几类随机模型及其在金融中的应用
|
摘要
本篇论文主要包含了以下三个随机模型。第一个模型是一个有交互作用的随机偏微分方程。我们证明了方程的弱解可由一对给定生灭率和转移率的马尔科夫链逼近。基于此逼近结果,我们可求得方程的解。第二个模型是一个以马氏调节的布朗运动作为输入流的存储过程。我们重点分析了它们负荷的极限性质。第三,作为随机模型在金融中的应用,我们考建立了一类依赖于波动类型的随机偏微分方程的远期利率模型,并将其用于信用违约互换等衍生品的定价。
具体地,本篇论文由以下三章组成：
在第一章中,我们研究了一类具有交互作用分支扰动的随机偏微分方程,此方程也称之为竞争的随机Lotka-Volterra方程。通过用一对时空尺度变换的粒子系统对方程进行逼近,我们证明了方程弱解的存在性。具体地,我们由给定的生灭率构造了一对马氏链,进而通过合适的时空尺度变换和Dynkin公式,我们得到了一对取值于离散函数空间的随机微分方程。而此方程的鞅部分根据跳的构造可以分解为反应、扩散和分支跳三项和。我们首先证明了关于它们上界和收敛的一些结果。利用这些结果我们可以证明方程各个构成项在合适的Sobolev空间的胎紧性。基于这些胎紧性结果,应用Prohorov定理和Skorohod表示定理等,我们可以得到一个一对收敛到方程的子序列,证明了方程弱解的存在性。
在第二章中,我们考虑了一个由马氏调节的布朗运动作为输入流的存储过程。近年来,马氏调节的布朗运动在金融模型和排队模型中有着广泛的应用。本章研究了关于此存储过程的两个重要性质。首先我们证明了在某些技术性条件下,过程平稳分布的存在性。进而假设过程依此平稳分布作为初始分布,我们分析了其运行极大过程的渐进增长率。证明了其以对数速度增长。
在第三章中,我们提出了一个对远期利率期限结构进行建模的新方法。避开即期利率模型,Heath et al. (1992)转而对远期利率的期限结构直接进行建模。进而金融学者们对此模型在许多方面进行了推广,这其中包含无穷维随机模型和带跳的模型。我们发现以上模型在处理两个时间指标的方式上是不同的：一个作为变量而另一个作为参数。在本章中,我们用一波动类型的随机偏微分方程来对远期利率的期限结构进行建模。在此模型中我们可以相对一致的处理这两个指标。使用波动类型方程的另一方面考虑是,方程本身能很自然地描述随机扰动性质。进而,我们推导出了在此模型下,市场是无套利的充要条件。与HJM条件类似,此条件可由模型的漂移项表示。进而,应用此远期利率对应的债券,我们考虑了几类可违约衍生品的定价问题,例如信用违约掉期。
In this doctorial dissertation, the main content is made up of three types of stochastic models. The first one is a pair of interacting stochastic partial differential equations (abbr. SPDEs). We proved the week (mild) solution of the system can be approximated by a pair of Markov chains with specified birth, death and transition rates. Based on this approximation, the solution obtained. The second model is a storage model fed by a Markov modulated Brownian motion (abbr. BM). The main effort is devoted to proving some limit properties of its loads as time goes to infinity. For the third model, as an application of stochastic models in finance, we establish a new term structure of forward rate modeled by a hyperbolic SPDE, and then this term structure model is applied to pricing CDS.
More specially, the dissertation is made up of the following three chapters.
In Chapter 1, we prove the existence of weak (mild) solutions to a pair of inter-acting SPDEs with branching noises, which is also known as a competitive stochastic Lotka-Volterra system, by approximating the equations with a pair of space-time rescaled particle systems. More accurately, we start with a pair of Markov chains with specified birth-death rates. Then by appropriate space-time rescaling, we apply Dynkin's formula to arrive at a pair of HN-valued stochastic differential equations in mild form, where HN is the space of all step functions taking constants on each interval [κ-1/N,κ/N).Note that the martingale components of the system can be de-composed into three parts according to the reaction, diffusion and branching jumps, respectively. We first present some up bound estimations and convergence results concerning the components of the system. Then the tightness for these processes is proved under some appropriate Sobolev spaces. Based on the tightness results and employing Prohorov's theorem and Skorohod representation theorem, we can prove that there is a subsequence which converges to the weak (mild) solution to the system.
