) with \(\bar{a}(\cdot )\) being estimated on the fly using an empirical average of the original slow coefficients \(a(\cdot )\) with respect to numerical solutions of the fast processes. This is one of our motivations.
For another significant motivation, a substantial body of work has been done concerning multiscale integration scheme for fast-slow SDEs. Most of the existing research theories discuss the convergence in \(L^{p}\) (\(0< p\leqslant 2\)), even in a weaker sense [4, 5, 8–10]. Nevertheless, convergence in a stronger sense is what we want. In 2007, the \(L^{2}\) averaging principle was proposed for a system, in which slow and fast dynamics were driven by Brownian noises and Poisson noises in [1]. Subsequently, the authors gave a multiscale integration scheme for the result in [9]. In 2015, Xu and Miao extended the result of [1] to the \(L^{p}\) (\(p>2\)) case under assumptions (H1)–(H5) in [2]. A natural question is as follows: Can we also establish the \(L^{p}\) (\(p>2\)) averaging principle by the multiscale integration scheme? It is well known that \(L^{1}\) convergence and \(L^{2}\) convergence cannot conclude \(L^{p}\) (\(p>2\)) convergence. However, \(L^{1}\) convergence and \(L^{2}\) convergence can be deduced by \(L^{p}\) (\(p>2\)) convergence. Once the \(L^{p}\) (\(p>2\)) convergence has been established, then a much bigger degree of freedom for parameter q in research of \(L^{q}\) (\(0< q< p\)) convergence would be obtained.
- (H1):
-
The measurable functions a, b, c, f, g, and h satisfy the global Lipschitz conditions, i.e., there is a positive constant L such thatfor all \(u_{i}\in \mathbb{R}^{n}\), \(v_{i}\in \mathbb{R}^{m}\), \(i=1,2\). Here and below we use \(|\cdot |\) to denote both the Euclidean vector norm and the Frobenius matrix norm.$$ \begin{aligned} & \bigl\vert a(u_{1},v_{1})-a(u_{2},v_{2}) \bigr\vert ^{2}+ \bigl\vert b(u_{1})-b(u_{2}) \bigr\vert ^{2} + \bigl\vert c(u _{1})-c(u_{2}) \bigr\vert ^{2} \\ &\qquad {}+ \bigl\vert f(u_{1},v_{1})-f(u_{2},v_{2}) \bigr\vert ^{2}+ \bigl\vert g(u_{1},v_{1})-g(u_{2}, v _{2}) \bigr\vert ^{2}+ \bigl\vert h(u_{1},v_{1})-h(u_{2},v_{2}) \bigr\vert ^{2} \\ &\quad \leqslant L\bigl( \vert u_{1}-u_{2} \vert ^{2}+ \vert v_{1}-v_{2} \vert ^{2}\bigr) \end{aligned} $$
Remark 1.1
- (H2):
-
a, g, and h are globally bounded.
Remark 1.2
By (H1) and (H2), it is easy to derive that ā in (2) is bounded and satisfies the Lipschitz condition [11].
