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Almost sure exponential stability of an explicit stochastic orthogonal RungeKuttaChebyshev method for stochastic delay differential equations
Advances in Difference Equations volume 2015, Article number: 304 (2015)
Abstract
Compared with EulerMaruyama type schemes, there is a lack of studies on the stability of RungeKutta type methods applied to stochastic delay differential equations (SDDEs). This paper is concerned with filling this imbalance. The focus is on the almost sure exponential stability of an explicit stochastic RungeKuttaChebyshev (SROCK) method for an Itôtype linear test equation, which is analyzed by applying the techniques based on a discrete semimartingale convergence theorem.
Introduction
Stability analysis of numerical methods for stochastic differential equations (SDEs) has recently attracted an increasing interest. Most researchers are concerned with two kinds of stability, i.e., almost sure stability [1–7] and moment stability [8–13], of the numerical solutions to SDEs as well as SDDEs. Generally, almost sure stability is less restrictive than moment stability, and almost sure stability results are more difficult to establish if deriving from the moment stability by the Chebyshev inequality and the BorelCantelli lemma. The situation has been improved since the martingale techniques were introduced to investigate the almost sure stability. By the discrete semimartingale convergence theorem (cf. [1]), the numerical stability of SDDEs has been examined, for example, by [4, 5].
To the best knowledge of authors, there is no similar result about almost sure stability of RungeKutta type methods for SDDEs, and nearly all existing results concerned with EulerMaruyama type schemes. Recently, stabilized explicit RungeKutta schemes have proved successful for solving SDEs, which are called SROCK (stochastic orthogonal RungeKuttaChebyshev) method; see, for example, [14, 15]. In this paper, we investigate the almost sure stability of the SROCK method applied to SDDEs. Consider Itô SDDEs of the form
for every \(t \geq0\). Here time delay \(\tau> 0\). The initial function \(y(t)=\psi(t)\) when \(t \in[{\tau},0]\). We further assume that the initial data is independent of Wiener measure driving the equation and \(w(t)\) is a scalar Brownian motion on the complete probability space \((\Omega ,\mathcal{F},{\mathcal{F}}_{t\geq0},\mathbb{P})\) with a filtration satisfying the usual conditions. Moreover, \(f,g: \mathbb {R}\times\mathbb{R} \to\mathbb{R}\) are Borelmeasurable functions.
The rest of this paper is organized as follows. In the next section, we propose the SROCK method for SDDEs. Our main stability results will be derived in Section 3.
The SROCK method and preliminary results
In the following, we employ an equidistant step points \(\mathcal{I}_{\Delta t} = \{t_{0}, t_{1}, \ldots, t_{N}\}\) where the time step size is a submultiple of the delay τ, i.e., \(\Delta t={\tau}/{m}\) for a given positive integer m, and the nth step point is denoted by \(t_{n} = n \Delta t\) for \(0 \leq n \leq N\). The numerical approximation of \(y(t)\) at \(t_{n}\) is denoted by \(Y_{n}\), and we denote the increment \(w(t_{n+1})  w(t_{n})\) by \(J_{n}\). Next we introduce the SROCK method for solving SDDEs (1.1), which is given by
where \(\{ K_{n}^{(i)} \}\) and \(\{ Z_{n}^{(i)} \}\) are the stage values defined by
and
Here parameters \(\alpha\in[0,1/2]\) and \(\beta_{i} =(i1)^{2}/\nu^{2}\) for \(i=1,\ldots, \nu\).
Let \(C([{\tau},0]; \mathbb{R})\) be the family of continuous functions φ from \([ \tau,0]\) to \(\mathbb{R}\), equipped with the supremum norm \(\\varphi\=\sup_{ \tau\leq\theta\leq0}\varphi(\theta )\). Also, denote by \({C}_{\mathcal{F}_{0}}^{b}([ \tau,0];\mathbb{R})\) the family of bounded, \(\mathcal{F}_{0}\)measurable, \(C([{\tau},0]; \mathbb{R})\)valued random variables.
Now we give some definitions on the almost sure exponential stability of exact and numerical solutions to SDDEs (cf. [16]).
Definition 2.1
The solution \(y(t,\psi)\) to SDDEs (1.1) is said to be almost surely exponentially stable if there exists a constant \(\eta> 0\) such that
for any initial data \(\psi\in {C}_{\mathcal{F}_{0}}^{b}([ \tau ,0];\mathbb{R})\).
Definition 2.2
The solution \(Y_{n}\) to numerical scheme (2.1) is said to be almost surely exponentially stable if there exists a constant \(\gamma> 0\) such that
for any bounded variables \(\psi(\vartheta\Delta t)\) when \(\vartheta \Delta t \in[{\tau},0]\).
For the purpose of stability, we assume that \(f(0,0)=g(0,0)=0\), which implies that (1.1) admits the equilibrium solution \(y(t) = 0\) corresponding to the initial condition \(\psi(t) = 0\) for \(t\in[\tau, 0]\). As a standing hypothesis, we shall impose the following local Lipschitz condition (cf. [7]) on the coefficients f and g.

