基于非线性概率的一些大偏差结果及金融保险中的应用
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摘要
大偏差理论是概率论的极限理论中极富成果的一个分支,它处理和中心极限定理不同的另一类极限问题,是大数定律的精密化.在数理统计、分析和物理中都有重要的应用,它的理论溯源于Khintchine(1929),Cram(?)r(1938)和Chernoff(1952),Cram(?)r研究了关于独立随机变量和大偏差估计的先驱工作,之后,Schilder(1966)也开始在这个领域研究,七八十年代,Donsker和Varadhan在马氏过程方面工作以及Wentzell和Freidlin在动态系统随机扰动方面的工作使得大偏差理论得到了巨大的发展.Varadhan因创立了统一的大偏差原理而在2007年获得了Abel奖.
彭老师和陈老师在倒向随机微分方程领域的研究为我们也提供了一个新的环境,随着非线性数学期望、非线性概率理论的完善,大偏差理论如何应用到这一方面给我们提出了—个新的挑战.
第一章:这一章主要给出了本文所需要的预备知识及相关的讨论.通过一个统计中的例子,我们给出了大偏差及中偏差的定义.进而引出了Varadhan[9]提出的大偏差原理.
定义1.1.4 (大偏差原理(Varadhan[9]))
令I为X上的速率函数,称一列随机变量{ξ_n}在X上满足速率函数为I的大偏差原理,如果下列两个条件成立,
(a)大偏差上界.对X中任意闭集F,有
(b)大偏差下界.对X中任意开集G,有同时我们给出了由倒向随机微分方程引出的g-期望及g-概率的定义。
定义1.2.1假设g满足条件(H1),(H2)and(H3),且ξ∈L~2(Ω,F,P)。记(y_s,z_s)是倒向随机微分方程的解.
称ξ_g[ξ]为随机变量ξ的g-期望,定义为
称ξ_g[ξ|F_t]为随机变量ξ的条件g-期望,定义为
定义1.2.3假设g满足条件(H1),(H2)and(H3),给定A∈F,A的g-概率定义为
第二章:这一章是我的主要结果.第一部分讨论了—类特殊的非线性概率.
给定两个线性概率P,Q,分别满足速率函数为I_1(x),I_2(x)的大偏差原理,我们考虑经典均值问题.其中,对任意i,ξ_i,i=1,2,…是独立同分布的随机变量.
为了方便,我们假设
我们定义一个新的概率μ=P∧Q,并且考虑这个非线性概率μ的大偏差问题。
命题2.1.1令μ=P∧Q,其中P,Q是两个概率,且分别满足速率函数为I_1(x),I_2(x)的大偏差原理,则μ满足速率函数为I(x)的大偏差原理,其中
本章的第二部分讨论了g-期望的大偏差问题。首先,我们给出了g-概率下随机变量相互独立和同分布的定义:
定义2.2.1随机变量X_1,X_2,…,X_n在g概率P_g下是相互独立的,如果下式成立:且
定义2.2.2随机变量X_1,X_2,…,X_n在g概率P_g下是同分布的,如果下式成立:且
最后,我们得到如下命题:
命题2.2.4(g-期望下的Cram(?)r定理)
X_1,X_2,…,X_n为一列随机变量,使得对于任意i,X_i,i=1,2,…n,…在g概率P_μ下是独立同分布的,其中μ＞0.假设则其中即速率函数为
第三章:这一章主要讲了保险金融中的大偏差的应用及方法,分别介绍了保险模型、Cram(?)r-Lundberg估计及保险金融模型中的大偏差问题.
Laxge deviation theory, which deals with limit problems that are different from central limit theory, is a very fruitful branch of limit theory of probability theory, and it is the precision of the law of large number theory, and also has importantapplications in mathematical statistic, analysis and physics. The theory of large deviations is traced to Khintchine (1929) Cramer (1938) and Chernoff (1952). Cram(?)r's theorem is about the large deviations of the sample mean of a sequence of independent identically distributed random variables. After this, Ventcel and Preidlin theory on small random perturbations of ordinary differential equations by an It(?) type stochastic term is taken up. Donsker and Varadhan developed large deviation problems for the empirical distributions of time homogeneous Markov chains, and Varadhan receives the Abel prize "for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviation" in 2007.
