Abstract
In this paper, our aim is to develop a compensated split-step θ (CSSθ) method for nonlinear jump-diffusion systems. First, we prove the convergence of the proposed method under a one-sided Lipschitz condition on the drift coefficient, and global Lipschitz condition on the diffusion and jump coefficients. Then we further show that the optimal strong convergence rate of CSSθ can be recovered, if the drift coefficient satisfies a polynomial growth condition. At last, a nonlinear test equation is simulated to verify the results obtained from theory. The results show that the CSSθ method is efficient for simulating the nonlinear jump-diffusion systems.
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1 Introduction
The aim of this paper is to study the strong convergence of the CSSθ method for the following nonlinear jump-diffusion systems:
for \(t>0\), with \(X(0^{-})=X_{0}\in \mathbb{R}^{n}\), where \(X(t^{-})\) denotes \(\lim_{s\rightarrow t^{-}} X(s)\), \(f:\mathbb{R} ^{n} \to \mathbb{R}^{n}\), \(g:\mathbb{R}^{n} \to \mathbb{R}^{n \times m}\), \(h:\mathbb{R}^{n} \to \mathbb{R}^{n}\), \(W ( t ) \) is an m-dimensional Wiener process, and \(N ( t ) \) is a scalar Poisson process with intensity λ.
Most of the studies concerned with numerical analysis for stochastic differential equations with jumps (SDEwJs) are based on the assumption of globally Lipschitz continuous coefficients, for example, [1–6]. However, they cannot be applied to many real-world models, such as financial models [7] and biology models [8], which violate the global Lipschitz assumptions. Hence, the development of numerical methods for SDEwJs under a non-globally Lipschitz condition has become a focus point.
Firstly, we review some achievements of the numerical analysis for highly nonlinear SDEs. Here, we highlight work by Higham et al. [9], Hutzenthaler et al. [10], Szpruch and Mao [11], Mao and Szpruch [12], Huang [13], Zong et al. [14, 15].
However, the development of numerical methods for nonlinear jump-diffusion systems with non-globally Lipschitz continuous coefficients is not as fast as nonlinear SDEs. There are only few results on the numerical methods for nonlinear SDEwJs. For example, Higham and Kloeden proved the strong convergence and its order of the split-step backward Euler (SSBE) method and compensated split-step backward Euler (CSSBE) method for nonlinear jump-diffusion system in [16, 17]. Huang applied the split-step θ (SSθ) method to SDEwJs, but he only studied the mean-square stability of the SSθ method for SDEwJs in [13]. To the best of our knowledge, there is no result about the strong convergence of the CSSθ method for SDEwJs with non-globally Lipschitz continuous coefficients. The main difference of this paper from our previous work [5] is that we deal with SDEwJs with non-globally Lipschitz condition on the drift coefficient f.
The outline of the paper is as follows. In Section 2, we introduce some notions and assumptions for SDEwJs. In Section 3, we construct the CSSθ method for nonlinear SDEwJs. In Section 4, the strong convergence of the numerical solutions produced by the CSSθ method is investigated. The convergence rate is studied in Section 5. Finally, a nonlinear numerical experiment is given to verify the convergence and efficiency of the proposed method.
2 Conditions on the SDEwJs
Let \(( \Omega ,\mathcal{F},\mathbb{P} ) \) be a complete probability space with a filtration \(\{ \mathcal{F}_{t} \} _{t\geq 0}\), which satisfies the usual conditions, i.e., the filtration is continuous on the right and \(\mathcal{F}_{0}\) contains all \(\mathbb{P}\)-null sets. Let \(\langle \cdot ,\cdot \rangle \) denote the Euclidean scalar product, and \(\vert \cdot \vert \) denote both the Euclidean vector norm in \(\mathbb{R}^{n}\) and the Frobenius matrix norm in \(\mathbb{R}^{n\times m}\). For simplicity, we also denote \(a \wedge b = \min \{ a,b \} \), \(a \vee b = \max \{ a,b \} \).
