鞅变换及其相关问题
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摘要
本文主要考虑了右过程的鞅变换的相关问题。首先，我们考虑了满足一定条件的右过程在Girsanov变换下的转移概率密度的表达式问题；其次，我们考虑了由Markov调制的Lévy过程的最小相对熵鞅测度的问题，证明了其最小相对熵鞅测度是某个状态转换Esscher变换；最后，我们考虑了右过程的不变测度及其遍历性的问题。
本文的具体安排如下：
本文主要分为五章。第一章介绍了第三、四、五这三章我们所研究问题的背景以及我们得到的结果。第二章简单介绍了后面三章里所用到的重要概念和重要的定理以及一些重要的性质，有关第二章的内容可参见[1]，[15]，[16]，[25]，[26]，[31]。
在第三章里，我们得到了右过程在Girsanov变换下的转移概率密度表示公式。Qian和Zheng([24])建立了由一个向量场扰动的扩散过程的转移概率密度表示公式，他们所考虑的过程是扩散过程，不需要考虑Lévy系。在这一章里，我们考虑更一般的情况，也就是带有跳的右过程，由于我们所考虑的右过程具有跳，因此需要计算Markov桥(首次在[11]中出现它的定义)和变换后过程的Lévy系，我们得到的右过程在Girsanov变换下的转移概率密度表示公式，对于获得由漂移变换后过程的转移密度函数的信息非常有用，因此其本身具有重要的理论和实际价值。此外，我们还得到了右过程在Essche变换下的转移密度表示公式及变换后过程的无穷小生成元。
在第四章里，我们得到了Markov交换Lévy过程的最小熵鞅测度。Fujiwara和Miyahara[12]与Esche和Schweizer([9])分别研究了几何Lévy过程与Lévy过程的最小熵鞅测度。Elliott，Chan和Siu([6])研究了当风险资产是由Markov调制的几何Brown运动驱动的期权定价问题，采用了状态转换Esscher变换(是文献[28]中介绍的随机Esscher变换的修正)，得到了由Markov调制的几何Brown运动的最小熵鞅测度。Elliott和Osakwe([8])研究了具有Markov交换补偿子的纯跳过程的期权定价问题。在这一章里，我们研究了当风险资产是由Markov调制的Lévy过程的随机指数所驱动的不完备市场下的期权定价问题，证明了最小相对熵鞅测度是某个状态转换Esscher变换。
在第五章里，我们得到了常返右过程的不变测度的存在性、唯一性及其遍历性。对于正常返的Markov链而言，它存在唯一的不变测度，在什么条件下一个Markov过程存在不变测度是非常有趣的问题，这个问题已经被一些学者研究过。文献[21]和文献[29]，证明了一维常返扩散过程存在唯一的不变测度。Khas'minskii([19])研究了σ-紧完备距离空间上常返扩散过程的遍历性。Maruyama和Tanaka([22])研究了N-维欧氏空间上常返且具有强Markov性的Markov过程的遍历性问题。在这一章里，我们考虑了Polish空间上的常返右过程的不变测度的存在性及其遍历性的问题。
The present Ph.D. thesis mainly deals with the the martingale transform for right processes and related problems. Firstly, we consider the problem of the representation formula for the transition probability density of a right process under Girsanov transform. Secondly,we consider the problem of the MEMMs (minimal entropy martingale measure)for Markov switching Levy processes, and justify that the minimal entropy martingale measure is obtained by some regime switching Esscher transform. At last,we consider the problem of the invariant measure and the ergodic property of recurrent right processes.
This paper includes 5 chapters. In chapter 1, we introduce the background of the third,the fourth and the fifth chapters and our results. In chapter 2, we introduce some important concepts,theorems and properties which are used in the following three chapters. For the content of the second chapter, we can see [1],[15], [16], [25],[26],[31].
In chapter 3, we obtain the representation formula for the transition probability density of a right process under Girsanov transform. Qian and Zheng ([24]) established a representation formula for the transition probability density of a diffusion perturbed by a vector field. In the case of [24], they considered the diffusion processes where the Levy systems disappeared. In this chapter, we shall consider a much more general case that is a right process with jumps.While in our case, since the process has jumps, we have to compute Levy systems of the Markovian bridge (which is firstly defined in [11]) and the transformed process. The representation formula for the transition probability density of a right process under Girsanov transform we obtained is very useful in obtaining information about the density functions perturbed by a drift transform. Therefore it has theoretic and practical values by its own. We also obtain the representation formula for the transition probability density of a right process under Esscher transform and the infinitesimal generator of the transformed process.