In Chapter 2, we explore-a storage model fed by a Markov modulated Brownian motion (abbr. MMBM). In recent years, MMBMs have been becoming increasingly popular in describing various financial and queue models, see, e.g., Asmussen (2003). Two indicative properties about the process are studied in this chapter. Firstly, we prove the existence of the stationary distribution under some mild conditions. When we assume that the initial distribution of the process is this stationary law, we give the growth rate of the running maximum process, which is proved to grow like log t when the time t goes to infinity.
In Chapter 3, a new term structure of the instantaneous forward rate is proposed. Rather than describing the dynamics of short rates, Heath et al (1992) proposed a new model to formulate the term structure of forward rate directly. The model had been extensively studied in the literature. While the two time arguments are treated by different means:one as a variable and the other as a parameter. In this chapter, a wave-typed SPDE is used to formulate the term structure of forward rate. The inconsistent treatment of the arguments can be overcome under this model. Another advantage to use the wave-typed equation is that they can capture the stochastic shocks. Further we derive the conditions such that the model is arbitrage-free. The conditions are expressed by the drift term of the forward rate similar to that of H JM conditions. Finally applying the bonds corresponding to this forward rate, several defaultable derivatives are priced, such as credit default swaps.
具体地,本篇论文由以下三章组成：
在第一章中,我们研究了一类具有交互作用分支扰动的随机偏微分方程,此方程也称之为竞争的随机Lotka-Volterra方程。通过用一对时空尺度变换的粒子系统对方程进行逼近,我们证明了方程弱解的存在性。具体地,我们由给定的生灭率构造了一对马氏链,进而通过合适的时空尺度变换和Dynkin公式,我们得到了一对取值于离散函数空间的随机微分方程。而此方程的鞅部分根据跳的构造可以分解为反应、扩散和分支跳三项和。我们首先证明了关于它们上界和收敛的一些结果。利用这些结果我们可以证明方程各个构成项在合适的Sobolev空间的胎紧性。基于这些胎紧性结果,应用Prohorov定理和Skorohod表示定理等,我们可以得到一个一对收敛到方程的子序列,证明了方程弱解的存在性。
在第二章中,我们考虑了一个由马氏调节的布朗运动作为输入流的存储过程。近年来,马氏调节的布朗运动在金融模型和排队模型中有着广泛的应用。本章研究了关于此存储过程的两个重要性质。首先我们证明了在某些技术性条件下,过程平稳分布的存在性。进而假设过程依此平稳分布作为初始分布,我们分析了其运行极大过程的渐进增长率。证明了其以对数速度增长。
在第三章中,我们提出了一个对远期利率期限结构进行建模的新方法。避开即期利率模型,Heath et al. (1992)转而对远期利率的期限结构直接进行建模。进而金融学者们对此模型在许多方面进行了推广,这其中包含无穷维随机模型和带跳的模型。我们发现以上模型在处理两个时间指标的方式上是不同的：一个作为变量而另一个作为参数。在本章中,我们用一波动类型的随机偏微分方程来对远期利率的期限结构进行建模。在此模型中我们可以相对一致的处理这两个指标。使用波动类型方程的另一方面考虑是,方程本身能很自然地描述随机扰动性质。进而,我们推导出了在此模型下,市场是无套利的充要条件。与HJM条件类似,此条件可由模型的漂移项表示。进而,应用此远期利率对应的债券,我们考虑了几类可违约衍生品的定价问题,例如信用违约掉期。
In this doctorial dissertation, the main content is made up of three types of stochastic models. The first one is a pair of interacting stochastic partial differential equations (abbr. SPDEs). We proved the week (mild) solution of the system can be approximated by a pair of Markov chains with specified birth, death and transition rates. Based on this approximation, the solution obtained. The second model is a storage model fed by a Markov modulated Brownian motion (abbr. BM). The main effort is devoted to proving some limit properties of its loads as time goes to infinity. For the third model, as an application of stochastic models in finance, we establish a new term structure of forward rate modeled by a hyperbolic SPDE, and then this term structure model is applied to pricing CDS.
