In this article, we propose a space-time Multi-Index Monte Carlo （MIMC） estimator for a one-dimensional parabolic stochastic partial differential equation （SPDE） of Zakai type. We compare the complexity with the M...In this article, we propose a space-time Multi-Index Monte Carlo （MIMC） estimator for a one-dimensional parabolic stochastic partial differential equation （SPDE） of Zakai type. We compare the complexity with the Multilevel Monte Carlo （MLMC） method of Giles and Reisinger （2012）, and find, by means of Fourier analysis, that the MIMC method： （i） has suboptimal complexity of 0（ε^-21 ｜ogε｜） for a root mean square error （RMSE） z if the same spatial discretisation as in the MLMC method is used; （ii） has a better complexity of 0（ε^-21 ｜ogε｜） if a carefully adapted discretisation is used; （iii） has to be adapted for non-smooth functionals. Numerical tests confirm these findings empirically.展开更多
In this paper we present an L2-theory for a class of stochastic partial differential equations driven by Levy processes. The coefficients of the equations are random functions depending on time and space variables, an...In this paper we present an L2-theory for a class of stochastic partial differential equations driven by Levy processes. The coefficients of the equations are random functions depending on time and space variables, and no smoothness assumption of the coefficients is assumed.展开更多
文摘In this article, we propose a space-time Multi-Index Monte Carlo （MIMC） estimator for a one-dimensional parabolic stochastic partial differential equation （SPDE） of Zakai type. We compare the complexity with the Multilevel Monte Carlo （MLMC） method of Giles and Reisinger （2012）, and find, by means of Fourier analysis, that the MIMC method： （i） has suboptimal complexity of 0（ε^-21 ｜ogε｜） for a root mean square error （RMSE） z if the same spatial discretisation as in the MLMC method is used; （ii） has a better complexity of 0（ε^-21 ｜ogε｜） if a carefully adapted discretisation is used; （iii） has to be adapted for non-smooth functionals. Numerical tests confirm these findings empirically.
基金supported by National Science Foundation of US (Grant No. DMS-0906743)the National Research Foundation of Korea (Grant No. 20110027230)
文摘In this paper we present an L2-theory for a class of stochastic partial differential equations driven by Levy processes. The coefficients of the equations are random functions depending on time and space variables, and no smoothness assumption of the coefficients is assumed.