Now, we give the following assumptions on the coefficients f, g and h.
Assumption 2.1
The functions f, g, h in (1.1) are \(C^{1}\), there exist constants K, \(L_{g}\) and \(L_{h}>0\), such that the drift coefficient f satisfies a one-sided Lipschitz condition,
and the diffusion and jump coefficients satisfy the global Lipschitz conditions,
We also assume that all moments of the initial solution are bounded, that is, for any \(p\in [1,+\infty )\) there exists a positive constant C, such that
Lemma 2.1
Under Assumption 2.1, equation (1.1) has a unique cadlag solution on \([0,+\infty )\).
Proof
See [16], and for a more relaxed conditions see [18]. □
From Assumption 2.1, we have the following estimates:
It follows that
where \(L=\max \{ (K+\frac{1}{2}),2L_{g},2L_{h},\frac{1}{2} \vert f(0) \vert ^{2},2 \vert g(0) \vert ^{2},2 \vert h(0) \vert ^{2} \} \).
3 The compensated split-step θ-method
First defining
we can rewrite the jump-diffusion system (1.1) in the following form:
where
is the compensated Poisson process.
Note that \(f_{\lambda }\) satisfies the one-sided Lipschitz condition with lager constant; that is,
Then we can get
where \(L_{\lambda }=\max \{ (K+\lambda \sqrt{L_{h}}+\frac{1}{2}),2L _{g},2L_{h},\frac{1}{2} \vert f_{\lambda }(0) \vert ^{2},2 \vert g(0) \vert ^{2},2 \vert h(0) \vert ^{2} \} \).
Now we define the CSSθ method for the jump-diffusion system (1.1) by \(Y_{0}=X(0)\) and
where \(\theta \in [ 0,1 ] \), \(\Delta t>0\), \(Y_{n}\) is the numerical approximation of \(X ( t_{n} ) \) with \(t_{n} = n \cdot \Delta t\). Moreover, \(\Delta W_{n}: = W ( t_{n + 1} ) - W ( t_{n} ) \), \(\Delta \tilde{N} _{n}: = \tilde{N} ( t_{n + 1} ) - \tilde{N} ( t_{n} ) \) are independent increments of the Wiener process and Poisson process, respectively.
If we have \(\theta =1\), the CSSθ method becomes the CSSBE method in [16].
Since the CSSθ method is an implicit scheme, we need to make sure that equation (3.4) has a unique solution \(Y_{n}^{*}\) given \(Y_{n}\).
In fact, under the one-sided Lipschitz condition (3.2) with \(\theta \Delta t K_{\lambda }< 1\), equation (3.4) admits a unique solution (see [12]). Meanwhile, if \(K_{\lambda }< 0\), then \(\theta \Delta t K_{\lambda }< 1\) holds for any \(\Delta t > 0\). Hence, we define
From now on we always assume that \(\Delta t \le \Delta \).
4 Strong convergence on finite time interval \([0,T]\)
First, for \(t \in [ t_{n},t_{n + 1} ) \), we define the step function:
where \(N_{T}\) is the largest number such that \(N_{T}\Delta t\leq T\), and \(I_{A}\) is the indicator function for the set A, i.e.,
Then we define the continuous-time approximations
Thus we can rewrite (4.2) in integral form:
It is easy to verify that \(\overline{Y}(t_{n})=Y_{n}\), that is, \(\overline{Y}(t)\) is a continuous-time extension of the discrete approximation \(\{ Y_{n} \} \).
Now we will prove the strong convergence of the CSSθ method. The main technique of the following proof is based on the fundamental papers [9, 13, 16], we refer to them for a fuller description of some of the technical details.
The following two lemmas show the pth moment properties of the true solutions and numerical solutions.
