-
Previous Article
Spatio-temporal models of synthetic genetic oscillators
- MBE Home
- This Issue
-
Next Article
On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach
Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion
Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland |
We consider nonlinear stochastic wave equations driven by one-dimensional white noise with respect to time. The existence of solutions is proved by means of Picard iterations. Next we apply Newton's method. Moreover, a second-order convergence in a probabilistic sense is demonstrated.
References:
| [1] |
K. Amano,
Newton's method for stochastic differential equations and its probabilistic second-order error estimate, Electron. J. Differential Equations, 2012 (2012), 1-8.
|
| [2] |
Z. Brzeźniak and M. Ondreját,
Weak solutions to stochastic wave equations with values in Riemannian manifolds, Commun. Part. Diff. Eq., 36 (2011), 1624-1653.
|
| [3] |
P. Caithamer,
The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise, Stoch. Dynam., 5 (2005), 45-64.
doi: 10.1142/S0219493705001286. |
| [4] |
P.-L. Chow,
Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 361-381.
doi: 10.1214/aoap/1015961168. |
| [5] |
D. Conus and R. C. Dalang,
The non-linear stochastic wave equation in high dimension, Electron. J. Probab., 13 (2008), 629-670.
doi: 10.1214/EJP.v13-500. |
| [6] |
R. C. Dalang,
Extending martingale measure stochastic integral with applications to spatially homogeneous spdes, Electron. J. Probab., 4 (1999), 1-29.
doi: 10.1214/EJP.v4-43. |
| [7] |
R. C. Dalang, The stochastic wave equation, A Minicourse on Stochastic Partial Differential Equations, in Lecture Notes in Math., 1962 (2009), Springer Berlin, 39-71.
doi: 10.1007/978-3-540-85994-9_2. |
| [8] |
R. C. Dalang, C. Mueller and R. Tribe,
A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other P.D.E.s, Trans. Amer. Math. Soc., 360 (2008), 4681-4703.
doi: 10.1090/S0002-9947-08-04351-1. |
| [9] |
R. C. Dalang and M. Sanz-Solé,
Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., 199 (2009), 1-70.
doi: 10.1090/memo/0931. |
| [10] |
E. Hausenblas,
Weak approximation of the stochastic wave equation, J. Comput. Appl. Math., 235 (2010), 33-58.
doi: 10.1016/j.cam.2010.03.026. |
| [11] |
J. Huang, Y. Hu and D. Nualart,
On Hölder continuity of the solution of stochastic wave equations, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014), 353-407.
doi: 10.1007/s40072-014-0035-5. |
| [12] |
S. Kawabata and T. Yamada, On Newton's method for stochastic differential equations, in Séminaire de Probabilités XXV, Lecture Notes in Math., 1485 (1991), Springer Berlin, 121-137.
doi: 10.1007/BFb0100852. |
| [13] |
J. U. Kim,
On the stochastic wave equation with nonlinear damping, Appl. Math. Optim., 58 (2008), 29-67.
doi: 10.1007/s00245-007-9029-2. |
| [14] |
C. Marinelli and L. Quer-Sardanyons,
Existence of weak solutions for a class of semilinear stochastic wave equations, Siam J. Math. Anal., 44 (2012), 906-925.
doi: 10.1137/110826667. |
| [15] |
A. Millet and M. Sanz-Solé,
A stochastic wave equation in two space dimensions: Smoothness of the law, Ann. Probab., 27 (1999), 803-844.
doi: 10.1214/aop/1022677387. |
| [16] |
M. Nedeljkov and D. Rajter,
A note on a one-dimensional nonlinear stochastic wave equation, Novi Sad Journal of Mathematics, 32 (2002), 73-83.
|
| [17] |
S. Peszat,
The Cauchy problem for a nonlinear stochastic wave equation in any dimension, J. Evol. Equ., 2 (2002), 383-394.
doi: 10.1007/PL00013197. |
| [18] |
L. Quer-Sardanyons and M. Sanz-Solé,
Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation, J. Funct. Anal., 206 (2004), 1-32.
doi: 10.1016/S0022-1236(03)00065-X. |
| [19] |
L. Quer-Sardanyons and M. Sanz-Solé,
Space semi-discretisations for a stochastic wave equation, Potential Anal., 24 (2006), 303-332.
doi: 10.1007/s11118-005-9002-0. |
| [20] |
M. Sanz-Solé and A. Suess,
The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity, Electron. J. Probab., 18 (2013), 1-28.