(A1)
For each integer D, there exists a positive constant \(K_{D}\) such that, for all \(y_{1}, y_{2}, z_{1}, z_{2}\in R\) with \(y_{1} \veey_{2} \veez_{1} \veez_{2} \leq D\), \(f(y_{1},z_{1})f(y_{2},z_{2})^{2}\veeg(y_{1},z_{1})g(y_{2},z_{2})^{2}\leq K_{D}(y_{1}y_{2}^{2}+z_{1}z_{2}^{2})\), where ∨ is the maximal operator.
In what follows we introduce the result of almost sure stability of SDDEs (1.1). The proof of the following lemma can be found in [4].
Lemma 2.3
Let Assumptions (A1) hold. Assume that there are four nonnegative constants \(\lambda_{1},\dots ,\lambda_{4}\) such that
for \(y,z\in\mathbb{R}\). If
then the trivial solution of (1.1) is almost surely exponentially stable.
To explain our idea, we cite the discrete semimartingale convergence theorem as follows (see also [4]).
Theorem 2.4
Let \(\{A_{j}\}\), \(\{U_{j}\}\) be two sequences of nonnegative random variables such that both \(A_{j}\) and \(U_{j}\) are \(\mathcal{F}_{j1}\)measurable for \(j=1,2,\dots\), and \(A_{0}=U_{0}=0\) a.s. Let \(\mathcal{M}_{j}\) be a realvalue local martingale with \(\mathcal {M}_{0}=0\) a.s. Let ζ be a nonnegative \(\mathcal{F}_{0}\)measurable random variable. Assume that \(\{X_{j}\}\) is a nonnegative semimartingale with the DoobMayer decomposition
If \(\lim_{j\rightarrow+\infty}A_{j}<+\infty\) a.s. then for almost all \(\omega\in\Omega\),
Almost sure asymptotic exponential stability of numerical solution
Consider the linear SDDE
It seems that the stability of approximate solutions to (3.1) using RungeKutta type methods is still an open problem. Here we consider the almost sure stability of the linear equation
which is the same test model as [17]. In this section, our aim is to examine how the SROCK method can reproduce the almost sure exponential stability of the exact solution of (3.2). By applying Lemma 2.3, the exact solution of (3.2) is almost surely exponentially stable when \(a<(b^{2}+c^{2})\). Now we give a main result of the almost sure stability of the approximate solution (2.1).
Theorem 3.1
Suppose that the conditions of Lemma 2.3 are satisfied. Then the approximate solution (2.1) applied to test model (3.2) is almost surely exponentially stable if the step size Δt satisfies
where \(T_{\nu}(x)\) is defined as a Chebyshev polynomial of the first kind of degree ν.
Proof
Applying (2.2) to test model (3.2), we have \(K_{n}^{(1)}=T_{0} (1+\frac{a\Delta t}{\nu^{2}} )Y_{n}\) and \(K_{n}^{(2)}=T_{1} (1+\frac{a\Delta t}{\nu^{2}} )Y_{n}\). Next, by the threeterm recurrence relation for Chebyshev polynomials, it is easy to prove that
for \(i=1,\dots,\nu1\). Then, from (2.2), we have
Similarly, we have
Note that
where \(M_{n}=(J_{n}^{2}\Delta t) (1+2\frac{a\Delta t}{\nu^{2}}\alpha )^{2} [b T_{\nu2} (1+\frac{a\Delta t}{\nu^{2}} )Y_{n}+cZ_{n}^{(\nu 1)} ]^{2}+2 T_{\nu}(1+\frac{a\Delta t}{\nu^{2}}) (1+2\frac{a\Delta t}{\nu^{2}}\alpha ) [b T_{\nu2} (1+\frac{a\Delta t}{\nu^{2}} )Y_{n}+cZ_{n}^{(\nu1)} ] Y_{n} J_{n}\).
For any positive constant \(C > 1\), we have
Therefore, by (3.5) and (3.6), we obtain
for any nonnegative integer ℓ. Summing up both sides of inequality (3.7) from \(\ell=0\) to \(n1\) (\(n\geq1\)), we have
Let \(\mathcal{M}_{n} = \sum_{\ell=0}^{n1}C^{(\ell+1)\Delta t}M_{\ell}\). Note that the expectation values \(E(J_{n1}^{2}\Delta t) = 0\), \(E(J_{n1}) = 0\), moreover, \(Y_{n1}\) and \(Z_{n1}^{(\nu1)}\) are \(\mathcal{F}_{(n1)\Delta t}\)measurable, then we have
which implies that \(\mathcal{M}_{n}\) is a martingale with \(\mathcal{M}_{0}=0\).