The studies of Prof. Peng and Prof. Chen in the field of backward stochastic differential equations have provided us a new environment. With the development of the theory of nonlinear expectation and nonlinear probability, the application of the large deviation theory in this field is a new challenge for us.
Chapter 1: This Chapter is about the prior knowledge and related discussions.Through discussing a statistic problem, we give the definitions of large deviationand moderate deviation. In turn the large deviation principle of Varadhan [9] is given.
Definition 1.1.2 (Large Deviation Principle(Varadhan [9]))
Let I be a rate function on X. The sequence {ξ_n} is said to satisfy the large deviation principle on X with rate function I if the following two conditions hold.
(a)Large deviation upper bound. For each closed subset F of X
(b)Large deviation lower bound. For each open subset G of XAt the same time, we give the definitions of g-expectation and g-probability.
Definition 1.2.1 Assume that (H1),(H2) and (H3) hold on g andξ∈L~2(Ω,F,P). Let (y_s,z_s) be the solution of BSDE
ξ_g[ξ] is called the g-expectation of the random variableξ, defined by
ξ_g[ξ|F_t] is called the conditional g-expectation of the random variableξ, defined by
Definition 1.2.3 Assume that (H1),(H2) and (H3) hold on g. Given A∈F, the g-probability of A is defined by
Chapter 2: This chapter is my main results. The first part, discusses a special nonlinear probability.
Given two linear probabilities P, Q, which satisfy the large deviation principlewith rate functions I_1(x),I_2(x) respectively. We consider the classical mean problem.where for each i,ξ_i,i=1,2,…are independent, identically distributed (i.i.d) random variables.
For the sake of convenience, we suppose
We define the new probabilityμ=P∧Q, and consider the large deviation principle about the new nonlinear probabilityμ.
Proposition 2.1.1 Letμ=P∧Q, where P, Q are two probabilities and satisfy the large deviation principle with rate functions I_1(x),I_2(x) respectively, thenμsatisfies the large deviation with the rate function I(x), where
The second part the discussion on the large deviations of g-expectation. Firstly, we give the definitions of mutually independence and identically distribution of randomvariances under g-probability.
Definition 2.2.1 The random variables X_1,X_2,…,X_n are strongly, mutuallyindependent under g-probability P_g, ifand
Definition 2.2.2 The random variables X_1,X_2,…,X_n are strongly, identicallydistributed under g-probability P_g, ifandFinally, we get
Proposition 2.2.4 (Cram(?)r Theorem under g-Expectation)
Let X_1,X_2,…,X_n be a sequence of random variables, such that for each i, i,X_i,i=1,2,…n,…, are independent, identically distributed under g-probability P_μ, whereμ＞0. Assume thatThenwherei. e. the rate function is
Chapter 3: This chapter is about some applications and methods of large deviations in insurance and finance, we introduce the large deviation problems in the insurance model, the Cram(?)r-Lundberg estimate and the insurance-finance model.
彭老师和陈老师在倒向随机微分方程领域的研究为我们也提供了一个新的环境,随着非线性数学期望、非线性概率理论的完善,大偏差理论如何应用到这一方面给我们提出了—个新的挑战.
第一章:这一章主要给出了本文所需要的预备知识及相关的讨论.通过一个统计中的例子,我们给出了大偏差及中偏差的定义.进而引出了Varadhan[9]提出的大偏差原理.
定义1.1.4 (大偏差原理(Varadhan[9]))
令I为X上的速率函数,称一列随机变量{ξ_n}在X上满足速率函数为I的大偏差原理,如果下列两个条件成立,
(a)大偏差上界.对X中任意闭集F,有
(b)大偏差下界.对X中任意开集G,有同时我们给出了由倒向随机微分方程引出的g-期望及g-概率的定义。
定义1.2.1假设g满足条件(H1),(H2)and(H3),且ξ∈L~2(Ω,F,P)。记(y_s,z_s)是倒向随机微分方程的解.
称ξ_g[ξ]为随机变量ξ的g-期望,定义为
称ξ_g[ξ|F_t]为随机变量ξ的条件g-期望,定义为
定义1.2.3假设g满足条件(H1),(H2)and(H3),给定A∈F,A的g-概率定义为
第二章:这一章是我的主要结果.第一部分讨论了—类特殊的非线性概率.
