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# On the stability of $\vartheta$-methods for stochastic Volterra integral equations

• * Corresponding author

The work is supported by GNCS-Indam project

• The paper is focused on the analysis of stability properties of a family of numerical methods designed for the numerical solution of stochastic Volterra integral equations. Stability properties are provided with respect to the basic test equation, as well as to the convolution test equation. For each equation, stability properties are intended both in the mean-square and in the asymptotic sense. Stability regions are also provided for a selection of methods. Numerical experiments confirming the theoretical study are also given.

Mathematics Subject Classification: Primary: 65C30.

 Citation: • • Figure 1.  Mean-square stability regions in the ($x, y$)-plane with respect to the basic test equation (2).

Figure 2.  Asymptotic stability regions in the ($x, y$)-plane with respect to the basic test equation (2).

Figure 3.  Mean-square stability regions in the ($x, y$)-plane with respect to the basic test equation (2) for values of $\vartheta\geq 1$.

Figure 4.  Mean-square and asymptotic stability regions in the ($x, y$)-plane with respect to the convolution test equation (3) for the stochastic $\vartheta$-method (5) for several choices of $\vartheta$ and $z$.

Figure 5.  Mean-square and asymptotic stability regions in the ($x, y$)-plane with respect to the convolution test equation (3) for $z = -2$ and several choices of $\vartheta$.

Figure 6.  Mean-square of the numerical solution of problem (2), with $\lambda = -8$ and $\mu = 2\sqrt{2}$, obtained by applying methods (5) (blue), (9) (black), (10) (magenta) and (15) (red) with $\vartheta = 1/2$.

Figure 7.  Absolute value of the numerical solution of problem (2), with $\lambda = -8$ and $\mu = 4$, obtained by applying methods (5) (blue), (9) (black), (10) (magenta) and (15) (red) with $\vartheta = 1/2$.

Figure 8.  Mean-square of the numerical solution of problem (3), with $\lambda = -4$, $\mu = 2\sqrt{5}/5$ and $\sigma = -2/h^2$, obtained by applying methods (5) (blue), (9) (black), (10) (magenta) and (15) (red) with $\vartheta = 1$.

Figure 9.  Absolute value of the numerical solution of problem (3), with $\lambda = -1$, $\mu = \sqrt{6}$ and $\sigma = -2/h^2$, obtained by applying methods (5) (blue), (9) (black), (10) (magenta) and (15) (red) with $\vartheta = 1$.

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