在這本論文中，我們研究兩個獨立的基本Lévy隨機過程之極值過程。我們考慮的Lévy過程有布朗運動、卜松過程，以及此兩種隨機過程的和。
在第二章，我們對兩個獨立的布朗運動，找出其極值過程的期望值、平方期望值，進一步討論其機率性質：獨立增量、平穩增量、到達時間的分布和隨機積分等。統計分析則有似然比值、最大似然估計及漸近的不偏估計。
第三章探討的是兩個獨立的卜松過程，我們找出其極值過程的期望值、平方期望值、邊際分布及條件分布。機率性質的討論包含獨立增量、平穩增量、跳躍時間的分布和間隔時間的分布。統計分析則是同第二章，有似然比值、最大似然估計及近似的不偏估計。
第四章則是將布朗運動與卜松過程結合，我們對兩者之和所形成新的兩獨立過程，寫出其極值過程的期望值、平方期望值的表示式，討論其獨立增量、平穩增量、到達時間的分布等機率性質，統計分析同前兩章。
第五章，我們將對第二章到第四章的結論作整理與解析。

In this dissertation, we study the maximum and minimum of two independent
simple L$acute{mbox e}$vy processes. L$acute{mbox e}$vy processes considered are
Brownian motion with drift, Poisson process and their sum.
Mathematical or probabilistic properties discussed include stationary, martingale, diffusion,
stochastic integral, interarrival time, jumping time, first passage time, etc.
Statistical properties investigated cover likelihood ratio, maximum likelihood estimator and
asymptotically unbiased estimator.
This dissertation is structured as follows. In Chapter 2, we deal with the maximum and minimum of two independent
Brownian motions with drifts. Chapter 3 contains the works about the maximum and minimum of two independent
Poisson processes. Chapter 4 is covered by the results about the maximum and minimum of two independent
sums of Brownian motion with drift and Poisson process. In Chapter 5, we provide conclusions and comparisons.

[1] Aase, K. K. (1986a). New option pricing formulas when the stock market is a
combined continuous point process. Technical Report. No. 60. CCREMS, MIT,
Cambridge, MA.
[2] Aase, K. K. (1986b). Probabilistic solutions of option pricing and corporate
security valuation. Technical Report. No. 63. CCREMS, MIT, Cambridge, MA.
[3] Alili, L. and Chaumont L. (2001). A new fluctuation identity for L´evy processes
and some applications. Bernoulli. 7, 557-569.
[4] Alos, E. Mazet, O. and Nualart, D. (2000). Stochastic calculus with respect
to fractional Brownian motion with Hurst parameter less than 1/2. Stochastic
Process. Appl. 86, 121-139.
[5] Andersen, P. K. Borgan, O. Gill, R. D. and Keiding, N. (1993). Statistical
Methods on Counting Processes. Springer-Velag.
[6] Aven, T. and Jensen, U. (1999). Stochastic Models in Reliability. Springer.
[7] Bachelier, L. (1990). Theorie de la speculation. Ann Sci. Ecole Norm. Sup. 17,
20-86.
[8] Barndorff-Nielsen, O. E. (1988). Parametric Statistical Models and Likelihood.
Lecture Notes in Statistics 50, Springer-Verlag.
[9] Barndorff-Nielsen, O. E. and Sørensen, M. (1994). A review of some aspects of
asymptotic likelihood theory for stochastic processes. International Statistical
Review. 50, 145-159.
[10] Basawa, I. V. and Prabhu, N. U. (1994). Statistical inference in stochastic
processes. Journal of Statistical Planning and Inference. 39.
[11] Basawa, I. V. and Prakasa Rao, B. L. S. (1980). Statistical Inference for Stochastic
Processes. Academic Press, London.
[12] Beran, J. (1994). Statistical Methods for Long Memory Processes. Chapman and
Hall, London.
[13] Berger, J. O. and Wolpert, R. L. (1988). The Likelihood Principle. Lecture
Notes-Monograph Series, Volume 6, Institute of Mathematical Statistics.
[14] Bertoin, J. (1996). L´evy Processes. Laboratoire de Probabilit´es Universit´e Pierre
et Marie Curie.
[15] Billingsley, P. (1961). Statistical Inference for Markov Processes. University of
Chicago Press, Chicago.
[16] Bingham, N. H. (1975). Fluctuation theory in continuous time. Adv. Appl.
Probab. 7, 705-766.
[17] Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities.
Journal of Political Economy. 81, 637-659.
[18] Bodo, B. A., Thompson, M. E., and Unny, T. E. (1987). A review on stochastic
differential equations for applications in hydrology. Stochastic Hydrol. Hydraul.
1, 81-100.