In chapter 4,we obtain the MEMMs for the Markov switching Levy processes. The MEMMs for Levy processes and geometric Levy processes have been studied by [9]and [12]. Elliott,Chan and Siu ([6]) investigated the option pricing problem when the risky underlying assets were driven by Markov-modulated Geometric Brownian Motion. There they adopted the regime switching Esscher transform which was the modification of the random Esscher transform introduced by [28]. They justified their pricing result by the minimal entropy martingale measure. Elliott and Osakwe ([8])studied the option pricing for pure jump processes with Markov switching compensators. In this chapter,we investigate the option pricing problem when the risky underlying assets are driven by Markov-modulated Levy process and we justify that the minimal entropy martingale measure is obtained by some regime switching Esscher transform.
In chapter 5, we obtain the invariant measure and the ergodic property of recurrent right processes. For a positive recurrent Markov chain, there exists a unique invariant distribution. It is interesting to study that under what conditions a Markov process has an invariant measure. This question has been studied by many authors. [21] and [29] had proved the existence and uniqueness of an invariant measure and the ergodic property for one-dimentional recurrent diffusion processes. Khas'minskii ([19]) studied the ergodic properties of recurrent diffusion processes on aσ-compact complete metric space. Maruyama and Tanaka ([22])studied the same questions for recurrent Markov process in N-dimentional Euclidean space R~N which have right continuous paths and the strong Markov property. In this chapter, we consider the recurrent right process on Polish space.
本文的具体安排如下：
本文主要分为五章。第一章介绍了第三、四、五这三章我们所研究问题的背景以及我们得到的结果。第二章简单介绍了后面三章里所用到的重要概念和重要的定理以及一些重要的性质，有关第二章的内容可参见[1]，[15]，[16]，[25]，[26]，[31]。
在第三章里，我们得到了右过程在Girsanov变换下的转移概率密度表示公式。Qian和Zheng([24])建立了由一个向量场扰动的扩散过程的转移概率密度表示公式，他们所考虑的过程是扩散过程，不需要考虑Lévy系。在这一章里，我们考虑更一般的情况，也就是带有跳的右过程，由于我们所考虑的右过程具有跳，因此需要计算Markov桥(首次在[11]中出现它的定义)和变换后过程的Lévy系，我们得到的右过程在Girsanov变换下的转移概率密度表示公式，对于获得由漂移变换后过程的转移密度函数的信息非常有用，因此其本身具有重要的理论和实际价值。此外，我们还得到了右过程在Essche变换下的转移密度表示公式及变换后过程的无穷小生成元。
在第四章里，我们得到了Markov交换Lévy过程的最小熵鞅测度。Fujiwara和Miyahara[12]与Esche和Schweizer([9])分别研究了几何Lévy过程与Lévy过程的最小熵鞅测度。Elliott，Chan和Siu([6])研究了当风险资产是由Markov调制的几何Brown运动驱动的期权定价问题，采用了状态转换Esscher变换(是文献[28]中介绍的随机Esscher变换的修正)，得到了由Markov调制的几何Brown运动的最小熵鞅测度。Elliott和Osakwe([8])研究了具有Markov交换补偿子的纯跳过程的期权定价问题。在这一章里，我们研究了当风险资产是由Markov调制的Lévy过程的随机指数所驱动的不完备市场下的期权定价问题，证明了最小相对熵鞅测度是某个状态转换Esscher变换。
在第五章里，我们得到了常返右过程的不变测度的存在性、唯一性及其遍历性。对于正常返的Markov链而言，它存在唯一的不变测度，在什么条件下一个Markov过程存在不变测度是非常有趣的问题，这个问题已经被一些学者研究过。文献[21]和文献[29]，证明了一维常返扩散过程存在唯一的不变测度。Khas'minskii([19])研究了σ-紧完备距离空间上常返扩散过程的遍历性。Maruyama和Tanaka([22])研究了N-维欧氏空间上常返且具有强Markov性的Markov过程的遍历性问题。在这一章里，我们考虑了Polish空间上的常返右过程的不变测度的存在性及其遍历性的问题。
The present Ph.D. thesis mainly deals with the the martingale transform for right processes and related problems. Firstly, we consider the problem of the representation formula for the transition probability density of a right process under Girsanov transform. Secondly,we consider the problem of the MEMMs (minimal entropy martingale measure)for Markov switching Levy processes, and justify that the minimal entropy martingale measure is obtained by some regime switching Esscher transform. At last,we consider the problem of the invariant measure and the ergodic property of recurrent right processes.
