Fast $\theta$-Maruyama scheme for stochastic Volterra integral equations of convolution type: mean-square stability and strong convergence analysis

Abstract

In this paper, a fast $\theta$-Maruyama method is proposed for solving stochastic Volterra integral equations of convolution type with singular and Hölder continuous kernels based on the sum-of-exponentials approximation. Furthermore, the average storage O(N) and the calculation cost $O(N^2)$ of $\theta$-Maruyama scheme are reduced to $O(\log N)$ and $O(N\log N)$ for $T\gg 1$ or $O(\log ^2 N)$ and $O(N\log ^2 N)$ for $T\approx 1$, respectively, which implies that the fast $\theta$-Maruyama scheme is confirmed to improve the computational efficiency of the $\theta$-Maruyama method. Under the local Lipschitz and linear growth conditions, strong convergence of the given numerical scheme are obtained. Then, for the linear test equation, we show the asymptotic behavior of solutions in mean square sense. Further, we obtain the explicit structure of the stability matrices and some numerical results of the mean-square stability for the fast $\theta$-Maruyama method applied to the linear test equation. Finally, some numerical experiments are also given to illustrate the effectiveness of the method.

Appendix

Before going to the proof of the Theorem 4.1, we need some preparatory lemmas. First, we establish a variation of constant formula for Eq. (4.1).

Lemma 8.1

(A Variation of Constant Formula For SVIEs) For Eq. (4.1) with $\alpha >-1$ and $\beta >-\frac{1}{2}$, the following statement:

$$\begin{aligned} x(t)&=E_{(\alpha +1),1}(\lambda \Gamma (\alpha +1)t^{(\alpha +1)})x_0\nonumber \\&\quad +\mu \Gamma (\beta +1)\int _0^t(t-s)^{\beta } E_{(\alpha +1),(\beta +1)}(\lambda \Gamma (\alpha +1)(t-s)^{(\alpha +1)})x(s)\text{ d }B(s) \end{aligned}$$

(8.1)

holds for all $t\in [0,T]$.

Proof

With the help of [Theorem 3.1, Li and Peng 2011] and [Theorem 2.3, Anh et al. 2019], the proof can be obtained easily. $\square$

To prove Theorem 4.1, we are going to give the following results about the estimates of the Mittag-Leffler function.

Lemma 8.2

Suppose that $\lambda \ne 0$, $-1<\alpha <1$ and $\beta$ is a real number. There exist positive constants $M(\alpha ,\beta )$ and $\bar{M}(\alpha )$ depending on $\alpha$ and $\beta$ such that for any $t\ge 0$, it holds that if $\beta \ne \alpha$, then

$$\begin{aligned} |E_{(\alpha +1),(\beta +1)}(\lambda t^{\alpha +1})|\le \frac{M(\alpha ,\beta )}{|\lambda |\max \{1,t^{\alpha +1}\}}. \end{aligned}$$

Moreover, if $\beta =\alpha$, then we can obtain

$$\begin{aligned} |E_{(\alpha +1),(\alpha +1)}(\lambda t^{\alpha +1})|\le \frac{\bar{M}(\alpha )}{\lambda ^2\max \{1,t^{2(\alpha +1)}\}}. \end{aligned}$$

Proof

See [Theorems 1.3 and 1.4, Podlubny 1999] and [Theorem 2, Cong et al. 2018] or [Lemma 2, Tuan 2017]. $\square$

Lemma 8.3

Suppose that

$$\begin{aligned} (\mu \Gamma (\beta +1))^2\int _0^{\infty }s^{2\beta } (E_{(\alpha +1),(\beta +1)}(\lambda \Gamma (\alpha +1)s^{(\alpha +1)}))^2\text{ d }s<1. \end{aligned}$$

If $\beta \ne \alpha$, let $2\alpha -2\beta +1>0$ and $\eta \in (0,\alpha -\beta +\frac{1}{2})$ be arbitrary, then

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }\sup (\mu \Gamma (\beta +1))^2&\int _0^{t}(t-s)^{2\beta }\cdot \\&(E_{(\alpha +1),(\beta +1)}(\lambda \Gamma (\alpha +1)(t-s)^{(\alpha +1)}))^2\frac{\max \{1,t^{2\eta }\}}{\max \{1,s^{2\eta }\}}\text{ d }s<1. \end{aligned}$$

Specially, If $\beta =\alpha$, then the condition of $\eta$ expands to $(0,\alpha +1)$.

