The main subject of the book is stochastic analysis and its various applications to mathematical finance and statistics of random processes. The main purpose of the book is to present, in a short and sufficiently self-contained form, the methods and results of the contemporary theory of stochastic analysis and to show how these methods and results work in mathematical finance and statistics of random processes. The book can be considered as a textbook for both senior undergraduate and graduate courses on this subject. The book can be helpful for undergraduate and graduate students, instructors and specialists on stochastic analysis and its applications.
Author(s): Alexander Melnikov
Series: CMS/CAIMS Books in Mathematics, 6
Edition: 1
Publisher: Springer
Year: 2023
Language: English
Pages: 213
City: Cham
Tags: Stochastic Processes, Martingales, Stochastic Differential Equations, Diffusion Processes, Mathematical Finance
Preface
Contents
Acronyms and Notation
1 Probabilistic Foundations
1.1 Classical theory and the Kolmogorov axiomatics
1.2 Probabilistic distributions and the Kolmogorov consistency theorem
2 Random variables and their quantitative characteristics
2.1 Distributions of random variables
2.2 Expectations of random variables
3 Expectations and convergence of sequences of random variables
3.1 Limit behavior of sequences of random variables in terms of their expected values
3.2 Probabilistic inequalities and interconnections …
4 Weak convergence of sequences of random variables
4.1 Weak convergence and its description in terms of distributions
4.2 Weak convergence and Central Limit Theorem
5 Absolute continuity of probability measures and conditional expectations
5.1 Absolute continuity of measures and the Radon-Nikodym theorem
5.2 Conditional expectations and their properties
6 Discrete time stochastic analysis: basic results
6.1 Basic notions: stochastic basis, predictability and martingales
6.2 Martingales on finite time interval
6.3 Martingales on infinite time interval
7 Discrete time stochastic analysis: further results and applications
7.1 Limiting behavior of martingales with statistical applications
7.2 Martingales and absolute continuity of measures. Discrete time Girsanov theorem and its financial application
7.3 Asymptotic martingales and other extensions of martingales
8 Elements of classical theory of stochastic processes
8.1 Stochastic processes: definitions, properties and classical examples
8.2 Stochastic integrals with respect to a Wiener process
8.3 The Ito processes: Formula of changing of variables, theorem of Girsanov, representation of martingales
9 Stochastic differential equations, diffusion processes and their applications
9.1 Stochastic differential equations
9.2 Diffusion processes and their connection with SDEs and PDEs
9.3 Applications to Mathematical Finance and Statistics of Random Processes
9.4 Controlled diffusion processes and applications to option pricing
10 General theory of stochastic processes under ``usual conditions''
10.1 Basic elements of martingale theory
10.2 Extension of martingale theory by localization of stochastic processes
10.3 On stochastic calculus for semimartingales
10.4 The Doob-Meyer decomposition: proof and related remarks
11 General theory of stochastic processes in applications
11.1 Stochastic mathematical finance
11.2 Stochastic Regression Analysis
12 Supplementary problems
References
Index