More specially, the dissertation is made up of the following three chapters.
In Chapter 1, we prove the existence of weak (mild) solutions to a pair of inter-acting SPDEs with branching noises, which is also known as a competitive stochastic Lotka-Volterra system, by approximating the equations with a pair of space-time rescaled particle systems. More accurately, we start with a pair of Markov chains with specified birth-death rates. Then by appropriate space-time rescaling, we apply Dynkin's formula to arrive at a pair of HN-valued stochastic differential equations in mild form, where HN is the space of all step functions taking constants on each interval [κ-1/N,κ/N).Note that the martingale components of the system can be de-composed into three parts according to the reaction, diffusion and branching jumps, respectively. We first present some up bound estimations and convergence results concerning the components of the system. Then the tightness for these processes is proved under some appropriate Sobolev spaces. Based on the tightness results and employing Prohorov's theorem and Skorohod representation theorem, we can prove that there is a subsequence which converges to the weak (mild) solution to the system.
In Chapter 2, we explore-a storage model fed by a Markov modulated Brownian motion (abbr. MMBM). In recent years, MMBMs have been becoming increasingly popular in describing various financial and queue models, see, e.g., Asmussen (2003). Two indicative properties about the process are studied in this chapter. Firstly, we prove the existence of the stationary distribution under some mild conditions. When we assume that the initial distribution of the process is this stationary law, we give the growth rate of the running maximum process, which is proved to grow like log t when the time t goes to infinity.
In Chapter 3, a new term structure of the instantaneous forward rate is proposed. Rather than describing the dynamics of short rates, Heath et al (1992) proposed a new model to formulate the term structure of forward rate directly. The model had been extensively studied in the literature. While the two time arguments are treated by different means:one as a variable and the other as a parameter. In this chapter, a wave-typed SPDE is used to formulate the term structure of forward rate. The inconsistent treatment of the arguments can be overcome under this model. Another advantage to use the wave-typed equation is that they can capture the stochastic shocks. Further we derive the conditions such that the model is arbitrage-free. The conditions are expressed by the drift term of the forward rate similar to that of H JM conditions. Finally applying the bonds corresponding to this forward rate, several defaultable derivatives are priced, such as credit default swaps.
引文
[1]Adams, R. (1978):Sobolev Space. Pure and Applied Mathematics,65, Aca-demic Press.
[2]Ang, E., Barria, J. (2000):The Markov modulated regulated Brownian motion: a second-order fluid flow of a finite buffer. Queueing Systems 35,263-287.
[3]Asmussen, S. (1995):Stationary distributions for fluid flow models with or wothout Brownian noise. Comm. Stat. Stoch. Models.11(1),21-49.
[4]Asmussen, S. (2003):Applied probability and queues,2nd ed. Springer-Verlag, New York.
[5]Applebauml, D. (2009):Levy Processes and Stochastic Calculus, Second Edi-tion. Cambridge University Press.
[6]Arnold, L. and Theodosopulu, M. (1980):Deterministic limit of the stochastic model of chemical reactions with diffusion. Advances in Applied Probability.12, 367-379.
[7]Belov, LA. (2005):On the computation of the probability density function of a-stable distributions. Mathematical Modelling and Analysis.2,333-341.
[8]Bielecki, T. and Rutkowski, M. (2002):Credit Risk:Modeling, Valuation and Hedging. Springer-Verlag, Germany.
[9]Black, F. and Scholes, M. (1973):The pricing of options and corporate liabili-ties. Journal of Political Economy.81,637-653.
[10]Blount, D. (1991):Comparison of stochastic and deterministic models of a linear chemical reaction with diffusion. Annals of Probability.19,1440-1462.
[11]Blount, D. (1994):Density dependent limits for a nonlinear reaction-diffusion model. Annals of Probability.22,2040-2070.
[12]Blount, D. (1996):Diffusion limits for a nonlinear density dependent space-time population model. Annals of Probability.24,639-659.
[13]Borak, S. Hardle, W. and Weron, R. (2005):Stable distribution. SFB 649 Discussion Paper 2005-008
[14]Brandt, M.W. and Santa-Clara, P. (2002):Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets. Journal of Financial Economics.63,161-210.