[19] Bosq, D. (1998). Nonparametric Statistics for Stochastic Process. 2nd. ed., Lecture
Notes in Statistics 110, Springer.
[20] Brown, B. M. and Hewitt, J. T. (1975). Asymptotic likelihood theory for diffusion
processes. Journal of Applied Probability. 12, 228-238.
[21] Brown, R. (1827). A Brief Account of Microscopical Observations. London.
[22] Chaumont, L. (1996). Conditionings and path decompositions for L´evy processes.
Stochastic Process Appl. 64, 39-54.
[23] Cheridito, A. (2001). Mixed fractional Brownian motion. Bernoulli. 7, 913-934.
[24] Chow, Y. S. and Teicher, H. (1997). Probability Theory. 3rd. ed., Springer.
[25] Ciesielski, Z. (1961). Holder condition for realizations of Gaussian processes.
Trans. Amer. Math. Soc. 99, 403-413.
[26] Cinlar, E. (1975). Introduction to Stochastic Processes. Prentice Hall.
[27] Coutin, L. and Qian, Z. (2000). Stochastic differential equations for fractional
Brownian motions. C. R. Acad. Sci. Paris S´e. I Math. 331, 75-80.
[28] Cox, D. R. (1975). Partial likelihood. Biometrika. 62, 269-276.
[29] Cox, D. R. and Lewis, P. A. W. (1978). The Statistical Analysis of Series of
Events. Chapman and Hall.
[30] Dai, W. and Heyde, C. (1996). Itˆo’s formula with respect to fractional Brownian
motion and its application. J. Appl. Math. Stochastic Anal. 9, 439-448.
[31] Decreusefond, L. and ¨Ustunel, A. (1997) Stochastic analysis of the fractional
Brownian motion. Potential Anal. 10, 177-214.
[32] Dietz, H. M. (1989). Asymptotic Properties of Maximum Likelihood Estimators
in Diffusion Type Models, Part 1: General Statements. Preprint No. 228,
Humboldt-Universit¨at, Berlin.
[33] Doob, J. L. (1953). Stochastic Processes. New York Wiley.
[34] Durrett, R. (1999). Essentials of Stochastic Processes. Springer.
[35] Eberlein, E. (2001). Application of Generalized Hyperbolic L´evy Motions to
Finance. In L´evy Process: Theory and Applications. Birkh¨auser: Basel.
[36] Eberlein, E. and Raible, S. (1999). Term structure models driven by general
L´evy process. Math. Finance. 9, 31-53.
[37] Edwards, A. F. (1972). Likelihood. Cambridge, U.K.: Cambridge University
Press.
[38] Einstein, A. (1905). On the movement of small particles suspend in a stationary
liquid demanded by the molecular-kinetic theory of heat. Ann. Physik. 17.
[39] Evans, M., Fraser, D. A., and Monette, G. (1986). On principles and arguments
to Likelihood. Canadian Journal of Statistics. 14, 181-199.
[40] Feigin, P. (1976). Maximum likelihood estimation for stochastic processes−A
martingale approach. Advances in Applied Probability. 8, 712-736.
[41] Fleischmann, K. and Mueller, C. (1997). A super-Brownian motion with a locally
infinite catalytic mass. Probab. Theory Related Fields. 107, 325-357.
[42] Fleischmann, K. and Klenke, A. (1999). Smooth density field of catalytic super-
Brownian motion. Ann. Appl. Probab. 19, 198-318.
[43] Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival
Analysis. Wiley.
[44] Fristedt, E. (1974). Sample Functions of Stochastic Processes with Stationary,
Independent Increments. Advances in Probability. 3, 241-396. Dekker, New
York.
[45] Geyer, C. (1999). Likelihood inference for spatial point processes. in Stochastic
Geometry: Likelihood and Computation, eds. O. E. Barndorff-Nielsen, W. S.Kendall, and M. N. M. van Lieshout, London: Chapman and Hall/CRC, 79-
140.
[46] Gihman, I. I. and Skorohod, A. V. (1975). The Theory of Stochastic Processes
II. Springer, Berlin.
[47] Godambe, V. P. and Heyde, C. C. (1987). Quasi-likelihood and optimal estimation.
International Statistical Review. 55, 231-244.
[48] Grandell, J. (1997). Mixed Poisson Processes. Chapman and Hall.
[49] Grenander, U. (1981). Abstract Inference. Wiley.
[50] Guttorp, P. (1991). Statistical Inference for Branching Processes. Wiley.
[51] He, S. W., Wang, J. G., and Yan, J.A. (1992). Semimartingale Theory and
Stochastic Calculus. Science Press.
[52] Heyde, C. C. (1997). Quasi-Likelihood and its Application. Springer.