This paper includes 5 chapters. In chapter 1, we introduce the background of the third,the fourth and the fifth chapters and our results. In chapter 2, we introduce some important concepts,theorems and properties which are used in the following three chapters. For the content of the second chapter, we can see [1],[15], [16], [25],[26],[31].
In chapter 3, we obtain the representation formula for the transition probability density of a right process under Girsanov transform. Qian and Zheng ([24]) established a representation formula for the transition probability density of a diffusion perturbed by a vector field. In the case of [24], they considered the diffusion processes where the Levy systems disappeared. In this chapter, we shall consider a much more general case that is a right process with jumps.While in our case, since the process has jumps, we have to compute Levy systems of the Markovian bridge (which is firstly defined in [11]) and the transformed process. The representation formula for the transition probability density of a right process under Girsanov transform we obtained is very useful in obtaining information about the density functions perturbed by a drift transform. Therefore it has theoretic and practical values by its own. We also obtain the representation formula for the transition probability density of a right process under Esscher transform and the infinitesimal generator of the transformed process.
In chapter 4,we obtain the MEMMs for the Markov switching Levy processes. The MEMMs for Levy processes and geometric Levy processes have been studied by [9]and [12]. Elliott,Chan and Siu ([6]) investigated the option pricing problem when the risky underlying assets were driven by Markov-modulated Geometric Brownian Motion. There they adopted the regime switching Esscher transform which was the modification of the random Esscher transform introduced by [28]. They justified their pricing result by the minimal entropy martingale measure. Elliott and Osakwe ([8])studied the option pricing for pure jump processes with Markov switching compensators. In this chapter,we investigate the option pricing problem when the risky underlying assets are driven by Markov-modulated Levy process and we justify that the minimal entropy martingale measure is obtained by some regime switching Esscher transform.
In chapter 5, we obtain the invariant measure and the ergodic property of recurrent right processes. For a positive recurrent Markov chain, there exists a unique invariant distribution. It is interesting to study that under what conditions a Markov process has an invariant measure. This question has been studied by many authors. [21] and [29] had proved the existence and uniqueness of an invariant measure and the ergodic property for one-dimentional recurrent diffusion processes. Khas'minskii ([19]) studied the ergodic properties of recurrent diffusion processes on aσ-compact complete metric space. Maruyama and Tanaka ([22])studied the same questions for recurrent Markov process in N-dimentional Euclidean space R~N which have right continuous paths and the strong Markov property. In this chapter, we consider the recurrent right process on Polish space.
引文
[1] Bertoin,J.: Levy processes. Cambridge University Press,1996.
[2] Chan,T.:Pricing contingent claims on stocks driven by Levy processes.Ann.Appl.Probab.,9(2): PP.504-528, 1999.
[3] Delbean,F.;Grandits,P.;Rheinlander,T.;Samperi,D.;Schweizer,M.;Stricker,C.: Exponential hedging and entropic penalties. Mathematical Finance 12,PP.99-123,2002.
[4] Doob,J.L.:Asymptotic properties of Markov transition probabilities.Trans.Amer.Math.Soc, 63, 3, PP.393-421,1948.
[5] Elliott,R.J.;Aggoun,L.;Moore,J.B.:Hidden Markov models:estimation and control. Berlin Heiderberg New York: Springer 1994.
[6] Elliott,R.J.; Chan,L.; Siu,T.K.: Option pricing and Esscher transform under regime switching. Ann. Finance 1,PP.423-432,2005.
[7] Elliott,R.J.;Kopp,P.E.: Mathematics of financial markets.Berlin Heidelberg New York:Springer 1999.