- (H3):
-
There exist constants \(\beta _{1}>0\) and \(\beta _{j} \in \mathbb{R}\), \(j=2, 3, 4\), which are all independent of \((u_{1},v _{1},v_{2})\), such thatand$$\begin{aligned}& \begin{gathered} v_{1}\cdot f(u_{1},v_{1})\leqslant -\beta _{1} \vert v_{1} \vert ^{2}+\beta _{2}, \\ \bigl(f(u_{1},v_{1})-f(u_{1},v_{2}) \bigr) (v_{1}-v_{2})\leqslant \beta _{3} \vert v _{1}-v_{2} \vert ^{2}, \end{gathered} \end{aligned}$$(3)for all \(u_{1}\in \mathbb{R}^{n}\) and \(v_{1},v_{2}\in \mathbb{R}^{m}\).$$ \bigl(h(u_{1},v_{1})-h(u_{1},v_{2}) \bigr) (v_{1}-v_{2})\leqslant \beta _{4} \vert v _{1}-v_{2} \vert ^{2} $$(4)
- (H4):
-
\(\eta :=-(2\beta _{3}+2\lambda _{2}\beta _{4}+C_{g}+ \lambda _{2}C_{h})>0\), here \(\beta _{3}\) and \(\beta _{4}\) are taken from (3) and (4), \(\lambda _{2}\) is from \(N_{t}^{\epsilon }\) with intensity \(\lambda _{2}/\epsilon \), \(C_{g}\), and \(C_{h}\) are the Lipschitz coefficients for g and h, respectively, i.e.,and$$ \bigl\vert g(u_{1},v_{1})-g(u_{2},v_{2}) \bigr\vert ^{2}\leqslant C_{g}\bigl( \vert u_{1}-u_{2} \vert ^{2}+ \vert v _{1}-v_{2} \vert ^{2}\bigr) $$for all \(u_{1},u_{2}\in \mathbb{R}^{n}\), \(v_{1},v_{2}\in \mathbb{R}^{m}\).$$ \bigl\vert h(u_{1},v_{1})-h(u_{2},v_{2}) \bigr\vert ^{2}\leqslant C_{h}\bigl( \vert u_{1}-u_{2} \vert ^{2}+ \vert v_{1}-v_{2} \vert ^{2}\bigr) $$
- (H5):
-
There exists a constant \(\gamma >0\), which is independent of \((u,v)\), such thatfor all \((u,v)\in \mathbb{R}^{n}\times \mathbb{R}^{m}\).$$ v^{\mathrm{T}}g(u,v)g^{\mathrm{T}}(u,v)v\geqslant \gamma \vert v \vert ^{2} $$
An example that satisfies (H1)–(H5) is \(a(u,v)= \frac{1}{1+(u+v)^{2}}\), \(b(u)=e^{-u^{2}}\), \(c(u)=\sin u\), \(f(u,v)=-1.5( \lambda _{2}+1)\nu \), \(g(u,\nu )=\frac{3+\sin u+\sin \nu }{\sqrt{2}}\), and \(h(u,\nu )=\frac{\sin u+\sin \nu }{\sqrt{2}}\).
It is worth pointing out that the \(L^{p}\) (\(p>2\)) averaging principle under assumptions (H1)–(H5) had been established in [2].
Now, we will introduce the multiscale integration scheme. The scheme is made up of a macro solver to evolve (2) and a micro solver to simulate the fast dynamics in (1):
Concretely speaking, we are concentrating on estimating the \(L^{p}\)-strong error between the solution \(\bar{x}_{t}\) of the effective dynamics (2) and the solution \(X_{n}\) of the multiscale integration scheme (5), (6), and (8) in this paper. Furthermore, we may easily obtain that the solution \(X_{n}\) of the multiscale integration scheme can approximate the solution \(\bar{x} _{t}\) of the effective dynamics in both the sense of \(L^{q}\) (\(0< q< p\)) and the probability by Hölder’s inequality and Chebyshev’s inequality. Then the process of proving the main result can be divided into two parts: \((I')\) the difference between the process \(\bar{x}_{t_{n}}\) and the auxiliary process \(\bar{X}_{n}\) (see Lemma 2.4 below); \((\mathit{II}')\) the difference between the process \(X_{n}\) and the auxiliary process \(\bar{X}_{n}\) (see Lemma 3.8 below).
We now describe the structure of the present paper. In Sect. 2, we introduce some a priori estimates to testify the error between the process \(\bar{x}_{t_{n}}\) and the auxiliary process \(\bar{X}_{n}\). In Sect. 3, we devote ourselves to proving the error between the process \(X_{n}\) and the auxiliary process \(\bar{X}_{n}\). In Sect. 4, based on the above two estimates, we can derive our main result (see Theorem 4.1).
