Introduction to Stochastic Analysis. Integrals and Differential Equations - Vigirdas Mackevicius

"Introduction to Stochastic Analysis. Integrals and Differential Equations" - это введение в стохастическое интегрирование и стохастические дифференциальные уравнения, написанное понятным языком для широкой аудитории, от студентов математики до практикующих специалистов в биологии, химии, физике и финансах. Представление основано на наивном стохастическом интегрировании, а не на абстрактных теориях меры и стохастических процессов. Доказательства достаточно просты для практиков и в то же время достаточно строги для математиков. Представлены подробные примеры приложений в естественных науках и финансах. Особое внимание уделяется моделированию диффузионных процессов. Рассматриваются такие темы, как броуновское движение, мотивация стохастических моделей с броуновским движением, стохастические интегралы Ито и Стратоновича, формула Ито, стохастические дифференциальные уравнения (СДУ), решения СДУ как марковские процессы, примеры приложений в физических науках и финансах, моделирование решений СДУ (сильные и слабые приближения). Также предоставлены упражнения с подсказками и/или решениями.

This is an easy-to-understand introduction to stochastic calculus and stochastic differential equations, aimed at both scientists and practicing economists. It is based upon naive stochastic integration (rather than the areas of measure theory and stochastic processes behind abstract formulations), with easy-to -follow proofs for both scientists and mathematicians alike. More specifically, it covers: Brownian motion, motivation and applications for stochastic models derived through Brownian motion theory; Itô versus Stratonvich stochastic integration; Itō's formula and random differential equations; stochastic differential equation (stochastics) solutions as Markov processes, and their applications to physical sciences, finance, and especially its Monte Carlo simulation both with strong and weak approximation. A wide variety of real-world examples are explored and demonstrated along with accompanying exercises.

This book is an elementary introduction to the theory of stochastic integration. Instead of introducing general measure theoretic tools, the approach proposed here is more intuitive and rigid enough for applied scientists and engineers to carry through rigorous calculations. Details regarding the process of proofs reflect the balance between theoretical rigor and ease-of-interpretation aimed at providing mathematical background for those familiar with calculus. An abundance of carefully chosen examples illustrate applications in physics and finance. Moreover, special concern is devoted to questions dedicated to Monte Carlo simulation methods. Naturally arising topics include Brownian motions and their motivations for establishing stochastic models, derivations of Itô integral and Itô's formulas, solutions to associated stochastic differential equations, and a variety of applications in different scientific fields and financial nanotechnology. Exercises should assist the reader as well as new concepts, by providing hints, analytical solutions, or both.

Электронная Книга «Introduction to Stochastic Analysis. Integrals and Differential Equations» написана автором Vigirdas Mackevicius в году.

Минимальный возраст читателя: 0

Язык: Английский

ISBN: 9781118603314

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.