This paper examines the roles martingale property played in the use of optional stopping theorem (OST). It also examines the implication of this property in the use of optional stopping theorem for the determination of mean and variance of a stopping time. A simple example relating to betting system of a gambler with limited amount of money has been provided. The analysis of the betting system showed that the gambler leaves with the same amount of money as when he started and therefore satisfied martingale property. Linearity of expectation property was used as a reliable tool in the use of the martingale property.
| Published in |
Applied and Computational Mathematics (Volume 3, Issue 6-1)
This article belongs to the Special Issue Computational Finance |
| DOI | 10.11648/j.acm.s.2014030601.13 |
| Page(s) | 12-17 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Martingales, Gambler, Random Walk, Stopping Time, Optional Stopping Theorem, Mean, Variance
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| [2] | Hazewinkel M. (2001). “Martingale” (http://www.encyclopediaofmaths.org?/index.php?title=p/m062570, Encyclopedia of mathematics, Springer, ISBN 978-1-55608-010-4. |
| [3] | Kannan, D. (1997). “An introduction to Stochastic Process”. North Holland Series in Probability and Applied Mathematics. P. 196, (222-223), 24-25, 28-29. |
| [4] | Karlin S. and H. Taylor (1975). A first course in Stochastic Processes. Second Edition. Academic Press, Section 6.3, P. (253-262). |
| [5] | Shiu, E.S.W and Gerber H.U. (1994a). “Option Pricing by Esscher Transforms,” . Transactions, Society of Actuaries XLVI:99-140; Discussion 141-191. |
| [6] | Shiu, E.S.W. and Gerber H.U. (1994b). “Martingales Approach to Pricing Perpectual American Option,”. ASTIN Bulletin 24:195-220. |
| [7] | Shiu, E.S.W. and Gerber H.U. (1996a). “Martingales Approach to Perpectual American Options on Two Stocks, “. Mathematical Finance 6:303-322. |
| [8] | Shiu, E.S.W. and Gerber H.U. (1996b). “Actuarial Bridges to Dynamic Hedging and Option Pricing,” Insurance: Mathematics and Economics 18:183-218. Walk for Foreign Exchange Rate Determination. An unpublished M.Phil. dissertation, University of Ibadan, Ibadan. Pp 7,11,14,20-21,71-74 |
| [9] | Ugbebor O.O. and Ganiyu A.A. (2007). Martingales Associated with Random Walk Model for Foreign Exchange Rate Determination. Nigerian Mathematical Society Journal, Vol. 26 Pp (19-31). |
| [10] | [10 ] Ugbebor O.O., Ganiyu A.A. and Fakunle I. (2012). “Optional Stopping Theorem as an Indispensable Tool in the Determination of Ruin Probability and Expected Duration of a Game”. Journal of the Nigerian Association of Mathematical Physics”, Vol. 21, Pp 85-93. |
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APA Style
Ganiyu, A. A., Fakunle, I. (2014). On Martingales and the Use of Optional Stopping Theorem to Determine the Mean and Variance of a Stopping Time. Applied and Computational Mathematics, 3(6-1), 12-17. https://doi.org/10.11648/j.acm.s.2014030601.13
ACS Style
Ganiyu; A. A.; Fakunle; I. On Martingales and the Use of Optional Stopping Theorem to Determine the Mean and Variance of a Stopping Time. Appl. Comput. Math. 2014, 3(6-1), 12-17. doi: 10.11648/j.acm.s.2014030601.13
AMA Style
Ganiyu, A. A., Fakunle, I. On Martingales and the Use of Optional Stopping Theorem to Determine the Mean and Variance of a Stopping Time. Appl Comput Math. 2014;3(6-1):12-17. doi: 10.11648/j.acm.s.2014030601.13
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Download@article{10.11648/j.acm.s.2014030601.13,
author = {Ganiyu and A. A. and Fakunle and I.},
title = {On Martingales and the Use of Optional Stopping Theorem to Determine the Mean and Variance of a Stopping Time},
journal = {Applied and Computational Mathematics},
volume = {3},
number = {6-1},
pages = {12-17},
doi = {10.11648/j.acm.s.2014030601.13},
url = {https://doi.org/10.11648/j.acm.s.2014030601.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.s.2014030601.13},
abstract = {This paper examines the roles martingale property played in the use of optional stopping theorem (OST). It also examines the implication of this property in the use of optional stopping theorem for the determination of mean and variance of a stopping time. A simple example relating to betting system of a gambler with limited amount of money has been provided. The analysis of the betting system showed that the gambler leaves with the same amount of money as when he started and therefore satisfied martingale property. Linearity of expectation property was used as a reliable tool in the use of the martingale property.},
year = {2014}
}
TY - JOUR T1 - On Martingales and the Use of Optional Stopping Theorem to Determine the Mean and Variance of a Stopping Time AU - Ganiyu AU - A. A. AU - Fakunle AU - I. Y1 - 2014/09/05 PY - 2014 N1 - https://doi.org/10.11648/j.acm.s.2014030601.13 DO - 10.11648/j.acm.s.2014030601.13 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 12 EP - 17 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.s.2014030601.13 AB - This paper examines the roles martingale property played in the use of optional stopping theorem (OST). It also examines the implication of this property in the use of optional stopping theorem for the determination of mean and variance of a stopping time. A simple example relating to betting system of a gambler with limited amount of money has been provided. The analysis of the betting system showed that the gambler leaves with the same amount of money as when he started and therefore satisfied martingale property. Linearity of expectation property was used as a reliable tool in the use of the martingale property. VL - 3 IS - 6-1 ER -
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