给定两个线性概率P,Q,分别满足速率函数为I_1(x),I_2(x)的大偏差原理,我们考虑经典均值问题.其中,对任意i,ξ_i,i=1,2,…是独立同分布的随机变量.
为了方便,我们假设
我们定义一个新的概率μ=P∧Q,并且考虑这个非线性概率μ的大偏差问题。
命题2.1.1令μ=P∧Q,其中P,Q是两个概率,且分别满足速率函数为I_1(x),I_2(x)的大偏差原理,则μ满足速率函数为I(x)的大偏差原理,其中
本章的第二部分讨论了g-期望的大偏差问题。首先,我们给出了g-概率下随机变量相互独立和同分布的定义:
定义2.2.1随机变量X_1,X_2,…,X_n在g概率P_g下是相互独立的,如果下式成立:且
定义2.2.2随机变量X_1,X_2,…,X_n在g概率P_g下是同分布的,如果下式成立:且
最后,我们得到如下命题:
命题2.2.4(g-期望下的Cram(?)r定理)
X_1,X_2,…,X_n为一列随机变量,使得对于任意i,X_i,i=1,2,…n,…在g概率P_μ下是独立同分布的,其中μ＞0.假设则其中即速率函数为
第三章:这一章主要讲了保险金融中的大偏差的应用及方法,分别介绍了保险模型、Cram(?)r-Lundberg估计及保险金融模型中的大偏差问题.
Laxge deviation theory, which deals with limit problems that are different from central limit theory, is a very fruitful branch of limit theory of probability theory, and it is the precision of the law of large number theory, and also has importantapplications in mathematical statistic, analysis and physics. The theory of large deviations is traced to Khintchine (1929) Cramer (1938) and Chernoff (1952). Cram(?)r's theorem is about the large deviations of the sample mean of a sequence of independent identically distributed random variables. After this, Ventcel and Preidlin theory on small random perturbations of ordinary differential equations by an It(?) type stochastic term is taken up. Donsker and Varadhan developed large deviation problems for the empirical distributions of time homogeneous Markov chains, and Varadhan receives the Abel prize "for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviation" in 2007.
The studies of Prof. Peng and Prof. Chen in the field of backward stochastic differential equations have provided us a new environment. With the development of the theory of nonlinear expectation and nonlinear probability, the application of the large deviation theory in this field is a new challenge for us.
Chapter 1: This Chapter is about the prior knowledge and related discussions.Through discussing a statistic problem, we give the definitions of large deviationand moderate deviation. In turn the large deviation principle of Varadhan [9] is given.
Definition 1.1.2 (Large Deviation Principle(Varadhan [9]))
Let I be a rate function on X. The sequence {ξ_n} is said to satisfy the large deviation principle on X with rate function I if the following two conditions hold.
(a)Large deviation upper bound. For each closed subset F of X
(b)Large deviation lower bound. For each open subset G of XAt the same time, we give the definitions of g-expectation and g-probability.
Definition 1.2.1 Assume that (H1),(H2) and (H3) hold on g andξ∈L~2(Ω,F,P). Let (y_s,z_s) be the solution of BSDE
ξ_g[ξ] is called the g-expectation of the random variableξ, defined by
ξ_g[ξ|F_t] is called the conditional g-expectation of the random variableξ, defined by
Definition 1.2.3 Assume that (H1),(H2) and (H3) hold on g. Given A∈F, the g-probability of A is defined by
Chapter 2: This chapter is my main results. The first part, discusses a special nonlinear probability.
Given two linear probabilities P, Q, which satisfy the large deviation principlewith rate functions I_1(x),I_2(x) respectively. We consider the classical mean problem.where for each i,ξ_i,i=1,2,…are independent, identically distributed (i.i.d) random variables.
For the sake of convenience, we suppose
We define the new probabilityμ=P∧Q, and consider the large deviation principle about the new nonlinear probabilityμ.
Proposition 2.1.1 Letμ=P∧Q, where P, Q are two probabilities and satisfy the large deviation principle with rate functions I_1(x),I_2(x) respectively, thenμsatisfies the large deviation with the rate function I(x), where
The second part the discussion on the large deviations of g-expectation. Firstly, we give the definitions of mutually independence and identically distribution of randomvariances under g-probability.