[8] Elliott,R.J.; Osakwe, C.-J. U. :Option pricing for pure jump processes with Markov switching compensators. Finance and Stochastic 10, PP.250-275,2006.
[9] Esche,F.; Schweizer, M.: Minimal entropy preserves the Levy property: how and why. Stochastic Processes and their Applications 115,PP.299-327,2005.
[10] Esscher,F.: On the probability function in the collective theory of risk.Skandinavisk Aktuarietidskrift 15,PP.175-195,1932.
[11] Fitsimmons,P.; Pitman,J. ; Yor,M.: Markovian Bridges:Construction, Palm Interpretation, and Splicing. Seminar on Stochastic Processes,PP. 101-133,1992.
[12] Fujiwara,T.; Miyahara,Y.: The minimal entropy martingale measures for geometric Levy processes. Finance Stochastic 7,PP.509-531,2003.
[13] Frittelli,M.: The minimal entropy martingale measure and the valuation problem in incomplete markets.Math.Finance 10,PP.39-52,2000.
[14] Gerber,H.U.;Shiu,E.S.W.:Option opricing by Esscher transforms(with discussion).Trans.Soc.Actuaries 46,PP.99-191, 1994
[15] He,S.; Wang,J.; Yan,J. : Semimartingale Theory and Stochastic Calculus. CRC Press, 1992.
[16] Jacod,J. and Shiryaev,A.N.:Limit Theorems for Stochastic Processes. Springer-Verlag Berlin Heidelberg, 1987.
[17] Kabanov,Y.;Stricker,C.: On the optimal portfolio for the exponential utility maximization:Remarks to the six-author paper.Math.Fiance 12,PP.125-134,2002.
[18] Kallenberg,O.: Foundations of Modern Probability, Springer, 2001.
[19] Khas'minskii,R.Z.: Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parablic equations.Theory Prob.Applications, Vol.V, No.2,179-196, 1960.
[20] Kuchler,U.and Sφrensen,M.:Exponential families of stochastic processes. Springer-Verlag New York,Inc.1997.
[21] Maruyama,G. and Tanaka,H.: Some propertiesof one-dimentional diffusion processes. Mem.Fac.Sci. Kyushu Univ. A-11, 2, PP.117-421, 1957.
[22] Maruyama,G. and Tanaka,H.: Ergodic property of N-dimentional recurrent Markov processes. Mem.Fac.Sci.Kyushu Univ. , Ser.A.Vol.13, No.2,157-172, 1959.
[23] Meyn,S.P.and Tweedie,R.L.: Markov Chains and Stochastic Stability. Springer-Verlag 1993.
[24] Qian,Z.;Zheng,W., A representation formula for transition probability densities of diffusions and applications. Stochastic processes and their applications, 111,PP. 57-76,2004.
[25] Sato,K:Levy processes and infinitely divisible distributions. Cambridge:Cambridge University Press 1999.
[26] Sharpe,M., General Theory of Markov Processes. Academic Press.Inc.,1988.
[27] Schachermayer,W.:A super-martingale property of the optional portforlio process.Finance Stochast. 7,PP.433-456,2003.
[28] Siu,T.K., Tong,H., Yang,H.: Bayesian risk measures for derivatives via random Esscher transform.North American Actuarial Journal 5(3), PP. 78-91, 2001.
[29] Watanabe, Motoo, Ergodic property of recurrent diffusion processes. J.Math. Soc. Japan, 10, 3, PP.272-286,1958.
[30] Ying,J., Bivariate Revuz measures and the Feynman-Kac formula. Ann.Inst.Henri Poincare,Vol.32, No2, PP.251-287,1996.
[31] 应坚刚，金蒙伟：随机过程基础．复旦大学出版社，2005．
[2] Chan,T.:Pricing contingent claims on stocks driven by Levy processes.Ann.Appl.Probab.,9(2): PP.504-528, 1999.
[3] Delbean,F.;Grandits,P.;Rheinlander,T.;Samperi,D.;Schweizer,M.;Stricker,C.: Exponential hedging and entropic penalties. Mathematical Finance 12,PP.99-123,2002.
[4] Doob,J.L.:Asymptotic properties of Markov transition probabilities.Trans.Amer.Math.Soc, 63, 3, PP.393-421,1948.