doi: 10.1214/EJP.v18-2341. |
| [21] |
J. B. Walsh, An introduction to stochastic partial differential equations, in: É cole d'été de Probabilités de Saint-Flour XIV, Lecture Notes in Math, 1180 (1986), Springer Berlin, 265-439.
doi: 10.1007/BFb0074920. |
| [22] |
J. B. Walsh,
On numerical solutions of the stochastic wave equation, Illinois J. Math., 50 (2006), 991-1018.
|
| [23] |
M. Wrzosek,
Newton's method for stochastic functional differential equations, Electron. J.Differential Equations, 2012 (2012), 1-10.
|
| [24] |
M. Wrzosek,
Newton's method for parabolic stochastic functional partial differential equations, Functional Differential Equations, 20 (2013), 285-310.
|
| [25] |
M. Wrzosek,
Newton's method for first-order stochastic functional partial differential equations, Commentationes Mathematicae, 54 (2014), 51-64.
|
show all references
References:
| [1] |
K. Amano,
Newton's method for stochastic differential equations and its probabilistic second-order error estimate, Electron. J. Differential Equations, 2012 (2012), 1-8.
|
| [2] |
Z. Brzeźniak and M. Ondreját,
Weak solutions to stochastic wave equations with values in Riemannian manifolds, Commun. Part. Diff. Eq., 36 (2011), 1624-1653.
|
| [3] |
P. Caithamer,
The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise, Stoch. Dynam., 5 (2005), 45-64.
doi: 10.1142/S0219493705001286. |
| [4] |
P.-L. Chow,
Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12 (2002), 361-381.
doi: 10.1214/aoap/1015961168. |
| [5] |
D. Conus and R. C. Dalang,
The non-linear stochastic wave equation in high dimension, Electron. J. Probab., 13 (2008), 629-670.
doi: 10.1214/EJP.v13-500. |
| [6] |
R. C. Dalang,
Extending martingale measure stochastic integral with applications to spatially homogeneous spdes, Electron. J. Probab., 4 (1999), 1-29.
doi: 10.1214/EJP.v4-43. |
| [7] |
R. C. Dalang, The stochastic wave equation, A Minicourse on Stochastic Partial Differential Equations, in Lecture Notes in Math., 1962 (2009), Springer Berlin, 39-71.
doi: 10.1007/978-3-540-85994-9_2. |
| [8] |
R. C. Dalang, C. Mueller and R. Tribe,
A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other P.D.E.s, Trans. Amer. Math. Soc., 360 (2008), 4681-4703.
doi: 10.1090/S0002-9947-08-04351-1. |
| [9] |
R. C. Dalang and M. Sanz-Solé,
Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., 199 (2009), 1-70.
doi: 10.1090/memo/0931. |
| [10] |
E. Hausenblas,
Weak approximation of the stochastic wave equation, J. Comput. Appl. Math., 235 (2010), 33-58.
doi: 10.1016/j.cam.2010.03.026. |
| [11] |
J. Huang, Y. Hu and D. Nualart,
On Hölder continuity of the solution of stochastic wave equations, Stoch. Partial Differ. Equ. Anal. Comput., 2 (2014), 353-407.
doi: 10.1007/s40072-014-0035-5. |
| [12] |
S. Kawabata and T. Yamada, On Newton's method for stochastic differential equations, in Séminaire de Probabilités XXV, Lecture Notes in Math., 1485 (1991), Springer Berlin, 121-137.
doi: 10.1007/BFb0100852. |
| [13] |
J. U. Kim,
On the stochastic wave equation with nonlinear damping, Appl. Math. Optim., 58 (2008), 29-67.
doi: 10.1007/s00245-007-9029-2. |
| [14] |
C. Marinelli and L. Quer-Sardanyons,
Existence of weak solutions for a class of semilinear stochastic wave equations, Siam J. Math. Anal., 44 (2012), 906-925.
doi: 10.1137/110826667. |
| [15] |
A. Millet and M. Sanz-Solé,
A stochastic wave equation in two space dimensions: Smoothness of the law, Ann. Probab., 27 (1999), 803-844.
doi: 10.1214/aop/1022677387. |
| [16] |
M. Nedeljkov and D. Rajter,
A note on a one-dimensional nonlinear stochastic wave equation, Novi Sad Journal of Mathematics, 32 (2002), 73-83.
