September  2018, 23(7): 2695-2708. doi: 10.3934/dcdsb.2018087

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

1. 

Department of Mathematics, University of Salerno, Fisciano (SA), Italy

2. 

Department of Engineering and Computer Science and Mathematics, University of L'Aquila, L'Aquila (AQ), Italy

* Corresponding author

Received  October 2016 Revised  January 2018 Published  March 2018

Fund Project: 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.

Citation: Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster. On the stability of $\vartheta$-methods for stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2695-2708. doi: 10.3934/dcdsb.2018087
References:
[1]

H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, CWI Monographs 3. North-Holland, Amsterdam, 1986. Google Scholar

[2]

A. Bryden and D. J. Higham, On the boundedness of asymptotic stability regions for the stochastic theta method, BIT, 43 (2003), 1-6. doi: 10.1023/A:1023659813269. Google Scholar

[3]

E. Buckwar and T. Sickenberger, A comparative linear mean-square stability analysis of Maruyama-and Milstein-type methods, Math. Comput. Simul., 81 (2011), 1110-1127. doi: 10.1016/j.matcom.2010.09.015. Google Scholar

[4]

D. ConteZ. Jackiewicz and B. Paternoster, Two-step almost collocation methods for Volterra integral equations, Appl. Math. Comp., 204 (2008), 839-853. doi: 10.1016/j.amc.2008.07.026. Google Scholar

[5]

D. Conte and B. Paternoster, Multistep collocation methods for Volterra integral equations, Appl. Numer. Math., 59 (2009), 1721-1736. doi: 10.1016/j.apnum.2009.01.001. Google Scholar

[6]

D. ConteR. D'Ambrosio and B. Paternoster, Two-step diagonally-implicit collocation based methods for Volterra integral equations, Appl. Numer. Math., 62 (2012), 1312-1324. doi: 10.1016/j.apnum.2012.06.007. Google Scholar

[7]

X. DingQ. Ma and L. Zhang, Convergence and stability of the split-step-method for stochastic differential equations, Comput. Math. Appl., 60 (2010), 1310-1321. doi: 10.1016/j.camwa.2010.06.011. Google Scholar

[8]

D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38 (2000), 753-769. doi: 10.1137/S003614299834736X. Google Scholar

[9]

D. J. Higham, A-stability and stochastic mean-square stability, BIT, 40 (2000), 404-409. doi: 10.1023/A:1022355410570. Google Scholar

[10]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302. Google Scholar

[11]

P. Hu and C. Huang, The stochastic $\vartheta$-method for nonlinear stochastic Volterra integro-differential equations, Abs. Appl. Anal. , (2014), 583930, 13pp. Google Scholar

[12]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin Heidelberg, 1992. Google Scholar

[13]

Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267. doi: 10.1137/S0036142992228409. Google Scholar

[14]

C. Shi, Y. Xiao and C. Zhang, The convergence and MS stability of exponential Euler method for semilinear stochastic differential equations, Abs. Appl. Anal., 2012 (2012), Art. ID 350407, 19 pp. Google Scholar

[15]

Z. Wang, Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients, Stat. Prob. Lett., 78 (2008), 1062-1071. doi: 10.1016/j.spl.2007.10.007. Google Scholar

[16]

C. H. Wen and T. S. Zhang, Rectangular method on stochastic Volterra equations, Int. J. Appl. Math. Stat., 14 (2009), 12-26. Google Scholar

[17]

C. H. Wen and T. S. Zhang, Improved rectangular method on stochastic Volterra equations, J. Comput. Appl. Math., 235 (2011), 2492-2501. doi: 10.1016/j.cam.2010.11.002. Google Scholar

[18]

X. Zhang, Euler schemes and large deviations for stochastic Volterra equations with singular kernels, J. Diff. Eq., 244 (2008), 2226-2250. doi: 10.1016/j.jde.2008.02.019. Google Scholar

show all references

References:
[1]

H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, CWI Monographs 3. North-Holland, Amsterdam, 1986. Google Scholar

[2]

A. Bryden and D. J. Higham, On the boundedness of asymptotic stability regions for the stochastic theta method, BIT, 43 (2003), 1-6. doi: 10.1023/A:1023659813269. Google Scholar

[3]

E. Buckwar and T. Sickenberger, A comparative linear mean-square stability analysis of Maruyama-and Milstein-type methods, Math. Comput. Simul., 81 (2011), 1110-1127. doi: 10.1016/j.matcom.2010.09.015. Google Scholar

[4]

D. ConteZ. Jackiewicz and B. Paternoster, Two-step almost collocation methods for Volterra integral equations, Appl. Math. Comp., 204 (2008), 839-853. doi: 10.1016/j.amc.2008.07.026. Google Scholar

[5]

D. Conte and B. Paternoster, Multistep collocation methods for Volterra integral equations, Appl. Numer. Math., 59 (2009), 1721-1736. doi: 10.1016/j.apnum.2009.01.001. Google Scholar

[6]

D. ConteR. D'Ambrosio and B. Paternoster, Two-step diagonally-implicit collocation based methods for Volterra integral equations, Appl. Numer. Math., 62 (2012), 1312-1324. doi: 10.1016/j.apnum.2012.06.007. Google Scholar

[7]

X. DingQ. Ma and L. Zhang, Convergence and stability of the split-step-method for stochastic differential equations, Comput. Math. Appl., 60 (2010), 1310-1321. doi: 10.1016/j.camwa.2010.06.011. Google Scholar

[8]

D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38 (2000), 753-769. doi: 10.1137/S003614299834736X. Google Scholar

[9]

D. J. Higham, A-stability and stochastic mean-square stability, BIT, 40 (2000), 404-409. doi: 10.1023/A:1022355410570. Google Scholar

[10]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546. doi: 10.1137/S0036144500378302. Google Scholar

[11]

P. Hu and C. Huang, The stochastic $\vartheta$-method for nonlinear stochastic Volterra integro-differential equations, Abs. Appl. Anal. , (2014), 583930, 13pp. Google Scholar

[12]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin Heidelberg, 1992. Google Scholar

[13]

Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267. doi: 10.1137/S0036142992228409. Google Scholar

[14]

C. Shi, Y. Xiao and C. Zhang, The convergence and MS stability of exponential Euler method for semilinear stochastic differential equations, Abs. Appl. Anal., 2012 (2012), Art. ID 350407, 19 pp. Google Scholar

[15]

Z. Wang, Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients, Stat. Prob. Lett., 78 (2008), 1062-1071. doi: 10.1016/j.spl.2007.10.007. Google Scholar

[16]

C. H. Wen and T. S. Zhang, Rectangular method on stochastic Volterra equations, Int. J. Appl. Math. Stat., 14 (2009), 12-26. Google Scholar

[17]

C. H. Wen and T. S. Zhang, Improved rectangular method on stochastic Volterra equations, J. Comput. Appl. Math., 235 (2011), 2492-2501. doi: 10.1016/j.cam.2010.11.002. Google Scholar

[18]

X. Zhang, Euler schemes and large deviations for stochastic Volterra equations with singular kernels, J. Diff. Eq., 244 (2008), 2226-2250. doi: 10.1016/j.jde.2008.02.019. Google Scholar

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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