[5] Elliott,R.J.;Aggoun,L.;Moore,J.B.:Hidden Markov models:estimation and control. Berlin Heiderberg New York: Springer 1994.
[6] Elliott,R.J.; Chan,L.; Siu,T.K.: Option pricing and Esscher transform under regime switching. Ann. Finance 1,PP.423-432,2005.
[7] Elliott,R.J.;Kopp,P.E.: Mathematics of financial markets.Berlin Heidelberg New York:Springer 1999.
[8] Elliott,R.J.; Osakwe, C.-J. U. :Option pricing for pure jump processes with Markov switching compensators. Finance and Stochastic 10, PP.250-275,2006.
[9] Esche,F.; Schweizer, M.: Minimal entropy preserves the Levy property: how and why. Stochastic Processes and their Applications 115,PP.299-327,2005.
[10] Esscher,F.: On the probability function in the collective theory of risk.Skandinavisk Aktuarietidskrift 15,PP.175-195,1932.
[11] Fitsimmons,P.; Pitman,J. ; Yor,M.: Markovian Bridges:Construction, Palm Interpretation, and Splicing. Seminar on Stochastic Processes,PP. 101-133,1992.
[12] Fujiwara,T.; Miyahara,Y.: The minimal entropy martingale measures for geometric Levy processes. Finance Stochastic 7,PP.509-531,2003.
[13] Frittelli,M.: The minimal entropy martingale measure and the valuation problem in incomplete markets.Math.Finance 10,PP.39-52,2000.
[14] Gerber,H.U.;Shiu,E.S.W.:Option opricing by Esscher transforms(with discussion).Trans.Soc.Actuaries 46,PP.99-191, 1994
[15] He,S.; Wang,J.; Yan,J. : Semimartingale Theory and Stochastic Calculus. CRC Press, 1992.
[16] Jacod,J. and Shiryaev,A.N.:Limit Theorems for Stochastic Processes. Springer-Verlag Berlin Heidelberg, 1987.
[17] Kabanov,Y.;Stricker,C.: On the optimal portfolio for the exponential utility maximization:Remarks to the six-author paper.Math.Fiance 12,PP.125-134,2002.
[18] Kallenberg,O.: Foundations of Modern Probability, Springer, 2001.
[19] Khas'minskii,R.Z.: Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parablic equations.Theory Prob.Applications, Vol.V, No.2,179-196, 1960.
[20] Kuchler,U.and Sφrensen,M.:Exponential families of stochastic processes. Springer-Verlag New York,Inc.1997.
[21] Maruyama,G. and Tanaka,H.: Some propertiesof one-dimentional diffusion processes. Mem.Fac.Sci. Kyushu Univ. A-11, 2, PP.117-421, 1957.
[22] Maruyama,G. and Tanaka,H.: Ergodic property of N-dimentional recurrent Markov processes. Mem.Fac.Sci.Kyushu Univ. , Ser.A.Vol.13, No.2,157-172, 1959.
[23] Meyn,S.P.and Tweedie,R.L.: Markov Chains and Stochastic Stability. Springer-Verlag 1993.
[24] Qian,Z.;Zheng,W., A representation formula for transition probability densities of diffusions and applications. Stochastic processes and their applications, 111,PP. 57-76,2004.
[25] Sato,K:Levy processes and infinitely divisible distributions. Cambridge:Cambridge University Press 1999.
[26] Sharpe,M., General Theory of Markov Processes. Academic Press.Inc.,1988.
[27] Schachermayer,W.:A super-martingale property of the optional portforlio process.Finance Stochast. 7,PP.433-456,2003.
[28] Siu,T.K., Tong,H., Yang,H.: Bayesian risk measures for derivatives via random Esscher transform.North American Actuarial Journal 5(3), PP. 78-91, 2001.
[29] Watanabe, Motoo, Ergodic property of recurrent diffusion processes. J.Math. Soc. Japan, 10, 3, PP.272-286,1958.
[30] Ying,J., Bivariate Revuz measures and the Feynman-Kac formula. Ann.Inst.Henri Poincare,Vol.32, No2, PP.251-287,1996.
[31] 应坚刚，金蒙伟：随机过程基础．复旦大学出版社，2005．