Definition 2.2.1 The random variables X_1,X_2,…,X_n are strongly, mutuallyindependent under g-probability P_g, ifand
Definition 2.2.2 The random variables X_1,X_2,…,X_n are strongly, identicallydistributed under g-probability P_g, ifandFinally, we get
Proposition 2.2.4 (Cram(?)r Theorem under g-Expectation)
Let X_1,X_2,…,X_n be a sequence of random variables, such that for each i, i,X_i,i=1,2,…n,…, are independent, identically distributed under g-probability P_μ, whereμ＞0. Assume thatThenwherei. e. the rate function is
Chapter 3: This chapter is about some applications and methods of large deviations in insurance and finance, we introduce the large deviation problems in the insurance model, the Cram(?)r-Lundberg estimate and the insurance-finance model.
引文
[1] A. Dembo and O. Zeitouni. "Large Deviations Techniques and Applications". Springer, 2nd edition, 1998.
[2] Asmussen S. "Ruin Probabilities". World Scientific, 2000.
[3] A. Puhalskii. "Large Deviations and Idempotent Probability". Chapman & Hall/CRC, 2001.
[4] D. Duffie, "Dynamic Aasset Picing Theory". Princeton University Press, 1992.
[5] Djehiche B., "A Large Deviation Estimate for Ruin Probabilities". Scandinavian Actuarial Journal, 1(1993), 42-59.
[6] D.W. Stroock. "An Introduction to the Theory of Large Deviations ". Spring-Verlag, 1984.
[7] Enzo Olivieri, Maria Eul (?) lia Vares. "Large Deviations and Metastability". Cambridge University Press, 2005.
[8] E. Pardoux and S. Peng. "Adapted Solution of a Backward Stochastic Differential Equation". Systems and Control Letters, 14(1990), 55-61.
[9] F. Jin and Thomas G. Kurtz. "Large Deviations for Stochastic Processes". Mathematics Subject Classification, 2000.
[10] Huyen Pham. "Some Applications and Methods of Large Deviations in Finance and Insurance". In Paris-Princeton Lectures on Mathematical Finance, 2007.
[11] J. D. Deuschel and D. W. Stroock. "Large Deviations". Academic Press, 1989.
[12] J. D. Deuschel and D. W. Stroock. "Large Deviations". American Mathematical Society, second edition, 2001.
[13] J. M. Bismut, "Conjugate Convex Functions in Optimal Stochastic Control". J. Math. Anal. Apl., 44(1973), 384-404.
[14] Lundberg,F. "Approximerad framst(?)llning av sannoliklietsfunktionen ". (?)terf(?)rs(?)kring av kollektivrisker, 1903.
[15] M.I. Freidlin and A.D. Wentzell. "Random Perturbations of Dynamical Systems" . Springer, 2nd edition, 1998.
[16] N. El Karoui, S. Peng and M. C. Quenez. "Backward Stochastic Differential Equation in Finance". Math. Finance, 1997, 1-71.
[17] Olav Kallenberg.“现代概率论基础”.北京-科学出版社,2001.
[18] Paul Dupuis, Richard S. Ellis, "A Weak Convergence Approach To The Theory Of Large Deviations". Wiley-Interscience, 1997.
[19] P. Briand, F. Coquet, Y. Hu. J. M(?)min and S. Peng. "A Converse Comparison Theorem for BSDEs and Related Properties of g-Expectation". Elect. Comm. in Probab., 5(2000), 101-117.
[20] Paul Embrechts, Claudia Kluppelberg, Thomas Mikosch. "Modelling Extremal Events for Insurance and Finance". Springer, 2003.
[21] Richard S. Ellis, "Entropy, Large Deviations, and Statistical Mechanics". Springer-Verlag, 1992.
[22] Peter Whittle. "Probability via expectation". World Publishing Corporation, 1998.
[23] S. Peng. "BSDE and related g-expectation". Pitman Research Notes in Mathematics Series, 364(1997), 141-159.
[24] S. R. S. Varadhan. "Large Deviations and Applications". SIAM, 1984.
[25] William Feller. "An Introduction to Probability Theory and Its Applications". Wiley, 1957-1971.
[26] Y. Hu and S. Peng. "Solution of Forward-Backward Stochastic Differential Equations". Prob. Theory and Rel. Fields, 103(1995), 273-283.