[53] Hogg and Craig (1995). Introduction to Mathematical Statistics. 5th ed. Prentice
Hall.
[54] Hutton, J. E. and Nelson, P. I. (1986). Quasi-likelihood estimation for semimartingales.
Stochastic Processes and their Applications. 22, 245-257.
[55] Itˆo, K. (1961). Lectures on Stochastic Processes. Tata Institute of Fundamental
Research. Springer, Berlin.
[56] Jacod, J and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes.
Springer, Berlin.
[57] Jeanblanc, M., Pitman, J., and Yor, M. (2002). Self-similar processes with
independent increments associated with L´evy and Bessel processes. Stochastic
Processes and their Applications. 100, 223-231.
[58] Kao, E. P. C. (1997). An Introduction to Stochastic Processes. Duxbury Press.
[59] Karlin, S. (1967). A First Course in Stochastic Processes. Academic Press, New
York.
[60] Karlin, S. and Taylor, H. (1980). A Second Sourse in Stochastic Processes.
Academic Press, New York.
[61] Karr, A. F. (1991). Point Processes and their Statistical Inference. Marcel
Dekker, New York.
[62] Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.
[63] K¨uchler, U. (2004). On integrals with respect to L´evy processes. Statistics &
Probability Letters. 66, 145-151.
[64] K¨uchler, U. and Sørensen, M. (1997). Exponential Families of Stochastic Processes.
Springer.
[65] Kutoyants, Yu. A. (1984). Parameter Estimation for Stochastic Processes.
Translated and edited by Prakasa Rao, B. L. S. Heldermann, Verlag Berlin.
[66] Kutoyants, Yu. A. (2004). Statistical Inference for Ergodic Diffusion Processes.
Springer.
[67] LeBlanc, B. and Yor, M. (1998). L´evy Processes in finance: A remedy to the
non-stationarity of continuous martingales. Finance and Stochastics. 2, 399-408.
[68] Le´on, J. A., Sol´e, J. L., Utzet, F., and Vives, J. (2002). On L´evy processes,
Malliavin calculus and market models with jumps. Finance Stochast. 6, 197-
225.
[69] Leskow, J. and Rozanski, R. (1989). Maximum likelihood estimation of the drift
function for a diffusion process. Statistics and Decisions. 7, 243-262.
[70] L´evy, P. (1939). Sur certains processus stochastiques homogenes. Compositio
Math. 7, 283-339.
[71] L´evy, P. (1948). Processus Stochastiques et Mouvement Brownien. Gauthier-
Villars, Paris.
[72] L´evy, P. (1954). Th´eorie de L’addition des Variables Al´eatoires. 2nd edn.
Gauthier-Villars, Paris.
[73] Lin, S. (1995). Stochastic analysis of fractional Brownian motions. Stochastics
Stochastics Rep. 55, 121-140.
[74] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions,
fractional noises and applications. SIAM Rev. 10, 422-437.
[75] Mazo, R. M. (2002). Brownian Motion. Clarendon Press. Oxford.
[76] McCullagh, P. (1983). Quasi-Likelihood Functions. Annals of Statistics. 11, 59-
67.
[77] Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics
and Management Science. 4, 141-183.
[78] Mikosch, T., Resnick, S., Rootzen, H. and Stegeman, A. (2002). Is network
traffic approximated by stable L´evy motion or fractional Brownian motion?.
The Annals of Applied Probability. 12, 23-68.
[79] Mishra, M. N. and Bishwal, J. P. N. (1995). Approximate maximum likelihood
estimation for diffusion processes from discrete observations. Stochastics and
Stochastics Reports. 52, 1-13.
[80] Mishra, M. N. and Prakasa Rao, B. L. S. (1985a). Asymptotic study of the
maximum likelihood estimator for nonhomogeneous diffusion processes. Statistics
and Decisions. 3, 193-203.
[81] Mishra, M. N. and Prakasa Rao, B. L. S. (1985b). On the Berry-Esseen bound
for maximum likelihood estimator for linear homogeneous diffusion processes.
Sankhy¯a A. 47, 392-398.
[82] Morales, M. and Schoutens, W. (2003). A risk model driven by L´evy processes.
Appl. Stochastic Models Bus. Ind. 19, 147-167.
[83] Mtundu, N. D. and Koch, R. W. (1987). A stochastic differential equation
approach to soil moisture. Stochastic Hydrol. Hydraul. 1, 101-116.
[84] Muisela, M. and Rutkowski, R. (1998). Martingale Methods in Financial Modeling.
Springer.
[85] Nualart, D. and Schoutens, W. (2000). Chaotic and predictable representation
for L´evy processes. Stochastic Process Appl. 90, 109-122.
[86] Owen, A. B. (2001). Empirical Likelihood. Chapman and Hall.
