The Markov chain is well studied and widely applied in many areas.For some Markov chains,it is infeasible to obtain the explicit expressions of their corresponding finite-dimensional distributions and sometimes it is ...The Markov chain is well studied and widely applied in many areas.For some Markov chains,it is infeasible to obtain the explicit expressions of their corresponding finite-dimensional distributions and sometimes it is time-consuming for computation.In this paper,we propose an approximation method for Markov chains by applying the copula theory.For this purpose,we first discuss the checkerboard copula-based Markov chain,which is the Markov chain generated by the family of checkerboard copulas.This Markov chain has some appealing properties,such as self-similarity in copulas and having explicit forms of finite-dimensional distributions.Then we prove that each Markov chain can be approximated by a sequence of checkerboard copula-based Markov chains,and the error bounds of the approximate distributions are provided.Employing the checkerboard copula-based approximation method,we propose a sufficient condition for the geometric β-mixing of copula-based Markov chains.This condition allows copulas of Markov chains to be asymmetric.Finally,by applying the approximation method,analytical recurrence formulas are also derived for computing approximate distributions of both the first passage time and the occupation time of a Markov chain,and numerical results are listed to show the approximation errors.展开更多
A credit-linked note(CLN)is a note paying an enhanced coupon to investors for bearing the credit risk of a reference entity.In this paper,we study the counterparty risk on CLNs under a Markov chain framework,and intro...A credit-linked note(CLN)is a note paying an enhanced coupon to investors for bearing the credit risk of a reference entity.In this paper,we study the counterparty risk on CLNs under a Markov chain framework,and introduce a Markov copula model to describe joint defaults between the reference entity underlying the CLN and CLN issuer.Assuming that the respective default intensities are directly and inversely proportional to the interest rate,which follows a CIR process,we obtain the explicit formulae for CLN values through a PDE approach.Finally,credit valuation adjustment(CVA)formula is derived to price counterparty credit risk.展开更多
岩土工程可靠度分析和设计中,合理地选取随机场参数和相关函数,并准确地描述土性参数空间变异性十分困难。基于贝叶斯理论,本文提出了一套量化砂土有效内摩擦角空间变异性的方法。该方法根据先验信息和静力触探试验锥尖阻力数据,确定砂...岩土工程可靠度分析和设计中,合理地选取随机场参数和相关函数,并准确地描述土性参数空间变异性十分困难。基于贝叶斯理论,本文提出了一套量化砂土有效内摩擦角空间变异性的方法。该方法根据先验信息和静力触探试验锥尖阻力数据,确定砂土有效内摩擦角的随机场参数和相关函数。该方法合理地考虑了砂土有效内摩擦角与锥尖阻力间经验回归方程的不确定性。采用马尔科夫链蒙特卡洛模拟(Markov Chain Monte Carlo Simulation,MCMCS)获取服从后验分布的随机场参数样本。利用MCMCS样本构建随机场参数的Gaussian Copula函数求解后验分布。估计备选相关函数的概率,选择概率最大的为最可能的相关函数。最后,采用美国德州农工大学国家岩土工程砂土试验场的CPT数据算例验证了文中所提方法的有效性。结果表明:文中所提方法可以正确、合理地利用间接测量的锥尖阻力数据确定砂土有效内摩擦角的随机场参数和相关函数,准确量化其空间变异性。对于美国德州农工大学国家岩土工程砂土试验场的砂土有效内摩擦角,建议选用二阶自回归函数作为其最可能的相关函数。展开更多
基金supported by the National Key R&D Program of China(Grant No.2018YFA0703900)National Natural Science Foundation of China(Grant No.11671021)+1 种基金supported by National Natural Science Foundation of China(Grant Nos.11761051and 11561047)the Natural Science Foundation of Jiangxi Province(Grant Nos.20181BAB211003 and 20192BAB211006)。
文摘The Markov chain is well studied and widely applied in many areas.For some Markov chains,it is infeasible to obtain the explicit expressions of their corresponding finite-dimensional distributions and sometimes it is time-consuming for computation.In this paper,we propose an approximation method for Markov chains by applying the copula theory.For this purpose,we first discuss the checkerboard copula-based Markov chain,which is the Markov chain generated by the family of checkerboard copulas.This Markov chain has some appealing properties,such as self-similarity in copulas and having explicit forms of finite-dimensional distributions.Then we prove that each Markov chain can be approximated by a sequence of checkerboard copula-based Markov chains,and the error bounds of the approximate distributions are provided.Employing the checkerboard copula-based approximation method,we propose a sufficient condition for the geometric β-mixing of copula-based Markov chains.This condition allows copulas of Markov chains to be asymmetric.Finally,by applying the approximation method,analytical recurrence formulas are also derived for computing approximate distributions of both the first passage time and the occupation time of a Markov chain,and numerical results are listed to show the approximation errors.
基金the National Natural Science Foundation of China(11671291,71971031,U1811462).
文摘A credit-linked note(CLN)is a note paying an enhanced coupon to investors for bearing the credit risk of a reference entity.In this paper,we study the counterparty risk on CLNs under a Markov chain framework,and introduce a Markov copula model to describe joint defaults between the reference entity underlying the CLN and CLN issuer.Assuming that the respective default intensities are directly and inversely proportional to the interest rate,which follows a CIR process,we obtain the explicit formulae for CLN values through a PDE approach.Finally,credit valuation adjustment(CVA)formula is derived to price counterparty credit risk.
文摘岩土工程可靠度分析和设计中,合理地选取随机场参数和相关函数,并准确地描述土性参数空间变异性十分困难。基于贝叶斯理论,本文提出了一套量化砂土有效内摩擦角空间变异性的方法。该方法根据先验信息和静力触探试验锥尖阻力数据,确定砂土有效内摩擦角的随机场参数和相关函数。该方法合理地考虑了砂土有效内摩擦角与锥尖阻力间经验回归方程的不确定性。采用马尔科夫链蒙特卡洛模拟(Markov Chain Monte Carlo Simulation,MCMCS)获取服从后验分布的随机场参数样本。利用MCMCS样本构建随机场参数的Gaussian Copula函数求解后验分布。估计备选相关函数的概率,选择概率最大的为最可能的相关函数。最后,采用美国德州农工大学国家岩土工程砂土试验场的CPT数据算例验证了文中所提方法的有效性。结果表明:文中所提方法可以正确、合理地利用间接测量的锥尖阻力数据确定砂土有效内摩擦角的随机场参数和相关函数,准确量化其空间变异性。对于美国德州农工大学国家岩土工程砂土试验场的砂土有效内摩擦角,建议选用二阶自回归函数作为其最可能的相关函数。