[27] 严加安等著.“随机分析选讲”.北京-科学出版社,1997.6.
[28] Z. Chen, R. Kulperger and G. Wei. "A Comonotonic Theorem for BSDEs". Stochastic Processes and their Applications. 115(2005), 41-54.
[29] Z. Chen, T. Chen and M. Davison, "Choquet Expectation and Peng' s g-Expectation". Annals of Probability, Vol.33, No. 3 (2005) 1179-1199.
[2] Asmussen S. "Ruin Probabilities". World Scientific, 2000.
[3] A. Puhalskii. "Large Deviations and Idempotent Probability". Chapman & Hall/CRC, 2001.
[4] D. Duffie, "Dynamic Aasset Picing Theory". Princeton University Press, 1992.
[5] Djehiche B., "A Large Deviation Estimate for Ruin Probabilities". Scandinavian Actuarial Journal, 1(1993), 42-59.
[6] D.W. Stroock. "An Introduction to the Theory of Large Deviations ". Spring-Verlag, 1984.
[7] Enzo Olivieri, Maria Eul (?) lia Vares. "Large Deviations and Metastability". Cambridge University Press, 2005.
[8] E. Pardoux and S. Peng. "Adapted Solution of a Backward Stochastic Differential Equation". Systems and Control Letters, 14(1990), 55-61.
[9] F. Jin and Thomas G. Kurtz. "Large Deviations for Stochastic Processes". Mathematics Subject Classification, 2000.
[10] Huyen Pham. "Some Applications and Methods of Large Deviations in Finance and Insurance". In Paris-Princeton Lectures on Mathematical Finance, 2007.
[11] J. D. Deuschel and D. W. Stroock. "Large Deviations". Academic Press, 1989.
[12] J. D. Deuschel and D. W. Stroock. "Large Deviations". American Mathematical Society, second edition, 2001.
[13] J. M. Bismut, "Conjugate Convex Functions in Optimal Stochastic Control". J. Math. Anal. Apl., 44(1973), 384-404.
[14] Lundberg,F. "Approximerad framst(?)llning av sannoliklietsfunktionen ". (?)terf(?)rs(?)kring av kollektivrisker, 1903.
[15] M.I. Freidlin and A.D. Wentzell. "Random Perturbations of Dynamical Systems" . Springer, 2nd edition, 1998.
[16] N. El Karoui, S. Peng and M. C. Quenez. "Backward Stochastic Differential Equation in Finance". Math. Finance, 1997, 1-71.
[17] Olav Kallenberg.“现代概率论基础”.北京-科学出版社,2001.
[18] Paul Dupuis, Richard S. Ellis, "A Weak Convergence Approach To The Theory Of Large Deviations". Wiley-Interscience, 1997.
[19] P. Briand, F. Coquet, Y. Hu. J. M(?)min and S. Peng. "A Converse Comparison Theorem for BSDEs and Related Properties of g-Expectation". Elect. Comm. in Probab., 5(2000), 101-117.
[20] Paul Embrechts, Claudia Kluppelberg, Thomas Mikosch. "Modelling Extremal Events for Insurance and Finance". Springer, 2003.
[21] Richard S. Ellis, "Entropy, Large Deviations, and Statistical Mechanics". Springer-Verlag, 1992.
[22] Peter Whittle. "Probability via expectation". World Publishing Corporation, 1998.
[23] S. Peng. "BSDE and related g-expectation". Pitman Research Notes in Mathematics Series, 364(1997), 141-159.
[24] S. R. S. Varadhan. "Large Deviations and Applications". SIAM, 1984.
[25] William Feller. "An Introduction to Probability Theory and Its Applications". Wiley, 1957-1971.
[26] Y. Hu and S. Peng. "Solution of Forward-Backward Stochastic Differential Equations". Prob. Theory and Rel. Fields, 103(1995), 273-283.
[27] 严加安等著.“随机分析选讲”.北京-科学出版社,1997.6.
[28] Z. Chen, R. Kulperger and G. Wei. "A Comonotonic Theorem for BSDEs". Stochastic Processes and their Applications. 115(2005), 41-54.
[29] Z. Chen, T. Chen and M. Davison, "Choquet Expectation and Peng' s g-Expectation". Annals of Probability, Vol.33, No. 3 (2005) 1179-1199.