|
| [17] |
S. Peszat,
The Cauchy problem for a nonlinear stochastic wave equation in any dimension, J. Evol. Equ., 2 (2002), 383-394.
doi: 10.1007/PL00013197. |
| [18] |
L. Quer-Sardanyons and M. Sanz-Solé,
Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation, J. Funct. Anal., 206 (2004), 1-32.
doi: 10.1016/S0022-1236(03)00065-X. |
| [19] |
L. Quer-Sardanyons and M. Sanz-Solé,
Space semi-discretisations for a stochastic wave equation, Potential Anal., 24 (2006), 303-332.
doi: 10.1007/s11118-005-9002-0. |
| [20] |
M. Sanz-Solé and A. Suess,
The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity, Electron. J. Probab., 18 (2013), 1-28.
doi: 10.1214/EJP.v18-2341. |
| [21] |
J. B. Walsh, An introduction to stochastic partial differential equations, in: É cole d'été de Probabilités de Saint-Flour XIV, Lecture Notes in Math, 1180 (1986), Springer Berlin, 265-439.
doi: 10.1007/BFb0074920. |
| [22] |
J. B. Walsh,
On numerical solutions of the stochastic wave equation, Illinois J. Math., 50 (2006), 991-1018.
|
| [23] |
M. Wrzosek,
Newton's method for stochastic functional differential equations, Electron. J.Differential Equations, 2012 (2012), 1-10.
|
| [24] |
M. Wrzosek,
Newton's method for parabolic stochastic functional partial differential equations, Functional Differential Equations, 20 (2013), 285-310.
|
| [25] |
M. Wrzosek,
Newton's method for first-order stochastic functional partial differential equations, Commentationes Mathematicae, 54 (2014), 51-64.
|
| [1] |
Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203 |
| [2] |
Lars Grüne, Peter E. Kloeden, Stefan Siegmund, Fabian R. Wirth. Lyapunov's second method for nonautonomous differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 375-403. doi: 10.3934/dcds.2007.18.375 |
| [3] |
Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 |
| [4] |
Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667 |
| [5] |
Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23 |
| [6] |
Juan C. Cortés, Sandra E. Delgadillo-Alemán, Roberto A. Kú-Carrillo, Rafael J. Villanueva. Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022079 |
| [7] |
R. Baier, M. Dellnitz, M. Hessel-von Molo, S. Sertl, I. G. Kevrekidis. The computation of convex invariant sets via Newton's method. Journal of Computational Dynamics, 2014, 1 (1) : 39-69. doi: 10.3934/jcd.2014.1.39 |
| [8] |
Liqun Qi, Zheng yan, Hongxia Yin. Semismooth reformulation and Newton's method for the security region problem of power systems. Journal of Industrial and Management Optimization, 2008, 4 (1) : 143-153. doi: 10.3934/jimo.2008.4.143 |
| [9] |
Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2081-2100. doi: 10.3934/cpaa.2020092 |
| [10] |
H. A. Erbay, S. Erbay, A. Erkip. The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6101-6116. doi: 10.3934/dcds.2016066 |
| [11] |
Xavier Blanc, Claude Le Bris, Pierre-Louis Lions. From the Newton equation to the wave equation in some simple cases. Networks and Heterogeneous Media, 2012, 7 (1) : 1-41. doi: 10.3934/nhm.2012.7.1 |
| [12] |
Anatoli Babin, Alexander Figotin. Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1685-1718. doi: 10.3934/cpaa.2014.13.1685 |
| [13] |
Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic and Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044 |
| [14] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
| [15] |
Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47 |
| [16] |
Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051 |
| [17] |
Junhao Hu, Chenggui Yuan. Strong convergence of neutral stochastic functional differential equations with two time-scales. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5831-5848. doi: 10.3934/dcdsb.2019108 |
| [18] |
Changfeng Gui, Zhenbu Zhang. Spike solutions to a nonlocal differential equation. Communications on Pure and Applied Analysis, 2006, 5 (1) : 85-95. doi: 10.3934/cpaa.2006.5.85 |
| [19] |
Jiequn Han, Ruimeng Hu, Jihao Long. Convergence of deep fictitious play for stochastic differential games. Frontiers of Mathematical Finance, 2022, 1 (2) : 287-319. doi: 10.3934/fmf.2021011 |
| [20] |
Marat Akhmet. Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Communications on Pure and Applied Analysis, 2014, 13 (2) : 929-947. doi: 10.3934/cpaa.2014.13.929 |
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]