When \(t_{n}\leq\tau\), from (2.3) and (3.8), we have
According to Theorem 2.4, we denote the right side of inequality (3.9) by \(X_{n}\). Then, let \(\zeta=Y_{0}^{2} + 2 c^{2}\Delta t (1+2\frac{a\Delta t}{\nu^{2}}\alpha )^{2} \sum_{\ell=0}^{n1}C^{(\ell +1)\Delta t} \psi^{2}(t_{\ell}+\beta_{\nu1}\Delta t\tau)\), \(U_{n}=0\), and
where \(H_{1}(C)=C^{\Delta t}+ [T_{\nu}(1+\frac{a\Delta t}{\nu^{2}}) ]^{2}+2 b^{2}\Delta t (1+2\frac{a\Delta t}{\nu^{2}}\alpha )^{2} [T_{\nu2}(1+\frac{a\Delta t}{\nu^{2}}) ]^{2}\). There exists a unique \(C^{*}>1\) such that \(H_{1}(C^{*})=0\) if
Applying Theorem 2.4, we therefore have \(\lim_{n\rightarrow \infty} X_{n} < +\infty\), which means
When \(t_{n}>\tau\), that is, \(n>m\), we have
Then, by using (3.4) and (3.8), we have
and
where \(\zeta_{0}=Y_{0}^{2} + 2 c^{2}\Delta t (1+2\frac{a\Delta t}{\nu ^{2}}\alpha )^{2} \sum_{\ell=0}^{m1}C^{(\ell+1)\Delta t} \psi^{2}(t_{\ell}+\beta_{\nu1}\Delta t\tau)\), \(H_{2}(C)=H_{1}(C)+2 c^{2}\Delta t (1+2\frac{a\Delta t}{\nu^{2}}\alpha )^{2} C^{m\Delta t} [T_{\nu 2}(1+\frac{a\Delta t}{\nu^{2}}) ]^{2}\).
Note that
for any \(C>1\), and \(H_{2}(\infty)>0\).
Obviously, the condition (3.3) yields \(H_{2}(1)<0\), which implies that there exists a unique \(C^{*}>1\) such that \(H_{2}(C^{*})=0\). We therefore have \(\lim_{n\rightarrow\infty} X_{n} < +\infty\) with Theorem 2.4, which means
by (3.12). Choose the \(\gamma> 0\), such that \(C^{*}=e^{\gamma}\) and hence
We therefore obtain
as required.
Finally, (3.3) also implies (3.10). This completes the proof of Theorem 3.1. □
Next, we state how to choose a parameter α and the stage number ν to obtain almost surely stable numerical solution based on Theorem 3.1.
Corollary 3.2
Suppose that conditions of Lemma 2.3 are satisfied. The approximate solution (2.1) applied to test model (3.2) is almost surely exponentially stable if we choose the parameter \(\alpha=\frac{\nu ^{2}}{2a\Delta t}\) and \(a\Delta t\) satisfies
where stage number \(\nu\geq2\).
Proof
The inequality \(a\Delta t \leq{\nu^{2}}\) guarantees that \(\frac{\nu ^{2}}{2a\Delta t}\in(0,1/2]\), hence choosing \(\alpha=\frac{\nu ^{2}}{2a\Delta t}\) satisfies the definition of the SROCK method and also simplifies the left hand side of (3.3) into \([T_{\nu}(1+\frac{a\Delta t}{\nu^{2}} ) ]^{2}\). Finally, \([T_{\nu}(1+\frac{a\Delta t}{\nu^{2}} ) ]^{2}<1\) if \(2\nu^{2}< a\Delta t<0\) and \(T_{\nu}(1+\frac{a\Delta t}{\nu^{2}} )\neq\pm1\) such that the inequality (3.3) is valid. This completes the proof of Corollary 3.2 by using Theorem 3.1. □
Now we consider the case \(4< a\Delta t < 0\).
To guarantee the sufficient condition (3.3), \((1+2\frac {a\Delta t}{\nu^{2}}\alpha )^{2}\) should be as small as possible such that (3.3) is valid. Therefore, it is a good choice to set \(\alpha=1/2\) and a small ν because \((1+2\frac{a\Delta t}{\nu^{2}}\alpha )^{2}\) is a monotonically decreasing and continuous function of the parameter α on \([0,1/2]\).
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Acknowledgements
This work was partially supported by EInstitutes of Shanghai Municipal Education Commission (No. E03004), Natural Science Foundation of Shanghai (No. 14ZR1431300) and Innovation Program of Shanghai Municipal Education Commission (No. 14YZ078).
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Authors’ contributions
The main idea of this paper was proposed by QG and QG wrote the paper. JZ participated in the proof of Theorem 3.1 and helped to draft the manuscript. All authors read and approved the final manuscript.
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Guo, Q., Zhong, J. Almost sure exponential stability of an explicit stochastic orthogonal RungeKuttaChebyshev method for stochastic delay differential equations. Adv Differ Equ 2015, 304 (2015). https://doi.org/10.1186/s1366201506420
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MSC
 65C30
 60H10
 65L06
Keywords
 stochastic delay differential equations
 discrete semimartingale convergence theorem
 almost sure stability
 Chebyshev method
 RungeKutta method
 explicit schemes