Research paper
Improved ϑ-methods for stochastic Volterra integral equations

https://doi.org/10.1016/j.cnsns.2020.105528Get rights and content

Highlights

  • Theta-methods improve the stability properties of existing direct quadrature methods.

  • The class of methods here introduced improves the stability properties of theta-methods with respect to the convolution test problem.

  • Novel methods inherit the stability properties from those of theta-methods for SDEs.

Abstract

The paper introduces improved stochastic ϑ-methods for the numerical integration of stochastic Volterra integral equations. Such methods, compared to those introduced by the authors in Conte et al. (2018)[14], have better stability properties. This is here made possible by inheriting the stability properties of the corresponding methods for systems of stochastic differential equations. Such a superiority is confirmed by a comparison of the stability regions.

Introduction

We focus our attention on the discretization of stochastic Volterra integral equations (SVIEs)Xt=X0+0ta(t,s,Xs)ds+0tb(t,s,Xs)dWs,t[0,T],where the functions a and b are assumed to have suitable regularity properties in such a way that existence and uniqueness of solutions can be admitted [24], [27], [31]. As regards the right-hand side of (1.1), we assume that the second integral is an Ito^ integral taken with respect to the Brownian motion Ws [21]. The problem is relevant in many applications, especially those concerning stochastic dynamical systems with memory, such as in economy (general stock, insurance, portfolio, financial markets) [1], [2], [32], [40] and engineering [37]. Their numerical approximation has raised a significant interest in the recent literature, through various numerical techniques: for instance, stochastic collocation [9], [36] and spline interpolation methods [28], wavelets based numerical schemes [19], [25], [26], Petrov-Galerkin methods [20], direct quadrature methods via rectangular rule [33], [34]. For most of the methods in the existing literature, the stability issues were almost unexplored.

In order to further improve direct quadrature methods, the authors have recently introduced in Cardone et al. [10], Conte et al. [14] the so-called stochastic ϑ-method for (1.1), defined as followsYn=Y0+hi=0n1(ϑa(tn,ti+1,Yi+1)+(1ϑ)a(tn,ti,Yi))+hi=0n1b(tn,ti,Yi)Vi,where Y0=X0, h=tn+1tn, n=0,1,,N and Vi is a standard Gaussian random variable, i.e., it is N(0,1)-distributed. Under suitable regularity assumptions on the coefficients a and b of (1.1), the stochastic ϑ-method (1.2) is convergent of order 1/2, i.e., there exists a real constant C such thatE[(X(tn)Yn)]Ch12,for any fixed tn=nh[0,T] and sufficiently small values of h, where E denotes the expected value. In particular, the stability analysis of the ϑ-method and some proposed variants has been carried out in Conte et al. [14]. This analysis also showed the stability properties of existing direct quadrature methods [33], [34] and the improvement gained by ϑ-methods (1.2).

Here we propose a class of revised ϑ-methods in order to further improve their stability properties, by inheriting the mean-square stability properties of the corresponding ϑ-method for stochastic differential equations (SDEs).

The paper is organized as follows: Section 2 provides the stability analysis of the exact solution to the proposed test equations and recalls the main stability results given in Conte et al. [14] for (1.2); Section 3 analyzes the connections among ϑ-methods for stochastic integral and differential equations when applied to the same test equation; in Section 4 we introduce an improved version of (1.2) for SVIEs inheriting the mean-square stability properties from the analogous methods for SDEs and provide examples of stability regions. The effectiveness of the improved methods is discussed in Section 5 through selected numerical experiments on given nonlinear problems; some conclusions are given in Section 6.

Section snippets

Stability issues

The stability analysis is here provided with respect to the following test equations introduced in Conte et al. [14]: the basic test equationXt=X0+0tλXsds+0tμXsdWs,λ,μRand the convolution test equationXt=X0+0t(λ+σ(ts))Xsds+0tμXsdWs,λ,μ,σR.Such equations derive from the basic and convolution test equations employed in the stability analysis of numerical methods for deterministic VIEs [3], [15], [16], by including an additional stochastic term.

Let us now analyze the stability properties of

Connection with SDEs

There is a deep connection among test equations for SDEs and for SVIEs. Indeed, as observed in the previous section, the basic test Eq. (2.1) is equivalent to the linear test equation for SDEs (2.3). As concerns the convolution test Eq. (2.2), it can be written as the 2 by 2 linear system of SDEs (2.4).

We observe that the recurrence relation (2.6) corresponds to the recurrence relation of stochastic ϑ-method for SDEs applied to the linear test Eq. (2.3), see [22]. However, if we apply the

Improved stochastic ϑ-method

As we have highlighted in the previous section, the recurrence relation (3.3) provides better stability properties with respect to (2.7). Our goal is now revising the ϑ-method (1.2) in order to develop a family of methods showing (3.3) as recurrence relation when applied to the convolution test Eq. (2.2). To achieve the purpose, we propose a novel quadrature rule for the approximation of the deterministic integral in (1.1). Indeed, we evaluate (1.1) in tn and split the deterministic integral as

Numerical evidence on nonlinear problems

This section is focused on providing the numerical evidence originating from the application of the improved methods introduced in Section 4 on a selection of nonlinear problems. Specifically, we consider

  • the nonlinear SVIE [25]:Xt=11018000ttanh(Xs)sech2(Xs)ds+1200tsech(Xs)dWs,for t[0,0.55], whose exact solution isX(t)=arcsinh(120Wt+sinh(110));

  • the nonlinear problem [35], [38], [39]Xt=1+0te(ts)sin(Xs)ds+0te(ts)cos(Xs)dWs,for t[0,1].

The results, contained in Figs. 7 and 8 confirm the

Conclusions

We have introduced the family of improved ϑ-methods (4.3) for the numerical solution of SVIEs (1.1). The improvement lies in achieving better stability properties with respect to the convolution test problem (2.2), namely the improved method applied to (2.2) provides the same recurrence relation of the ϑ-method for SDEs, applied to the equivalent system of SDEs (2.4). The improved method is obtained by the quadrature formula (4.2) for the approximation of the deterministic integral in (1.1). A

Author contribution

We declare that the contribution given by the authors to the manuscript “Improved theta-methods for stochastic Volterra integral equations” has been equal in each part of the process.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors are grateful to the anonymous reviewers for their precious comments. This work is supported by GNCS-INDAM project and by PRIN2017-MIUR project. The authors are members of the INdAM Research group GNCS.

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      Jensen’s Inequality [37] Quite recently, we noted that Conte et al. [34] also studied the mean square stability of the convolution test problem (4.1). Compared to the results in [34], here we obtain explicit conditions for the mean square stability.

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