Stochastic Numerics for Mathematical Physics

1. Front Matter

   Pages i-xxv

   PDF

2. Mean-Square Approximation for Stochastic Differential Equations

   • Grigori N. Milstein, Michael V. Tretyakov

   Pages 1-94

3. Weak Approximation for Stochastic Differential Equations: Foundations

   • Grigori N. Milstein, Michael V. Tretyakov

   Pages 95-191

4. Weak Approximation for Stochastic Differential Equations: Special Cases

   • Grigori N. Milstein, Michael V. Tretyakov

   Pages 193-269

5. Numerical Methods for SDEs with Small Noise

   • Grigori N. Milstein, Michael V. Tretyakov

   Pages 271-312

6. Geometric Integrators and Computing Ergodic Limits

   • Grigori N. Milstein, Michael V. Tretyakov

   Pages 313-420

7. Simulation of Space and Space-Time Bounded Diffusions

   • Grigori N. Milstein, Michael V. Tretyakov

   Pages 421-481

8. Random Walks for Linear Boundary Value Problems

   • Grigori N. Milstein, Michael V. Tretyakov

   Pages 483-555

9. Probabilistic Approach to Numerical Solution of the Cauchy Problem for Nonlinear Parabolic Equations

   • Grigori N. Milstein, Michael V. Tretyakov

   Pages 557-604

10. Numerical Solution of the Nonlinear Dirichlet and Neumann Problems Based on the Probabilistic Approach

    • Grigori N. Milstein, Michael V. Tretyakov

    Pages 605-643

11. Solving FBSDEs Using Layer Methods

    • Grigori N. Milstein, Michael V. Tretyakov

    Pages 645-677

12. Solving Parabolic SPDEs by Averaging Over Characteristics

    • Grigori N. Milstein, Michael V. Tretyakov

    Pages 679-710

13. Back Matter

    Pages 711-736

    PDF

This book is a substantially revised and expanded edition reflecting major developments in stochastic numerics since the first edition was published in 2004. The new topics, in particular, include mean-square and weak approximations in the case of nonglobally Lipschitz coefficients of Stochastic Differential Equations (SDEs) including the concept of rejecting trajectories; conditional probabilistic representations and their application to practical variance reduction using regression methods; multi-level Monte Carlo method; computing ergodic limits and additional classes of geometric integrators used in molecular dynamics; numerical methods for FBSDEs; approximation of parabolic SPDEs and nonlinear filtering problem based on the method of characteristics.

SDEs have many applications in the natural sciences and in finance. Besides, the employment of probabilistic representations together with the Monte Carlo technique allows us to reduce the solution of multi-dimensional problems for partial differential equations to the integration of stochastic equations. This approach leads to powerful computational mathematics that is presented in the treatise. Many special schemes for SDEs are presented. In the second part of the book numerical methods for solving complicated problems for partial differential equations occurring in practical applications, both linear and nonlinear, are constructed. All the methods are presented with proofs and hence founded on rigorous reasoning, thus giving the book textbook potential. An overwhelming majority of the methods are accompanied by the corresponding numerical algorithms which are ready for implementation in practice. The book addresses researchers and graduate students in numerical analysis, applied probability, physics, chemistry, and engineering as well as mathematical biology and financial mathematics.

• Stochastic Differential Equations
• SDEs
• backward SDEs
• computing ergodic limits
• geometric integration
• nonglobal Lipshitz coefficients
• multi-level Monte Carlo methods
• variance reduction
• stochastic PDEs
• stochastic Hamiltonian systems
• Langevin equation
• nonlinear parabolic equations
• Cauchy problem
• Strong and Weak Approximation for SDE
• mathematical biology
• financial mathematics

• Chappaqua, USA

  Grigori N. Milstein

• University of Nottingham, School of Mathematical Sciences, Nottingham, UK

  Michael V. Tretyakov

Professor G.N. Milstein received his undergraduate degree in mathematics from the Ural State University (UrGU; Sverdlovsk, USSR), which is now Ural Federal University (Ekaterinburg, Russia).  He completed his PhD studies at the same University. Professor Milstein has been an assistant professor, associate professor and, after defending his DSc thesis, professor at the Faculty of Mathematics and Mechanics of UrGU (then URFU). For a number of years, he worked as a senior researcher at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS; Berlin, Germany). He was also a Visiting Professor at the University of Leicester (UK) and the University of Manchester (UK).  Professor Milstein has a world-leading expertise in stochastic numerics, estimation, control, stability, financial mathematics. Milstein's early pioneering papers on numerical methods for stochastic differential equations are the cornerstones of the modern stochastic numerics.Professor M.V. Tretyakov received his undergraduate degree in mathematics from the Ural State University (UrGU; Sverdlovsk, USSR).  He completed his PhD studies at the same University. 
Professor Tretyakov has gained experience in stochastic numerics during his stay at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS, Berlin) as a DAAD Research Fellow and then a Research Fellow of the Alexander von Humboldt Foundation. He worked as senior researcher at the Institute of Mathematics and Mechanics (Russian Academy of Sciences, Ekaterinburg) and at UrGU. He was a lecturer at Swansea University (UK) and a lecturer, reader and professor at the University of Leicester (UK). Since 2012 he is a professor at the University of Nottingham (UK). He has served on editorial boards of numerical analysis and scientific computing journals. His research has been supported by the Leverhulme Trust, EPSRC, BBSRC, and Royal Society. Professor Tretyakov has extensive world-class expertise in stochastic numerical analysis. He also conducts high quality research in financial mathematics, stochastic dynamics, and uncertainty quantification.