[15]Brace, A. and Musiela, M. (1994):A multi-factor Gauss Markov implementa-tion of Heath Jarrow and Morton. Mathematical Finance 4(3),563-576.
[16]Brennan, M. and Schwartz, E. (1979):A continuous time approach to the pricing of bonds. Journal of Banking and Finance.3,133-155.
[17]Burtschell, X., Gregory, J. and Laurent. J.P. (2009):A comparative analysis of CDO pricing models. Working Paper.
[18]Carmona, R. and Nualart, D. (1988):Random non-linear wave equations: smoothness of the solutions. Probability Theory and Related Fields.79,469-508.
[19]Carr, P., Geman, H., Madan, D. and Yor, M. (2002):The fine structure of asset returns:An empirical investigation. Journal of Business.75(2),302-332.
[20]Chen, L. (1996):Stochastic mean and stochastic volatility:a three-factor model of the term structure of interest rates and its application to the pricing of interest rate derivatives. London:Blackwell Publishers.
[21]Cont, R. and Tankov, P. (2003):Financial Modelling with Jump Porcesses. Chapman and Hall/CRC.
[22]Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985):A theory of the term structure of interest rates. Econometrica.53,385-407.
[23]David L. (2003):Credti risk modeling:theory and application. Princeton Uni-versity Press.
[24]Dawson, D. and Perkins, E.(1998):Long term behavior and coexistence in a mutually catalytic branching model. Annals of Probability.26,1088-1138.
[25]Duffie, D. (1998):First-to-default valuation. Working Paper.
[26]Duffie, D., Pan, J. and Singleton, K. (2000):Transform analysis and asset pricing for affine jump-diffusions. Econometrica.68(6),1343-1376.
[27]Duffie. D., Filipovic, D. and Schachermayer, W. (2003):Affine processes and application in finance. The Annals of Applied Probability.13(3),984-1053.
[28]Durrett, R., Mytnik, L. and Perkins, E.(2005):Competing super-Brownian motions as limits of interacting particle systems. Elect. J. Probability.10,1147-1220.
[29]Either, S. and Kurtz, T. (1986):Markov processes, Characterization and Con-vergence. Wiley, New York.
[30]Glynn, P., Whitt, W. (1994):Logarithmic asymptotics for steady-state tail probabilities in a single-server queue.J. Appl. Prob.31A,131-156.
[31]Giesecke, K. (2008a):An overview of credit derivatives. Working Paper.
[32]Giesecke, K. and Goldberg, L.R. (2008b):The market price of credit risk:the impact of asymmetric information. Working Paper.
[33]Geman, H. (2002):Pure jump Levy processes for asset price modelling. Journal of Banking and Finance.26(7),1297-1316.
[34]Glasserman, P. and Ruiz-Mata, J. (2005):Computing the credit loss distribu-tion in the Gaussian copula model:a comparison of methods. Working Paper, Columbia University.
[35]Goldstein, R.S. (2000):The term structure of interest rates as a random field. The Review of Financial Studies.13(2),365-384.
[36]Gugg, C. and Duan, J. (2004):A Markov jump process approximation of the stochastic Burgers equations. Stochastic and Dynamics.4,245-264.
[37]Hamza K. and Klebaner, F. (2006):On solutions of first order stochastic partial differential equations. Far East Journal of Theory Statistics.20(1),13-25.
[38]Harrison, J. (1985):Brownian motion and stochastic flow ststems. Wiley, New York.
[39]Haworth, H., Reisinger, C. and Shaw. W. (2007):Modeling bonds and credit default swaps using a structural model with contagion. Working Paper.
[40]Heath, D.C. Jarrow, R.A. and Morton, A. (1990):Bond pricing and the term structure of interest rates:A discret time approximation. Journal of Financial and Quantitative Analysis.25,419-440.
[41]Heath, D.C. Jarrow, R.A. and Morton, A. (1992):Bond pricing and the term structure of interest rates:A new methodology for contingent claims valuation. Econometrica.60,77-105.
[42]Heston, S. (1993):A closed-form solution for options with stochastic volatility with applications to bonds and currency options. Review of Financial Studies. 9,326-343.
