商品描述
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."
商品描述(中文翻譯)
1 隨機微分方程的均方近似。- 2 隨機微分方程的弱近似。- 3 小噪聲隨機微分方程的數值方法。- 4 隨機哈密頓系統和朗之萬型方程。- 5 空間和時空有界擴散的模擬。- 6 線性邊界值問題的隨機遊走。- 7 非線性拋物方程的Cauchy問題的數值解的概率方法。- 8 基於概率方法的非線性Dirichlet和Neumann問題的數值解。- 9 隨機數值方法在隨機共振模型和布朗隨機齒輪中的應用。- 附錄：隨機數值方法實施的實用指導。- A.1 均方方法。- A.2 弱方法和蒙地卡羅技術。- A.3 有界擴散的算法。- A.4 線性邊界值問題的隨機遊走。- A.5 非線性偏微分方程。- A.6 其他。- 參考文獻。
在更新版中，我們計劃包含以下新材料：
(i) 針對反向隨機微分方程的數值方法，將專門設立一章；
(ii) 我們將通過有關隨機微分方程的幾何積分和計算遍歷極限（隨機微分方程的長時間積分）的新結果來擴展第4章；
(iii) 我們將在第1章和第2章中添加針對非全局Lipshitz係數的隨機微分方程的最新結果；
(iv) 我們將新增一章或擴展第2章，以包括自2008年以來發展的多層蒙地卡羅方法及其在方差減少方面的新結果。
我們還將探索包含一些隨機偏微分方程材料的可能性。我們將刪除第9章，並刪除或轉換第8章。此外，在新版本的工作過程中將會發生自然變化。
作者簡介
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.
作者簡介(中文翻譯)
G.N. Milstein 教授在烏拉爾國立大學（UrGU；斯維爾德洛夫斯克，前蘇聯）獲得數學學士學位，該校現為烏拉爾聯邦大學（俄羅斯葉卡捷琳堡）。他在同一所大學完成了博士學位。Milstein 教授曾擔任烏拉爾國立大學（後來的 URFU）數學與力學系的助理教授、副教授，並在完成 DSc 論文答辯後成為教授。多年來，他在魏爾施特拉斯應用分析與隨機學研究所（WIAS；德國柏林）擔任高級研究員。他還曾擔任萊斯特大學（英國）和曼徹斯特大學（英國）的訪問教授。Milstein 教授在隨機數值分析、估計、控制、穩定性和金融數學方面擁有世界領先的專業知識。Milstein 早期在隨機微分方程數值方法方面的開創性論文是現代隨機數值分析的基石。
M.V. Tretyakov 教授在烏拉爾國立大學（UrGU；斯維爾德洛夫斯克，前蘇聯）獲得數學學士學位，並在同一所大學完成博士學位。Tretyakov 教授在魏爾施特拉斯應用分析與隨機學研究所（WIAS，柏林）擔任 DAAD 研究員期間，獲得了隨機數值分析的經驗，之後又成為亞歷山大·馮·洪堡基金會的研究員。他曾在數學與力學研究所（俄羅斯科學院，葉卡捷琳堡）和烏拉爾國立大學擔任高級研究員。他曾在斯旺西大學（英國）擔任講師，並在萊斯特大學（英國）擔任講師、讀者和教授。自 2012 年以來，他在諾丁漢大學（英國）擔任教授。他曾在數值分析和科學計算期刊的編輯委員會任職。他的研究得到了 Leverhulme Trust、EPSRC、BBSRC 和皇家學會的支持。Tretyakov 教授在隨機數值分析方面擁有廣泛的世界級專業知識，並在金融數學、隨機動力學和不確定性量化方面進行高質量的研究。
