Stochastic Calculus and Brownian Motion
By Tejas Thakur
Length11h 52m
About this audiobook
"Stochastic Calculus and Brownian Motion" is a comprehensive guide crafted for students and professionals in mathematical sciences, focusing on stochastic processes and their real-world applications in finance, physics, and engineering. We explore key concepts and mathematical foundations of random movements and their practical implications.At its core, the book delves into Brownian motion, the random movement of particles suspended in a fluid, as described by Robert Brown in the 19th century. This phenomenon forms a cornerstone of modern probability theory and serves as a model for randomness in physical systems and financial models describing stock market behaviors.We also cover martingales, mathematical sequences where future values depend on present values, akin to a fair game in gambling. The book demonstrates how martingales are used to model stochastic processes and their calibration in real-world scenarios.Stochastic calculus extends these ideas into continuous time, integrating calculus with random processes. Our guide provides the tools to understand and apply Itô calculus, crucial for advanced financial models like pricing derivatives and managing risks.Written clearly and systematically, the book includes examples and exercises to reinforce concepts and showcase their real-world applications. It serves as an invaluable resource for students, educators, and professionals globally.
Audiobook details
GenreScience and Nature, Education and Learning
Length11 hrs 52 mins
Narrated byListen with 1,000+ voices
FormateBook with Audio
Publish dateFeb 20, 2025
LanguageEnglish
Table of contents
1Introduction to Stochastic Processes
21.1 Overview of Stochastic Processes
31.1.1 Definition and Basic Concepts
41.1.2 Classification of Stochastic Processes
51.1.3 Key Properties and Examples
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61.1.4 Applications and Implications
71.1.5 Challenges and Solutions
8Educational and Collaborative Efforts
91.2 Key Concepts and Definitions
101.2.1 Stochastic Process
111.2.2 Random Variable
121.2.3 Probability Space
131.2.4 Filtration
141.2.5 Adapted Process
151.2.6 Expectation
161.2.7 Independence
171.2.8 Stationarity
181.3 Importance in Finance and Science
191.3.1 Applications in Financial Markets
201.3.2 Influence in Scientific Research: 1.3.3 Interdisciplinary Approaches and Innovations
21Probability Theory Basics
222.1 Probability Spaces
232.1.1 Definition of a Probability Space
242.1.2 Properties of Probability Measures
252.1.3 Examples of Probability Spaces: 2.1.4 Importance in Stochastic Processes
262.2 Random Variables and Expectations
272.2.1 Definition of a Random Variable
282.2.2 Probability Distribution of a Random Variable: 2.2.2 Probability Distribution of a Random Variable
292.2.3 Expectation of Random Variables
302.2.4 Properties of Expectation
312.2.5 Conditional Expectation
322.3 Convergence of Sequences of Random Variables
332.3.1 Types of Convergence
342.3.2 Importance and Implications: 2.3.3 Key Theorems and Results
352.3.4 Practical Considerations
362.3.5 Challenges and Limitations
37Basics of Brownian Motion
383.1 Definition and Properties of Brownian Motion
393.1.1 What is Brownian Motion?
403.1.2 Key Properties of Brownian Motion
413.1.3 Examples and Applications of Brownian Motion: 3.1.4 Mathematical Framework and Calculus for Brownian Motion
423.2 The Wiener Process
433.2.1 Definition of the Wiener Process
443.2.2 Properties of the Wiener Process
453.2.3 Simulation of the Wiener Process
463.2.4 Applications of the Wiener Process: 3.2.5 Theoretical Implications of the Wiener Process
473.3 Path Properties of Brownian Motion
483.3.1 Continuity of Paths
493.3.2 Non-differentiability of Paths
503.3.3 Self-Similarity: 3.3.4 Zero Crossing and Oscillation
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