[43]Hilber, N., Reich, N., Schwab, C. and Winter, C. (2009):Numerical methods for Levy processes. Finance and Stochastic.13,471-500.
[44]Ho, T. and Lee. S. (1986):Term structure movements and pricing interest rate contingent claims. Journal of Finance.41,1011-1029.
[45]Hudson, W.N. and Mason, J.D. (1976):Variational sums for additive processes. Proceedings of the American Mathematical Society.55,395-399.
[46]Hull, J. and White, A. (1990):Pricing interest-rate-derivative securities. The Review of Financial Studies.3,573-592.
[47]Hurn, S. and Lindsay, K. (1999):Estimating the parameters of stochastic dif-ferential equations. Mathematics and Computers in Simulation.48,373-384.
[48]Jacod, J. and Shiryaev, A. (1987):Limit Theorems for Stochastic Processes. Springer.
[49]Jarrow, R.A., Lando, D. and Turnbull, S.M. (1997):A Markov model for the term structure of credit risk spreads. The Review of Financial Studies.10(2), 481-523.
[50]Karatzas, I. and Shreve, S. (1991):Brownian Motion and Stochastic Calculus. Springer-Verlag, New York.
[51]Kella,O., Stadje, W. (2004):A Brownian motion with two reflecting barriers and Markov-modulated speed. J. Appl. Prob.41(4),1237-1242.
[52]Kennedy, D.P. (1994):The term stucture of interest rates as a Gaussian random field. Mathematical Finance,4,247-258.
[53]Kennedy, D.P. (1997):Characterizing Gaussian models of the term structure of interest rates. Mathematical Finance,7,107-118.
[54]Kimmel. R.L. (2001):Affine latent variable models of the term structure:em-pirical evidence and estimation. Working paper. Princeton University.
[55]Kimmel, R.L. (2004):Modeling the term structure of interest rates:a new approach. Journal of Financial Economics.72,143-183.
[56]Kloeden, P.E. and Platen, E. (2007):Numerical solution of stochastic differen-tial equations. Springer Press.
[57]Konno, N. and Shiga, T. (1988):Stochastic partial differential equations for some measure-valued diffusions. Probability Theory and The Related Fields 79, 201-225.
[58]Kotelenez, P. (1986a):Law of large numbers and central limit theorem for linear chemical reactions with diffusion. Annals of Probability.14,173-193.
[59]Kouritzin, M. and Long, H. (2002):Convergence of Markov chain approxima-tions to stochastic reation-diffusion equations. The Annals of Applied Probabil-ity.12,1039-1070.
[60]Kudryavtsev,O. and Levendorskii, S. (2008):Fast and accurate pricing of barrier options under Levy processes. Working Paper.
[61]Kurtz, T. (1970):Solutions of ordinary differential equations as limits of pure jump Markov processes. Journal of Applied Probability.7,45-58.
[62]Kyprianou, A.E. Introductory Lectures on Fluctuations of Levy Processes with Applications. Springer-Verlag Berlin Heidelberg.
[63]Lando, D. (2003):Credit Risk Modeling:Theory and Applications. Princeton University Press, Princeton.
[64]Leadbetter, M., Lindgren, G. and Rootzen, H. (1983):Extremes and related properties of random, sequences and processes. Springer-Verlag, New York.
[65]Longstaff. F. and Schwartz, E. (1992):Interest rate volatility and the term structure of interest rates:A two-factor general equilibrium model. Journal of Finance,47(4),1259-1282.
[66]Merton, R. (1974):On the pricing of corporate debt:the risk structure of interest rates. Journal of Finance.29,449-470.
[67]Ogawa, S. (1995):Some problems in the simulation of nonlinear diffusion pro-cesses. Mathematics and Computers in Simulations.28,217-223.
[68]Pazy, A. (1983):Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York.
[69]Peszat, S. Rusinek, A. and Zabczyk, J. (2007a):On a stochastic partial differ-ential equation of bond market. Preprint
[70]Peszat, S. and Zabczyk, J. (2007b):Heath-Jarrow-Morton-Musiela equation of bond market. Preprint
[71]Protter, P. (1990):Stochastic Integration and Differential Equations. Springer, New York.
