Abstract
Gambling with dice was very popular in medieval Europe. The study of problems involving probability associated with gambling led to the development of probability theory. However, it was not until the early twentieth century that probability theory was considered as a branch of mathematics. The mathematical foundation of modern probability theory was laid by Andrei N. Kolmogorov in 1933. He adopted Lebesgue’s framework of measure theory and created an axiomatic system for probability theory. This chapter introduces some basic concepts and results of modern probability theory, highlights the results related to the conditional mathematical expectation, and then introduces discrete-time martingale theory, including the martingale transform and the Snell envelope. We assume that the reader has basic knowledge of measure theory.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Meyer, P.A.: Martingales and Stochastic Integrals I. Lecturer Notes in Mathematics, vol. 284. Springer, Berlin (1972)
Yan, J.A.: Lectures on Measure Theory, 2nd edn. Science Press, Beijing (2009, in Chinese)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd. and Science Press
About this chapter
Cite this chapter
Yan, JA. (2018). Foundation of Probability Theory and Discrete-Time Martingales. In: Introduction to Stochastic Finance. Universitext. Springer, Singapore. https://doi.org/10.1007/978-981-13-1657-9_1
Download citation
DOI: https://doi.org/10.1007/978-981-13-1657-9_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-1656-2
Online ISBN: 978-981-13-1657-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)