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Stochastic Numerics for Mathematical Physics
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 no...
Applied Quantitative Finance
Recent years have witnessed a growing importance of quantitative methods in both financial research and industry. This development requires the use of advanced techniques on a theoretical and applied level, especially when it comes to the quantification of risk and the valuation of modern financial products. Applied Quantitative Finance (2nd edition) provides a comprehensive and state-of-the-art treatment of cutting-edge topics and methods. It provides solutions to and presents theoretical developments in many practical problems such as risk management, pricing of credit derivatives, quantification of volatility and copula modelling. The synthesis of theory and practice supported by computat...
An Introduction to the Numerical Simulation of Stochastic Di?erential Equations
This book provides a lively and accessible introduction to the numerical solution of stochastic differential equations with the aim of making this subject available to the widest possible readership. It presents an outline of the underlying convergence and stability theory while avoiding technical details. Key ideas are illustrated with numerous computational examples and computer code is listed at the end of each chapter. The authors include 150 exercises, with solutions available online, and 40 programming tasks. Although introductory, the book covers a range of modern research topics, including Itô versus Stratonovich calculus, implicit methods, stability theory, nonconvergence on nonlin...
Stochastic Analysis for Finance with Simulations
This book is an introduction to stochastic analysis and quantitative finance; it includes both theoretical and computational methods. Topics covered are stochastic calculus, option pricing, optimal portfolio investment, and interest rate models. Also included are simulations of stochastic phenomena, numerical solutions of the Black–Scholes–Merton equation, Monte Carlo methods, and time series. Basic measure theory is used as a tool to describe probabilistic phenomena. The level of familiarity with computer programming is kept to a minimum. To make the book accessible to a wider audience, some background mathematical facts are included in the first part of the book and also in the appendi...
Modeling and Analysis of Bio-molecular Networks
This book addresses a number of questions from the perspective of complex systems: How can we quantitatively understand the life phenomena? How can we model life systems as complex bio-molecular networks? Are there any methods to clarify the relationships among the structures, dynamics and functions of bio-molecular networks? How can we statistically analyse large-scale bio-molecular networks? Focusing on the modeling and analysis of bio-molecular networks, the book presents various sophisticated mathematical and statistical approaches. The life system can be described using various levels of bio-molecular networks, including gene regulatory networks, and protein-protein interaction networks...
Introduction to Stochastic Analysis
This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion processes. The topics covered include Brownian motion; motivation of stochastic models with Brownian motion; Itô and Stratonovich stochastic integrals, Itô’s formula; stochastic differential equations (SDEs); solutions of SDEs as Markov processes; application examples in physical sciences and finance; simulation of solutions of SDEs (strong and weak approximations). Exercises with hints and/or solutions are also provided.
Numerical Integration of Stochastic Differential Equations
This book is devoted to mean-square and weak approximations of solutions of stochastic differential equations (SDE). These approximations represent two fundamental aspects in the contemporary theory of SDE. Firstly, the construction of numerical methods for such systems is important as the solutions provided serve as characteristics for a number of mathematical physics problems. Secondly, the employment of probability representations together with a Monte Carlo method allows us to reduce the solution of complex multidimensional problems of mathematical physics to the integration of stochastic equations. Along with a general theory of numerical integrations of such systems, both in the mean-square and the weak sense, a number of concrete and sufficiently constructive numerical schemes are considered. Various applications and particularly the approximate calculation of Wiener integrals are also dealt with. This book is of interest to graduate students in the mathematical, physical and engineering sciences, and to specialists whose work involves differential equations, mathematical physics, numerical mathematics, the theory of random processes, estimation and control theory.
