Stochastic Numerics for Mathematical Physics

1 Mean-square approximation for stochastic differential equations.- 2 Weak approximation for stochastic differential equations.- 3 Numerical methods for SDEs with small noise.- 4 Stochastic Hamiltonian systems and Langevin-type equations.- 5 Simulation of space and space-time bounded diffusions.- 6 Random walks for linear boundary value problems.- 7 Probabilistic approach to numerical solution of the Cauchy problem for nonlinear parabolic equations.- 8 Numerical solution of the nonlinear Dirichlet and Neumann problems based on the probabilistic approach.- 9 Application of stochastic numerics to models with stochastic resonance and to Brownian ratchets.- A Appendix: Practical guidance to implementation of the stochastic numerical methods.- A.1 Mean-square methods.- A.2 Weak methods and the Monte Carlo technique.- A.3 Algorithms for bounded diffusions.- A.4 Random walks for linear boundary value problems.- A.5 Nonlinear PDEs.- A.6 Miscellaneous.- References.

new TOC"In the updated edition we are planning to  include the following new material:

(i) numerics for backward SDEs to which a new chapter will be dedicated;

(ii) we will extend chapter 4 by new results on Geometric Integration of SDEs and computing ergodic limits (long time integration of SDEs);

(iii) we will add recent results for  SDEs with nonglobal Lipshitz coefficients to Chapters 1 and 2;

(iv) we will add a new chapter or extend Chapter 2 to include multi-level Monte Carlo methods which has been developed since 2008 and new results on variance reduction.

We will also explore a possibility to include some material on stochastic PDEs. We will remove Chapter 9 and either remove or transform Chapter 8. Further, natural changes will occur during the work on the new edition."