Lemma 4.1
Let Assumption 2.1 hold, and \(0 <\theta \leq 1\), \(p\ge 1\), \(0 < \Delta t < \min \{ 1,\frac{1}{ 2\theta L_{\lambda }} \} \), then there exists a positive constant A independent of \(N_{T}\) such that
where \(Y_{n}^{*}\) and \(Y_{n}\) are produced by (3.4) and (3.5).
Proof
In the following we assume that M is a positive integer such that \(n\Delta t\leq M \Delta t \le T\).
Squaring both sides of (3.4), we find
and
Substituting (4.5) into (4.4), we have
which gives
where \(\alpha =\frac{2\theta \Delta t L_{\lambda }}{1-2\theta \Delta t L_{\lambda }}\), \(\beta =1+\alpha \). By (3.5) we have
Then by (3.3), (3.4) and (4.5), we get
Hence from (4.7) we have
By the recursive calculation, we can get
Raising both sides to the power p, we can obtain
Notice that
Thus, for \(0\leq M\leq N_{T}\), we obtain
To bound the fourth term on the right side of (4.14), we note that \(Y_{n}^{*}\in \mathcal{F}_{t_{n}}\), \(\Delta W_{n}\) is independent of \(\mathcal{F}_{t_{n}}\) and \(\mathbb{E} \vert \Delta W_{j} \vert ^{2p}\le c _{p} \Delta t^{p}\), where \(c_{p}\) is a constant. Meanwhile, letting \(C=C(p,T,L_{\lambda },\theta )\) be a constant that may change from line to line,
Using a similar approach to the fifth term and noticing that \(\mathbb{E} \vert \Delta \widetilde{N}_{j} \vert ^{2p}\le c_{p} \Delta t^{p}\), we have
Now we bound the sixth term in (4.14), using the Burkholder-Davis-Gundy inequality
Similar to the sixth term, we can bound the seventh term
Also similar to the sixth term, we can bound the eighth term in (4.14),
and the ninth term,
Finally we bound the tenth term in (4.14), by (4.15)-(4.16); we have
Combining (4.15)-(4.21) into (4.14), we obtain
Using the discrete-type Gronwall inequality and noting that \(M\Delta t \leq T\), we obtain
By (4.7), we find that \(\mathbb{E} \sup_{0 \leq n \leq M} \vert Y_{n}^{*} \vert ^{2p}\) is also bounded. □
Lemma 4.2
Let Assumption 2.1 hold, and \(0 < \theta \le 1\), \(p\ge 1\), \(0 < \Delta t < \min \{ 1,\frac{1}{ 2\theta L_{\lambda }} \} \), then the exact solution of (3.1) and the continuous-time extension (4.3) satisfy
where \(A_{1}\) is a positive constant independent of \(N_{T}\).
Proof
From Lemma 1 in [16], we can see that \(\mathbb{E} ( \sup_{ 0 \le t \le T} \vert X(t) \vert ^{2p} ) \) is bounded. Now we prove that \(\mathbb{E} ( \sup_{ 0 \le t \le T} \vert \overline{Y}(t) \vert ^{2p} ) \) is bounded.
From (4.2), we obtain
Let \(s\in [0, \Delta t)\), we have
where
However, \(Y_{n}^{*} = Y_{n}+\theta \Delta t f_{\lambda }(Y _{n}^{*})\) and so, for \(a=s/\Delta t\), we can rewrite equation (4.25) in the following form:
By (4.7), we have
Thus
Now using Doob’s martingale inequality, (3.3) and Lemma 4.1, we have
Since the \(\Delta \widetilde{N}_{j}(s)\) is also a martingale, by a similar method, we get
Then by (4.28), (4.29) and Lemma 4.1, combining \(N_{T}\Delta t\le T\), we have
Then we get the desired results. □
Now we use the above lemmas to prove a strong convergence result.