[87] Parzen, E. (1962). Stochastic Processes. Holden-Day.
[88] Pedersen, A. R. (1995). Consistency and asymptotic normality of an approximate
maximum likelihood estimator for discretely observed diffusion processes.
Bernoulli. 1, 257-279.
[89] Philippe, C., Laure, C. and G´erard M. (2003). Stochastic integration with respect
to fractional Brownian motion. Ann. I. H. Poincar´e-PR39. 1, 27-68.
[90] Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional
Brownian motion. Probab. Theory Related Fields. 251-291.
[91] Prabhu, N. U., ed. (1988). Statistical Inference from Stochastic Processes. (Contemporary
Mathematics, Vol. 80). American Mathematical Society, Providence,
RI.
[92] Prabhu, N. U. and Basawa, I. V. (1991). Statistical Inference in Stochastic
Processes. Marcel Dekker, New York.
[93] Prakasa Rao, B. L. S. (1972). Maximum likelihood estimation for Markov processes.
Annals of the Institute of Statistical Mathematics. 24, 333-345.
[94] Prakasa Rao, B. L. S. (1999a). Statistical Inference for Diffusion Type Process.
Arnold.
[95] Prakasa Rao, B. L. S. (1999b). Semimartingales and their Statistical Inference.
Chapman and Hall.
[96] Prakasa Rao, B. L. S. and Bhat, B. R. (1996). Stochastic Processes and Statistical
Inference. New Age International, New Delhi.
[97] Protter, P. (1990). Stochastic Integration and Differential Equations. Springer.
[98] Reid, N. (2000). Likelihood. Journal of the American Statistical Association.
95, 1335-1340.
[99] Ren, Y. (2001). Construction of super-Brownian motions. Stochastic Analysis
and Applications. 19, 103-114.
[100] Revuz, D. and You, M. (1994). Continuous Martingales and Brownian Motion.
Springer.
[101] Robins, J. and Wasserman, L. (2000). Conditioning, Likelihood, and Coherence:
A Review of Some Foundational Concepts. 95, 1340-1346.
[102] Rogers, L. C. G. (1997). Arbitrage with fractional Brownian motion. Math.
Finance. 7, 95-105.
[103] Rogers, L. C. G. and Williams, D. (1994). Diffusions Markov Processes, and
Martingales vol. I: Foundations. [1st ed. by Williams, D., 1979.] Wiley.
[104] Ross, S. M. (1997). Introduction to Probability Models. 6th. ed., Academic Press.
[105] Sato, K. (1990). Stochastic Processes with Stationary Independent Increments.
Kinokuniya, Tokyo.
[106] Sato, K. (1995). L´evy Processes on the Euclidean Spaces. University of Zurich.
[107] Sato, K. (1999). L´evy Processes and Infinitely Divisible Distributions. Cambridge
University Press.
[108] Schoutens, W. (2003). L´evy Processes in Finance: Pricing Financial Derivatives.
Wiley.
[109] Severini, T. A. (2000). Likelihood Methods in Statistics. Oxford Univ. Press.
[110] Shorack, G. and Wellner, J. A. (1986). Empirical Processes with Applications
to Statistics. Wiley, New York.
[111] Skorohod, A. V. (1991). Random Processes with Independent Increments.
Kluwer, Dordrecht, Netherlands.
[112] Sørensen, M. (1983). On maximum likelihood estimation in randomly stopped
diffusion-type processes. International Statistical Review. 51, 95-110.
[113] Stark, H. and Woods, J. W. (2002). Probability and Random Processes with
Applications to Signal Processing. 3rd. ed., Prentice Hall.
[114] Taniguchi, M. and Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference
for Time Series. Springer.
[115] Taylor, S. J. (1973). Sample path properties of processes with stationary independent
increments. Stochastic Analysis, 387-414. Wiley, London.
[116] Tsou, T. S. and Royall, R. M. (1995). Robust likelihood. JASA. 90, 316-320.
[117] Wang, Y. (2002). An alternative approach to super-Brownian motion with a
locally infinite branching mass. Stochastic Processes and Applications. 102, 221-
233.
[118] Welsh, A. H. (1996). Aspects of Statistical Inference. Wiley.
[119] Wiener, N. (1924). Un probl`eme de probabilit´es d´enombrables. Bull Soc. Math.
France. 52, 569-578.
[120] Wiener, R. (1923). Differential space. J. Math. Phys. 2, 131-174.
[121] Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes.
Springer.
[122] Yoshimoto, A., Shoji, I. and Yoshimoto, Y. (1996). Application of a stochastic
control model to a free market policy of rice. Statistical Aalysis of Time Series.
Research Report No. 90, Institute of Statistical Mathematics. Tokyo.