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 note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equ...In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.展开更多
A conditional log-Laplace functional （CLLF） for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching ...A conditional log-Laplace functional （CLLF） for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching and an interacting dynamic generated by the random environments. CLLF will play an important role in the investigation of branching processes and superprocesses with interaction.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 an NSERC granta startup fund of University of Albertasupported by the NSF grant DMS1613163
文摘In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.
文摘A conditional log-Laplace functional （CLLF） for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching and an interacting dynamic generated by the random environments. CLLF will play an important role in the investigation of branching processes and superprocesses with interaction.