Remark 4.1
Since \(f(x)\in C^{1}\), i.e. \(f'(x)\) is continuous, \(\vert f'(x) \vert \) is bounded locally. Then by the mean value theorem, there exists a positive constant \(L_{R}\) for each \(R>0\), such that
for all \(x, y,z\in \mathbb{R}^{n}\) with \(\vert x \vert \vee \vert y \vert \le R\).
We note that the function \(f_{\lambda }\) in (3.1) automatically inherits this condition, with a larger \(L_{R}\).
Theorem 4.3
Under Assumption 2.1, let \(0 < \theta \le 1\), \(0 < \Delta t < \min \{ 1,\frac{1}{2\theta L_{\lambda }} \} \), the continuous-time approximate solution \(\overline{Y}(t)\) defined by (4.3) will converge to the true solution \(X(t)\) of (3.1) in the mean-square sense, i.e.
Proof
First, we define
and let
Recall the Young inequality: for \(\frac{1}{p} + \frac{1}{q} = 1\) (\(p,q > 0\)), we have
Thus, for any \(\delta >0\), we have
By Lemma 4.2, then
Similarly, the result can be derived for \(\sigma_{d}\)
so that
Using the bounds of \(X(t)\) and \(\overline{Y}(t)\), we have
Substituting the above inequality into (4.32) leads to
Now we bound the first term on the right-hand side of (4.33). By the definitions of \(X(t)\) and \(\overline{Y}(t)\), combining the fact that \(0<\theta \le 1\), we have
For any \(\tau \in [0,T]\), using the Cauchy-Schwarz inequality and the Doob martingale inequality, we obtain
Applying the local Lipschitz condition (4.30) and Assumption 2.1, we get
To bound the first term inside the parentheses of (4.34), we denote by \(n_{s}\) the integer for which \(s\in [t_{n_{s}}, t_{n_{s+1}}]\) and note that
and hence that
Note that
Thus by the second moments of martingale increments and the moment bound on the numerical solution \(Y_{n}^{*}\), we can obtain
for a constant \(C_{1}=C_{1}(R, T, A)\). Substituting this bound into (4.34) and applying the continuous Gronwall inequality gives
for a constant \(C_{2}=C_{2}(R, T, A)\).
Now combining (4.35) with (4.33), we have
For any given \(\varepsilon >0\), we can choose δ sufficiently small for
and then choose d sufficient large for
and finally choose Δt so that
Thus \(\mathbb{E} [ \sup_{0 \le t \le T} \vert e ( t ) \vert ^{2} ] < \varepsilon \). The proof is completed. □
5 Convergence rate
To prove the convergence rate of the CSSθ method, we give the following assumption.
Assumption 5.1
There exist constants \(D\in \mathbb{R}^{+}\) and \(q\in \mathbb{Z}^{+}\) such that, for all \(a,b \in \mathbb{R}^{n}\),
Firstly, we establish Lemma 5.1 under Assumptions 2.1 and 5.1.
Lemma 5.1
Under Assumptions 2.1 and 5.1, let \(0 < \theta \le 1\), \(0 < \Delta t < \min \{ 1,\frac{1}{2\theta L_{\lambda }} \} \), for any given integer \(r\geq 2\), there exists a positive constant \(E=E(r)\) such that
Proof
Since for any given \(t\in [n\Delta t, (n+1)\Delta t]\), we have \(Y(t)=Y_{n}\), and then by the continuous-time approximate solution \(\overline{Y}(t)\) defined by (4.3), we can get
and hence for \(t-t_{n}\le \Delta t\)
By Assumption 5.1 on \(f_{\lambda }\), and linear growth condition (2.6)-(2.7) for g and h, we have
where \(C_{3}(r)\) and u are both integer constants depending on q from Assumption 5.1 and r. By Lemma 4.1, we obtain
where \(E=E(r)\) a positive constant which depends on r. □
Theorem 5.2
Under Assumptions 2.1 and 5.1, let \(0 < \theta \le 1\), \(0 < \Delta t < \min \{ 1,\frac{1}{2\theta L_{\lambda }} \} \), the continuous-time approximate solution \(\overline{Y}(t)\) defined by (4.3) will converge to the true solution \(X(t)\) of (3.1) with strong order of one half, i.e.
