近年來，在金融市場中，財務數學已經扮演著越來越重要的角色了。特別是應用在期貨、選擇權之定價、投資組合與避險套利問題之研究，現在嚴然成為機率論領域中最熱門的話題。基於這樣的研究風氣以及衡量目前台灣最熱門的樂透彩未來極有可能發展成為期貨商品之故。本研究建造了樂透期貨之數學模式並討論最佳停止時間之問題，目的在尋找能使投資人獲取最大的利益之投資時刻，亦即尋找最佳停止時間。

In recent years, financial mathematics has become a very popular subject. Also it is applied to study the problems of futures, pricing of options etc. On other hand the most popular thing in Taiwan is to buy lottery tickets. There is a tendency that Taiwan’s lottery will become an important income for local government. Moreover expanding Taiwan’s lottery to a form of derivative security may be possible in future. Although Taiwan’s lottery is not a derivative security now, we construct a mathematical model of the discrete time to study a visionary derivative security concerning Taiwan’s lottery.
Our purpose in this paper is to find the optimal stopping time.

1 Introduction.................................... 3
2 Definitions and preliminary theorems............ 5
3 The main results................................ 9
3.1 The case of (N < infinite).................. 9
3.2 The case of (N = infinite).................. 14
4 The conclusion.................................. 21

1. Chow, Y.S., Robbins, H. & Siegmund, D.(1991):
The theory of optimal stopping, Dover Publications, New York.
2. Durrett, R.(1996): Probability: Theory and examples.
Duxbury Press, New York.
3. Liptser, R.S. & Shiryayev, A.N.(1974): Statistics of Random Processes I, Springer-Verlag, Heidelberg Berlin.
4. Revuz, D.(1975):Markov Chains, North Holland, Amsterdam.
Van der Linden, W.J. & Hambleton, R.K.(1997):
Handbook of modern item response theory, Springer-Verlag, New York.
5. Lamberton, D. & Lapeyre, B.(1997):
Introduction to stochastic calculus applied to finance,
Chapman & Hall, New York.
6. Nishioma, K.(1981):Probability, JITSUGYO Publishing Company (in Japanese).
7. Black, F. and Scholes, M.(1972):
The valuation of option contracts and a test of market efficiency, {t Journal of Finance}, {27}, 399-417.
8. Black, F. and Scholes, M.(1973):
The pricing of options and corporate liabilities, { Jouranl of Political Economy}, {81}, 637-659.
9. Scott, L.O.(1997):
Pricing stock options in a jump-diffusion model with stochastic volatility and interest rates:
applications of Fourier inversion methods, {Mathematical Finance} {7}, 413-426.
10. Karoui, N. and Quenez, M.C.(1995):
Dynamic programming and pricing of comtingent claims in an incomplete market,
{ Journal of SIAM Control and Optimization}, {33}, 29-66.
11. Karatzas, I. and Yor, M.(1991):
Methods of Mathematical Finance, Springer-Verlag, New York.
12. Royden, H.L.(1963)
Real Analysis, Prentice, New Jersey.