Stability issues for selected stochastic evolutionary problems: a review. (English) Zbl 1432.65009
Summary: We review some recent contributions of the authors regarding the numerical approximation of stochastic problems, mostly based on stochastic differential equations modeling random damped oscillators and stochastic Volterra integral equations. The paper focuses on the analysis of selected stability issues, i.e., the preservation of the long-term character of stochastic oscillators over discretized dynamics and the analysis of mean-square and asymptotic stability properties of \(\vartheta\)-methods for Volterra integral equations.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 45D05 | Volterra integral equations |
Keywords:
stochastic differential equations; stochastic multistep methods; stochastic Volterra integral equations; mean-square stability; asymptotic stabilityReferences:
| [1] | Burrage, K.; Lenane, I.; Lythe, G.; Numerical methods for second-order stochastic differential equations; SIAM J. Sci. Comput.: 2007; Volume 29 ,245-264. · Zbl 1144.65004 |
| [2] | Burrage, K.; Lythe, G.; Accurate stationary densities with partitioned numerical methods for stochastic differential equations; SIAM J. Numer. Anal.: 2009; Volume 47 ,1601-1618. · Zbl 1197.60068 |
| [3] | D’Ambrosio, R.; Moccaldi, M.; Paternoster, B.; Numerical preservation of long-term dynamics by stochastic two-step methods; Discret. Cont. Dyn. Syst. B: 2018; Volume 23 ,2763-2773. · Zbl 1396.65011 |
| [4] | D’Ambrosio, R.; Moccaldi, M.; Paternoster, B.; Rossi, F.; Stochastic Numerical Models of Oscillatory Phenomena; Artificial Life and Evolutionary Computation: Berlin/Heidelberg, Germany 2018; ,59-69. |
| [5] | Schurz, H.; The invariance of asymptotic laws of linear stochastic systems under discretization; Z. Angew. Math. Mech.: 1999; Volume 6 ,375-382. · Zbl 0938.34072 |
| [6] | Strömmen Melbö, A.H.; Higham, D.J.; Numerical simulation of a linear stochastic oscillator with additive noise; Appl. Numer. Math.: 2004; Volume 51 ,89-99. · Zbl 1060.65007 |
| [7] | Burrage, P.M.; Burrage, K.; Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise; Numer. Algor.: 2014; Volume 65 ,519-532. · Zbl 1291.65017 |
| [8] | Burrage, P.M.; Burrage, K.; Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise; J. Comput. Appl. Math.: 2014; Volume 236 ,3920-3930. · Zbl 1245.65007 |
| [9] | Vilmart, G.; Weak second order multi-revolution composition methods for highly oscillatory stochastic differential equations with additive or multiplicative noise; SIAM J. Sci. Comput.: 2014; Volume 36 ,1770-1796. · Zbl 1320.65019 |
| [10] | Wen, C.H.; Zhang, T.S.; Rectangular method on stochastic Volterra equations; Int. J. Appl. Math. Stat.: 2009; Volume 14 ,12-26. |
| [11] | Wen, C.H.; Zhang, T.S.; Improved rectangular method on stochastic Volterra equations; J. Comput. Appl. Math.: 2011; Volume 235 ,2492-2501. · Zbl 1221.65023 |
| [12] | Zhang, X.; Euler schemes and large deviations for stochastic Volterra equations with singular kernels; J. Differ. Eq.: 2008; Volume 244 ,2226-2250. · Zbl 1139.60329 |
| [13] | Higham, D.J.; An algorithmic introduction to numerical simulation of stochastic differential equations; SIAM Rev.: 2011; Volume 43 ,525-546. · Zbl 0979.65007 |
| [14] | Kloeden, P.E.; Platen, E.; ; The Numerical Solution of Stochastic Differential Equations: Berlin/Heidelberg, Germany 1992; . · Zbl 0752.60043 |
| [15] | Conte, D.; D’Ambrosio, R.; Paternoster, B.; On the stability of ϑ-methods for stochastic Volterra integral equations; Discret. Cont. Dyn. Syst. B: 2018; Volume 23 ,2695-2708. · Zbl 1398.65348 |
| [16] | Buckwar, E.; Horvath-Bokor, R.; Winkler, R.; Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations; BIT Numer. Math.: 2006; Volume 46 ,261-282. · Zbl 1121.60071 |