Proof
Let
From the identity
and (4.3), we apply the Itô formula [17] to obtain
where
Using the Assumptions 2.1 and 5.1, and (3.2) we have
where we use \(D_{1}\) to denote a generic constant (independent of Δt) that may change from line to line.
Using the Lemma 4.1, Lemma 4.2 and Lemma 5.1, we have
Now, as in the proof of [9], the Burkholder-Davis-Gundy inequality can be used to get the estimate
Using this in (5.6), we obtain
The result follows from the continuous Gronwall inequality. □
6 Numerical experiments
We consider the following nonlinear stochastic different equation with jumps from [16]:
Define \(f(x(t))=-4x ( t ) -x^{3}(t)\), \(g(x(t))=x(t)\), \(h(x(t))=x(t)\). It is easy to compute that
which implies that \(f(x)\) satisfies the one-sided Lipschitz condition, \(g(x)\) and \(h(x)\) satisfy the global Lipschitz condition, then the Assumptions of Theorem 5.2 hold. That is to say, the numerical solution by our method will converge to the true solution of system (6.1).
To show the convergence of the CSSθ method for system (6.1), we fix \(\Delta t=2^{-14}\), \(T=2\), \(\lambda =1\), \(\theta =0.7\). Noting that the exact solution of nonlinear jump-diffusion system (6.1) is not available, we use the numerical solution by the SSBE method with step size \(\Delta t=2^{-14}\) as the ‘referenced exact solution’ (Theorem 2 in [16] can guarantee its strong convergence) in Figure 1.
CSS θ solution and ‘referenced exact solution’ for system ( 6.1 ).
In Figure 1, we show the numerical solution by the CSSθ method with step size \(\Delta t=2^{-10}\) and the ‘referenced exact solution’. we can easy to find that the CSSθ approximation and the ‘referenced exact solution’ make no major difference between both paths. That is to say the CSSθ method converges to the ‘referenced exact solution’ well. Hence our method is efficient for the nonlinear jump-diffusion systems.
To show the strong convergence order of the CSSθ method with different parameter θ, we fix \(T=2\), \(\lambda =1\) and change \(\theta =0.5, 0.7, 0.9\), respectively. The ‘referenced exact solution’ of system (6.1) is also used by the SSBE method with step size \(\Delta t=2^{-14}\). We simulate the numerical solutions with five different step sizes \(h=2^{p-1}\Delta t\) for \(1\leq p\leq 5\), \(\Delta t=2^{-14} \). The mean-square errors \(\varepsilon = 1/5{,}000\sum_{i = 1}^{5{,}000} \vert Y_{n} ( \omega _{i} ) - X ( T ) \vert ^{2}\), all measured at time \(T=2\), are estimated by trajectory averaging. We plot our approximation to \(\sqrt{\epsilon }\) against Δt on a log-log scale. For reference a dashed line of slope one-half is added.
In Figure 2, we see that the slopes of the four curves appear to match well. Therefore, the results verify the strong convergence order of the proposed method.
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Acknowledgements
This research is partial supported with funds provided by the National Natural Science Foundation of China (Grant Nos. 11501410, 11471243 and 11672207), and in part by Natural Science Foundation of Tianjin (No: 17JCQNJC03800). We thank the anonymous reviewers for their very valuable comments and helpful suggestions which improve this paper significantly.
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Tan, J., Men, W. Convergence of the compensated split-step θ-method for nonlinear jump-diffusion systems. Adv Differ Equ 2017, 189 (2017). https://doi.org/10.1186/s13662-017-1247-6
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DOI: https://doi.org/10.1186/s13662-017-1247-6