Throughout this paper, we will denote by C or K a generic positive constant which may change its value from line to line. In chains of inequalities, we will adopt C, \(C^{\prime }\), \(C^{\prime \prime }\), … or \(C_{1}\), \(C_{2}\), \(K_{1}\), \(K_{2}\), … to avoid confusion.
2 Some a priori estimates
In this section, we shall give some a priori estimates in the first three lemmas. Then we can apply the obtained results to estimate the difference between the process \(\bar{x}_{t_{n}}\) and the auxiliary process \(\bar{X}_{n}\).
Firstly, we show that the discrete numerical solution \(\bar{X}_{k}\) and the continuous approximation \(\bar{X}(t)\) have 2p bounded moments in the first two lemmas.
Lemma 2.1
Proof
Lemma 2.2
Proof
Secondly, we show that the continuous-time approximation remains close to the step functions Z(s) in a strong sense.
Lemma 2.3
Proof
Lastly, we prove a strong convergence result for \(\bar{X}(t)\).
Lemma 2.4
Proof
3 Strong convergence of the scheme
In this section, some a priori estimates would be provided in the first seven lemmas. Then we can use the established estimates to get the error between the process \(X_{n}\) and the auxiliary process \(\bar{X}_{n}\).
Now, we firstly show the 2pth moment estimates for the processes \(z_{t}^{n}\), \(X_{n}\), and \(Y_{m}^{n}\).
Lemma 3.1
Proof
The proof of the following lemma is similar to Sect. 2. We omit the details.
Lemma 3.2
Lemma 3.3
Proof
Next, we give the 2pth moment deviation between two successive iterations of the micro-solver.
Lemma 3.4
Proof
Lemma 2.1 in [9] shows that \(z_{t}^{n}\) is statistically equivalent to a shifted and rescaled version of \(y_{t}^{\epsilon }\), with x being a parameter, that is, \(z_{t}^{k}\sim y_{t-t_{k}/\epsilon }^{\epsilon }\).
Then we establish the mixing properties of the auxiliary processes \(z_{t}^{n}\). Note that \(\bar{a}(X_{n})\) is the average of \(a(X_{n},y)\) with respect to \(\mu ^{X_{n}}\), which is the invariant measure induced by \(z_{t}^{n}\). We denote \(z_{m}^{n}=z_{m\delta t}^{n}\).
Lemma 3.5
Proof
It remains to estimate the mean-square term, and the proof for the term is similar to the method in [9] (Lemma 2.6). We omit the details. Thus we obtain the desired result (41). □
Afterwards, we establish the 2pth moments deviation between (9) and its numerical approximation (8).
Lemma 3.6
Proof
Lemma 3.7
Proof
Finally, we estimate the difference between the process \(X_{n}\) and the auxiliary process \(\bar{X}_{n}\).
Lemma 3.8
Proof
4 Main result
Now we can state and prove our main theorem readily.
Theorem 4.1
Proof
5 Conclusions
In this paper, the \(L^{p}\) (\(p>2\))-strong convergence of the multiscale integration scheme has been studied for the two-time-scale jump-diffusion systems. By Lemmas 2.4 and 3.8, we obtained our desired main result. The results in [2, 9] are extended in this paper. First, we provide a numerical method for the \(L^{p}\) (\(p>2\)) averaging principle in [2]; second, in [9], the authors only studied \(L^{2}\) convergence of the multiscale integration scheme, and we extended the result into the \(L^{p}\) (\(p>2\)) case.
Declarations
Acknowledgements
The first author is very grateful to Associate Professor Jie Xu for his encouragement and useful discussions.
Availability of data and materials
Not applicable.
Funding
The author acknowledges the support provided by NSFs of China No. U1504620 and Youth Science Foundation of Henan Normal University Grant No. 2014QK02.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
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Authors’ Affiliations
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