A course of stochastic analysis Alexander Melnikov

Format:

Book

Author/Creator:

Melʹnikov, A. V., 1953- author.

Series:

CMS/CAIMS books in mathematics ; 2730-6518 v. 17

CMS/CAIMS books in mathematics 2730-6518 volume 17

Language:

English

Subjects (All):

Stochastic analysis.

Physical Description:

1 online resource

Edition:

Second edition

Place of Publication:

Cham Springer 2026

Summary:

This thoroughly updated second edition offers a unified, modern pathway from the Kolmogorov foundations of probability to the tools of stochastic calculus -- and on to applications in finance, statistics, and risk. With clarity and breadth, it develops martingale and semimartingale theory alongside stochastic differential equations, keeping both discrete- and continuous-time viewpoints in play. What's new in the 2nd Edition Optional Stochastic Analysis on non‑"usual" filtrations: the first textbook presentation of optional processes on stochastic bases beyond the standard right‑continuous, complete setting, with an accompanying optional stochastic calculus. Optional SDEs and stochastic exponentials/logarithms: existence-uniqueness theory and product/inverse rules, with financial modeling worked out in this optional‑semimartingale framework. New applications: Stochastic Regression Analysis and Risk Theory, showing how optional tools yield estimation results and ruin‑probability bounds in general settings. Expanded exercises with solutions: a substantially enlarged Supplement (Ch. 15) featuring problems that reinforce both core theory and applications. Designed for senior undergraduates, graduate students, and instructors, the book also serves researchers and practitioners who need a concise, example‑driven route from measure‑theoretic probability to the techniques used in finance, statistics, and risk modeling. Abundant worked examples and a comprehensive set of problems -- with hints and solutions -- make it ideal for self‑study or course adoption

Contents:

Probabilistic Foundations

Random variables and their quantitative characteristics

Expectations and convergence of sequences of random variables

Weak convergence of sequences of random variables

Absolute continuity of probability measures and conditional expectations

Discrete time stochastic analysis: basic results

Discrete time stochastic analysis: further results and applications

Elements of classical theory of stochastic processes

Stochastic differential equations, diffusion processes and their applications

General theory of stochastic processes under "usual conditions"

General theory of stochastic processes in applications

Basic elements of optional stochastic analysis

Optional stochastic differential equations and their applications

Optional semimartingales for stochastic regression analysis and risk theory

Supplementary problems

Notes:

Includes bibliographical references and index

Online resource; title from PDF title page (SpringerLink, viewed June 19, 2026)

Other Format:

Print version Melnikov, Alexander A Course of Stochastic Analysis

ISBN:

9783032204820

3032204828

OCLC:

1596922141

Access Restriction:

Restricted for use by site license