Research paperImproved ϑ-methods for stochastic Volterra integral equations
Introduction
We focus our attention on the discretization of stochastic Volterra integral equations (SVIEs)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 It 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 followswhere and Vi is a standard Gaussian random variable, i.e., it is -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 thatfor any fixed and sufficiently small values of h, where 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 equationand the convolution test equationSuch 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]:for whose exact solution is
- •
the nonlinear problem [35], [38], [39]for .
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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