Abstract
One of the most important concepts of modern probability theory is the martingale, which formalizes the notion of a fair game. In this chapter, we first lay the foundations for the treatment of general stochastic processes (filtrations, adapted processes, stopping times). We then introduce martingales and the discrete stochastic integral as well as the martingale representation theorem and the stability theorem for discrete martingales.
We close with an application to a model from mathematical finance.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Baxter, R. Rennie, Financial Calculus (Cambridge University Press, Cambridge, 1997)
F. Delbaen, W. Schachermayer, A general version of the fundamental theorem of asset pricing. Math. Ann. 300(3), 463–520 (1994)
R.J. Elliott, P.E. Kopp, Mathematics of Financial Markets, 2nd edn. Springer Finance (Springer, New York, 2005)
A. Etheridge, A Course in Financial Calculus (Cambridge University Press, Cambridge, 2002)
H. Föllmer, A. Schied, Stochastic Finance. de Gruyter Studies in Mathematics, vol. 27, 2nd edn. (Walter de Gruyter & Co., Berlin, 2004)
J.M. Harrison, S.R. Pliska, Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11(3), 215–260 (1981)
I. Karatzas, S.E. Shreve, Methods of Mathematical Finance. Applications of Mathematics, vol. 39 (Springer, New York, 1998)
R. Korn, E. Korn, Option Pricing and Portfolio Optimization. Graduate Studies in Mathematics, vol. 31 (American Mathematical Society, Providence, RI, 2001). Modern methods of financial mathematics, translated from the 1999 German original by the authors
M. Musiela, M. Rutkowski, Martingale Methods in Financial Modelling. Stochastic Modelling and Applied Probability, vol. 36, 2nd edn. (Springer, Berlin, 2005)
J.M. Steele, Stochastic Calculus and Financial Applications. Applications of Mathematics (New York), vol. 45 (Springer, New York, 2001)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Klenke, A. (2020). Martingales. In: Probability Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-56402-5_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-56402-5_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-56401-8
Online ISBN: 978-3-030-56402-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)