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  September 2018 Early access  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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[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.

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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