| [17] | Hairer, L.; Lubich, C.; Wanner, G.; ; Geometric Numerical Integration: Berlin/Heidelberg, Germany 2006; . · Zbl 1094.65125 |
| [18] | Brunner, H.; van der Houwen, P.J.; ; The Numerical Solution of Volterra Equations: Amsterdam, The Netherlands 1986; . · Zbl 0611.65092 |
| [19] | Conte, D.; D’Ambrosio, R.; Paternoster, B.; Two-step diagonally-implicit collocation based methods for Volterra integral equations; Appl. Numer. Math.: 2012; Volume 62 ,1312-1324. · Zbl 1251.65172 |
| [20] | Conte, D.; Jackiewicz, Z.; Paternoster, B.; Two-step almost collocation methods for Volterra integral equations; Appl. Math. Comput.: 2008; Volume 204 ,839-853. · Zbl 1160.65066 |
| [21] | Conte, D.; Paternoster, B.; Multistep collocation methods for Volterra integral equations; Appl. Numer. Math.: 2009; Volume 59 ,1721-1736. · Zbl 1172.65069 |
| [22] | Cardone, A.; Conte, D.; Multistep collocation methods for Volterra integro-differential equations; Appl. Math. Comput.: 2013; Volume 221 ,770-785. · Zbl 1329.65170 |
| [23] | Wang, Z.; Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients; Stat. Probab. Lett.: 2008; Volume 78 ,1062-1071. · Zbl 1140.60331 |
| [24] | Buckwar, E.; Sickenberger, T.; A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods; Math. Comput. Simul.: 2011; Volume 81 ,1110-1127. · Zbl 1219.60066 |
| [25] | Saito, Y.; Mitsui, T.; Stability analysis of numerical schemes for stochastic differential equations; SIAM J. Numer. Anal.: 1996; Volume 33 ,2254-2267. · Zbl 0869.60052 |
| [26] | Bryden, A.; Higham, D.J.; On the boundedness of asymptotic stability regions for the stochastic theta method; BIT: 2003; Volume 43 ,1-6. · Zbl 1026.65004 |
| [27] | Higham, D.J.; Mean-square and asymptotic stability of the stochastic theta method; SIAM J. Numer. Anal.: 2000; Volume 38 ,753-769. · Zbl 0982.60051 |
| [28] | Ding, X.; Ma, Q.; Zhang, L.; Convergence and stability of the split-step-method for stochastic differential equations; Comput. Math. Appl.: 2010; Volume 60 ,1310-1321. · Zbl 1201.65010 |
| [29] | Higham, D.J.; A-stability and stochastic mean-square stability; BIT: 2000; Volume 40 ,404-409. · Zbl 0961.65003 |
| [30] | Hu, P.; Huang, C.; The stochastic ϑ-method for nonlinear stochastic Volterra integro-differential equations; Abstr. Appl. Anal.: 2014; Volume 2014 ,583930. · Zbl 1474.65496 |
| [31] | Shi, C.; Xiao, Y.; Zhang, C.; The convergence and MS stability of exponential Euler method for semilinear stochastic differential equations; Abstr. Appl. Anal.: 2012; Volume 2012 ,350407. · Zbl 1253.65009 |
| [32] | Butcher, J.; D’Ambrosio, R.; Partitioned general linear methods for separable Hamiltonian problems; Appl. Numer. Math.: 2017; Volume 117 ,69-86. · Zbl 1365.65272 |
| [33] | D’Ambrosio, R.; De Martino, G.; Paternoster, B.; Numerical integration of Hamiltonian problems by G-symplectic methods; Adv. Comput. Math.: 2014; Volume 40 ,553-575. · Zbl 1301.65127 |
| [34] | D’Ambrosio, R.; Izzo, G.; Jackiewicz, Z.; Search for highly stable two-step Runge-Kutta methods; Appl. Numer. Math.: 2012; Volume 62 ,1361-1379. · Zbl 1252.65119 |
| [35] | Burrage, K.; Cardone, A.; D’Ambrosio, R.; Paternoster, B.; Numerical solution of time fractional diffusion systems; Appl. Numer. Math.: 2017; Volume 116 ,82-94. · Zbl 1372.65228 |
| [36] | D’Ambrosio, R.; De Martino, G.; Paternoster, B.; General Nystrom methods in Nordsieck form: Error analysis; J. Comput. Appl. Math.: 2016; Volume 292 ,694-702. · Zbl 1329.65150 |
| [37] | D’Ambrosio, R.; Paternoster, B.; Santomauro, G.; Revised exponentially fitted Runge-Kutta-Nyström methods; Appl. Math. Lett.: 2014; Volume 30 ,56-60. · Zbl 1311.65097 |
| [38] | D’Ambrosio, R.; Moccaldi, M.; Paternoster, B.; Adapted numerical methods for advection-reaction-diffusion problems generating periodic wavefronts; Comput. Math. Appl.: 2017; Volume 74 ,1029-1042. · Zbl 1448.65135 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
