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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.
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