[72]Rogers, L.C.G. and Williams, D. (2000):Diffusions, Markov Processes and Martingales. Cambridge University Press.
[73]Sato, K. (1999):Levy processes and infinitely divisible distribution. Cambridge University Press.
[74]Vasicek,O. (1977):An equilibrium characterization of the term structure. Jour-nal of Financial Economics.5,177-188.
[75]Walsh, J. (1986). An introduction to stochastic partial differential equations. Lecture Notes in Math.1180 265-439. Springer, Berlin.
[76]Wang, X.Q., Bo, L.J. and Wang, Y.J. (2009):From Markov jump systems to two species competitive Lotka-Volterra equations with diffusion. Acta Mathe-matica Sinica. English Series,25(1),157-170.
[77]Ward, A. and Glynn, P. (2003):A diffusion approximation for Markovian queue with reneging. Queueing Systems Theory and Application.43,103-128.
[78]Yosida, K. (1980):Functional Analysis.,6th Ed. Springer, Berlin Heidelberg New York.
[79]Zeevi, A., Glynn, P. (2000):On the maximum workload of a queue fed by fractional Brownian Motion. Ann. Appl. Prob.10(4),1084-1099.
[2]Ang, E., Barria, J. (2000):The Markov modulated regulated Brownian motion: a second-order fluid flow of a finite buffer. Queueing Systems 35,263-287.
[3]Asmussen, S. (1995):Stationary distributions for fluid flow models with or wothout Brownian noise. Comm. Stat. Stoch. Models.11(1),21-49.
[4]Asmussen, S. (2003):Applied probability and queues,2nd ed. Springer-Verlag, New York.
[5]Applebauml, D. (2009):Levy Processes and Stochastic Calculus, Second Edi-tion. Cambridge University Press.
[6]Arnold, L. and Theodosopulu, M. (1980):Deterministic limit of the stochastic model of chemical reactions with diffusion. Advances in Applied Probability.12, 367-379.
[7]Belov, LA. (2005):On the computation of the probability density function of a-stable distributions. Mathematical Modelling and Analysis.2,333-341.
[8]Bielecki, T. and Rutkowski, M. (2002):Credit Risk:Modeling, Valuation and Hedging. Springer-Verlag, Germany.
[9]Black, F. and Scholes, M. (1973):The pricing of options and corporate liabili-ties. Journal of Political Economy.81,637-653.
[10]Blount, D. (1991):Comparison of stochastic and deterministic models of a linear chemical reaction with diffusion. Annals of Probability.19,1440-1462.
[11]Blount, D. (1994):Density dependent limits for a nonlinear reaction-diffusion model. Annals of Probability.22,2040-2070.
[12]Blount, D. (1996):Diffusion limits for a nonlinear density dependent space-time population model. Annals of Probability.24,639-659.
[13]Borak, S. Hardle, W. and Weron, R. (2005):Stable distribution. SFB 649 Discussion Paper 2005-008
[14]Brandt, M.W. and Santa-Clara, P. (2002):Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets. Journal of Financial Economics.63,161-210.
[15]Brace, A. and Musiela, M. (1994):A multi-factor Gauss Markov implementa-tion of Heath Jarrow and Morton. Mathematical Finance 4(3),563-576.
[16]Brennan, M. and Schwartz, E. (1979):A continuous time approach to the pricing of bonds. Journal of Banking and Finance.3,133-155.
[17]Burtschell, X., Gregory, J. and Laurent. J.P. (2009):A comparative analysis of CDO pricing models. Working Paper.
[18]Carmona, R. and Nualart, D. (1988):Random non-linear wave equations: smoothness of the solutions. Probability Theory and Related Fields.79,469-508.
[19]Carr, P., Geman, H., Madan, D. and Yor, M. (2002):The fine structure of asset returns:An empirical investigation. Journal of Business.75(2),302-332.
[20]Chen, L. (1996):Stochastic mean and stochastic volatility:a three-factor model of the term structure of interest rates and its application to the pricing of interest rate derivatives. London:Blackwell Publishers.
[21]Cont, R. and Tankov, P. (2003):Financial Modelling with Jump Porcesses. Chapman and Hall/CRC.