Proof

The proof is similar to [Theorem 1.1, Cong et al. 2018]. $\square$

The proof of Theorem 4.1

Let us only consider the case of $\beta \ne \alpha$ because the proof for $\beta =\alpha$ is similar to [Theorem 3, Doan et al. (2020)]. Choose and fix an arbitrary $\hat{\eta }\in (\eta ,\alpha -\beta +\frac{1}{2})$. Then, it is sufficient to show that

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }\sup t^{2\hat{\eta }}\frac{{\mathbb {E}}|x(t)|^2}{{\mathbb {E}}|x_0|^2}<\infty . \end{aligned}$$

(8.2)

We are now proving the inequality (8.2) by contradiction. Then, there exists an increasing sequence $\{t_n\}$ tending to $\infty$ such that $\phi _n:=\max \{1,t^{2\hat{\eta }}_n\}\frac{{\mathbb {E}}|x(t)|^2}{{\mathbb {E}}|x_0|^2}$ satisfies

$$\begin{aligned} \phi _n=\max \Big \{\max \{1,t^{2\hat{\eta }}_n\}\frac{{\mathbb {E}}|x(t)|^2}{{\mathbb {E}}|x_0|^2}:\quad t\in [0,t_n] \Big \} \end{aligned}$$

and $\lim \limits _{n\rightarrow \infty }\phi _n=\infty$. Replacing $t=t_n$ in Eq. (8.1) and by virtue of Lemma 8.2, we arrive at

$$\begin{aligned} \frac{{\mathbb {E}}|x(t_n)|^2}{{\mathbb {E}}|x_0|^2}&\le \frac{M(\alpha )}{|\lambda |\max \{1,t_n^{2(\alpha +1)}\}}+(\mu \Gamma (\beta +1))^2\cdot \\&\int _0^{t_n}(t_n-s)^{2\beta } (E_{(\alpha +1),(\beta +1)}(\lambda \Gamma (\alpha +1)(t_n-s)^{(\alpha +1)}))^2\frac{{\mathbb {E}}|x(s)|^2}{{\mathbb {E}}|x_0|^2}\text{ d }s, \end{aligned}$$

which implies that

$$\begin{aligned} \phi _n&\le \frac{M(\alpha )\max \{1,t_n^{2\hat{\eta }}\}}{|\lambda |\max \{1,t_n^{2(\alpha +1)}\}}+\phi _n(\mu \Gamma (\beta +1))^2\cdot \\&\int _0^{t_n}(t_n-s)^{2\beta } (E_{(\alpha +1),(\beta +1)}(\lambda \Gamma (\alpha +1)(t_n-s)^{(\alpha +1)}))^2\frac{\max \{1,t_n^{2\hat{\eta }}\}}{\max \{1,s^{2\hat{\eta }}\}}\text{ d }s. \end{aligned}$$

Therefore,

$$\begin{aligned} \phi _n\Big (1-(\mu \Gamma (\beta +1))^2&\int _0^{t_n}(t_n-s)^{2\beta } (E_{(\alpha +1),(\beta +1)}(\lambda \Gamma (\alpha +1)(t_n-s)^{(\alpha +1)}))^2\cdot \\&\frac{\max \{1,t_n^{2\hat{\eta }}\}}{\max \{1,s^{2\hat{\eta }}\}}\text{ d }s\Big )\le \frac{M(\alpha )\max \{1,t_n^{2\hat{\eta }}\}}{|\lambda |\max \{1,t_n^{2(\alpha +1)}\}}. \end{aligned}$$

Since $2\hat{\eta }<2\alpha -2\beta +1<2(\alpha +1)$, it follows that

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }\sup \frac{M(\alpha )\max \{1,t_n^{2\hat{\eta }}\}}{|\lambda |\max \{1,t_n^{2(\alpha +1)}\}}=0, \end{aligned}$$

which contradicts to the definition of $\lim \limits _{n\rightarrow \infty }\phi _n=\infty$ and Lemma 8.3. The proof is complete. $\square$