### On Martingales and the Use of Optional Stopping Theorem to Determine the Mean and Variance of a Stopping Time

Received: 1 August 2014    Accepted: 6 August 2014    Published: 5 September 2014

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.

Keywords

Martingales, Gambler, Random Walk, Stopping Time, Optional Stopping Theorem, Mean, Variance

References
 [1] Ganiyu A. A. (2006). Theoretical Framework of Martingales Associated with Random Walk for Foreign Exchange Rate Determination. An unpublished M.Phil. dissertation, University of Ibadan, Ibadan. Pp 7,11,14,20-21,71-74 [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 An unpublished M.Phil.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. [11] http://nvis.neurodebates.com/on/Optional_stopping_theorem
• 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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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},
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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.},
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Author Information
• Sections