[22]Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985):A theory of the term structure of interest rates. Econometrica.53,385-407.
[23]David L. (2003):Credti risk modeling:theory and application. Princeton Uni-versity Press.
[24]Dawson, D. and Perkins, E.(1998):Long term behavior and coexistence in a mutually catalytic branching model. Annals of Probability.26,1088-1138.
[25]Duffie, D. (1998):First-to-default valuation. Working Paper.
[26]Duffie, D., Pan, J. and Singleton, K. (2000):Transform analysis and asset pricing for affine jump-diffusions. Econometrica.68(6),1343-1376.
[27]Duffie. D., Filipovic, D. and Schachermayer, W. (2003):Affine processes and application in finance. The Annals of Applied Probability.13(3),984-1053.
[28]Durrett, R., Mytnik, L. and Perkins, E.(2005):Competing super-Brownian motions as limits of interacting particle systems. Elect. J. Probability.10,1147-1220.
[29]Either, S. and Kurtz, T. (1986):Markov processes, Characterization and Con-vergence. Wiley, New York.
[30]Glynn, P., Whitt, W. (1994):Logarithmic asymptotics for steady-state tail probabilities in a single-server queue.J. Appl. Prob.31A,131-156.
[31]Giesecke, K. (2008a):An overview of credit derivatives. Working Paper.
[32]Giesecke, K. and Goldberg, L.R. (2008b):The market price of credit risk:the impact of asymmetric information. Working Paper.
[33]Geman, H. (2002):Pure jump Levy processes for asset price modelling. Journal of Banking and Finance.26(7),1297-1316.
[34]Glasserman, P. and Ruiz-Mata, J. (2005):Computing the credit loss distribu-tion in the Gaussian copula model:a comparison of methods. Working Paper, Columbia University.
[35]Goldstein, R.S. (2000):The term structure of interest rates as a random field. The Review of Financial Studies.13(2),365-384.
[36]Gugg, C. and Duan, J. (2004):A Markov jump process approximation of the stochastic Burgers equations. Stochastic and Dynamics.4,245-264.
[37]Hamza K. and Klebaner, F. (2006):On solutions of first order stochastic partial differential equations. Far East Journal of Theory Statistics.20(1),13-25.
[38]Harrison, J. (1985):Brownian motion and stochastic flow ststems. Wiley, New York.
[39]Haworth, H., Reisinger, C. and Shaw. W. (2007):Modeling bonds and credit default swaps using a structural model with contagion. Working Paper.
[40]Heath, D.C. Jarrow, R.A. and Morton, A. (1990):Bond pricing and the term structure of interest rates:A discret time approximation. Journal of Financial and Quantitative Analysis.25,419-440.
[41]Heath, D.C. Jarrow, R.A. and Morton, A. (1992):Bond pricing and the term structure of interest rates:A new methodology for contingent claims valuation. Econometrica.60,77-105.
[42]Heston, S. (1993):A closed-form solution for options with stochastic volatility with applications to bonds and currency options. Review of Financial Studies. 9,326-343.
[43]Hilber, N., Reich, N., Schwab, C. and Winter, C. (2009):Numerical methods for Levy processes. Finance and Stochastic.13,471-500.
[44]Ho, T. and Lee. S. (1986):Term structure movements and pricing interest rate contingent claims. Journal of Finance.41,1011-1029.
[45]Hudson, W.N. and Mason, J.D. (1976):Variational sums for additive processes. Proceedings of the American Mathematical Society.55,395-399.
[46]Hull, J. and White, A. (1990):Pricing interest-rate-derivative securities. The Review of Financial Studies.3,573-592.
[47]Hurn, S. and Lindsay, K. (1999):Estimating the parameters of stochastic dif-ferential equations. Mathematics and Computers in Simulation.48,373-384.
[48]Jacod, J. and Shiryaev, A. (1987):Limit Theorems for Stochastic Processes. Springer.
[49]Jarrow, R.A., Lando, D. and Turnbull, S.M. (1997):A Markov model for the term structure of credit risk spreads. The Review of Financial Studies.10(2), 481-523.
[50]Karatzas, I. and Shreve, S. (1991):Brownian Motion and Stochastic Calculus. Springer-Verlag, New York.
[51]Kella,O., Stadje, W. (2004):A Brownian motion with two reflecting barriers and Markov-modulated speed. J. Appl. Prob.41(4),1237-1242.
[52]Kennedy, D.P. (1994):The term stucture of interest rates as a Gaussian random field. Mathematical Finance,4,247-258.
[53]Kennedy, D.P. (1997):Characterizing Gaussian models of the term structure of interest rates. Mathematical Finance,7,107-118.
[54]Kimmel. R.L. (2001):Affine latent variable models of the term structure:em-pirical evidence and estimation. Working paper. Princeton University.
[55]Kimmel, R.L. (2004):Modeling the term structure of interest rates:a new approach. Journal of Financial Economics.72,143-183.
[56]Kloeden, P.E. and Platen, E. (2007):Numerical solution of stochastic differen-tial equations. Springer Press.
[57]Konno, N. and Shiga, T. (1988):Stochastic partial differential equations for some measure-valued diffusions. Probability Theory and The Related Fields 79, 201-225.
[58]Kotelenez, P. (1986a):Law of large numbers and central limit theorem for linear chemical reactions with diffusion. Annals of Probability.14,173-193.
[59]Kouritzin, M. and Long, H. (2002):Convergence of Markov chain approxima-tions to stochastic reation-diffusion equations. The Annals of Applied Probabil-ity.12,1039-1070.
[60]Kudryavtsev,O. and Levendorskii, S. (2008):Fast and accurate pricing of barrier options under Levy processes. Working Paper.
[61]Kurtz, T. (1970):Solutions of ordinary differential equations as limits of pure jump Markov processes. Journal of Applied Probability.7,45-58.
[62]Kyprianou, A.E. Introductory Lectures on Fluctuations of Levy Processes with Applications. Springer-Verlag Berlin Heidelberg.
[63]Lando, D. (2003):Credit Risk Modeling:Theory and Applications. Princeton University Press, Princeton.
[64]Leadbetter, M., Lindgren, G. and Rootzen, H. (1983):Extremes and related properties of random, sequences and processes. Springer-Verlag, New York.
[65]Longstaff. F. and Schwartz, E. (1992):Interest rate volatility and the term structure of interest rates:A two-factor general equilibrium model. Journal of Finance,47(4),1259-1282.
[66]Merton, R. (1974):On the pricing of corporate debt:the risk structure of interest rates. Journal of Finance.29,449-470.
[67]Ogawa, S. (1995):Some problems in the simulation of nonlinear diffusion pro-cesses. Mathematics and Computers in Simulations.28,217-223.
[68]Pazy, A. (1983):Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York.
[69]Peszat, S. Rusinek, A. and Zabczyk, J. (2007a):On a stochastic partial differ-ential equation of bond market. Preprint
[70]Peszat, S. and Zabczyk, J. (2007b):Heath-Jarrow-Morton-Musiela equation of bond market. Preprint
[71]Protter, P. (1990):Stochastic Integration and Differential Equations. Springer, New York.
[72]Rogers, L.C.G. and Williams, D. (2000):Diffusions, Markov Processes and Martingales. Cambridge University Press.
[73]Sato, K. (1999):Levy processes and infinitely divisible distribution. Cambridge University Press.
[74]Vasicek,O. (1977):An equilibrium characterization of the term structure. Jour-nal of Financial Economics.5,177-188.
[75]Walsh, J. (1986). An introduction to stochastic partial differential equations. Lecture Notes in Math.1180 265-439. Springer, Berlin.
[76]Wang, X.Q., Bo, L.J. and Wang, Y.J. (2009):From Markov jump systems to two species competitive Lotka-Volterra equations with diffusion. Acta Mathe-matica Sinica. English Series,25(1),157-170.
[77]Ward, A. and Glynn, P. (2003):A diffusion approximation for Markovian queue with reneging. Queueing Systems Theory and Application.43,103-128.
[78]Yosida, K. (1980):Functional Analysis.,6th Ed. Springer, Berlin Heidelberg New York.
[79]Zeevi, A., Glynn, P. (2000):On the maximum workload of a queue fed by fractional Brownian Motion. Ann. Appl. Prob.10(4),1084-1099.