Stochastic Equations And Their Applications In Credit Risk

This doctoral dissertation is made up of five chapters and concentrates on the study of the reflected stochastic differential equation(RSDE),stochastic parital differential equation(SPDE) and their applications in credit risk.It is known that the RSDE can be viewed as a Skorohod problem.Under the Lipschitz condition,the existence and uniquness of the strong solution was first to be demonstrated by Lions and Sznitman and subsequently the Lipschitz condition was extended to the one-sided Lipschitz condition and Yamada-Watanabe condition in Le Gall,Bass and Chen,Zhang and Marin and Real,respectively.In Section 1.1 of Chapter 1,we deal with the RSDE with the non-Lipschitz condition as in Fang and Zhang and prove the existence and uniquness of the strong solution. In particular,in the case of the one-dimension,a strong comparsion theorem is proved by employing the local time theory and the properties of the Fang and Zhang's non-Lipschitz condition.Since the solution of the RSDE can be restricted into some specified convex domain,the RSDE can be applied to model in the queueing theory and financial market,etc.In particular,Harrison used the reflected Brownian motion with two-sided barriers to construct the storage model.A drift rate control of a Brownian processing system descrided as the reflected O-U process with two-sided barriers was explored in Ata,Harrison and Shepp.Goldstein and Keirstead employed the RSDEs with one-sided barrier at zero to model the spot interest rate.Furthermore, they deduced the closed-form solutions of the derivatives' prices in the cases of the reflected Brownian motion and O-U process with the barrier zero.Ward and Glynn presented a result that the infinite capacity discrete queueing systems with reneging can diffusion approximate to the reflected O-U process with the one-sided barrier.The properties associated with the reflected O-U process with the one-sided barrier have been discussed in their successive paper Ward and Glynn.Motivated by the result, it is natural to prove that the same discrete queueing systems but with finite buffer capacity can diffusion approximate to the reflected O-U process with two one-sided barriers.In Section 1.2,we present an explicit formulae for the Laplace transform of the first passage time of the reflected O-U process with two-sided barriers.Using the transform,it is not difficult to deduce the moments of the first passage time for the reflected process and which provides the foundation for studying the parameter estimation and ergodicity of the system.Subsequently we consider a class of generalized zero mean reverting reflcted O-U process with two-sided barriers.The closed-form formulae for the Laplace transform of the integral functional of the strong solution till the first passage time is proved in Subsection 1.2.4.Furthermore,this result is applied to the pricings of the derivative with the defaultable risk and a digital option,respectively. In Section 1.3,we establish a large deviation principle(LDP) for the solutions of perturbed reflected diffusion processes.As for the perturbed reflected diffusion process, it is perturbed by an additional maximum process associated with the solution to the reflected diffusion process.The readers can refer to Doney and Zhang for more details.On the other hand,the estimates concerning the maximum perturbed terms are key in the proof of uniform Freidlin-Ventzell estimates of perturbed diffusion processes. Finally,we explore a hedge scheme for a defaultable claim with the recovery and dividend under local risk minimization.The proofs are based on the intensity-based reduced-form framework(see Appendix A) and the assumption 'Martingale Invariance Property'(MIP).In the proofs,we introduce a new filtration which is similar to the one used in Belanger,Shreve and Wong.To the best of my knowledge,it seems that it is the first time to consider the case of a defaultable claim with dividends in the topic.In Chapter 2,we are concerned with the optimal portfolio problem with the defanltable risk.The study of the optimal portfolio without the defaultable risk are owed to Merton.Recently,the optimal portfolio and the hedging with a defaultable security have aroused much more attentions.For the literature with the deafultable risk,a popular and tractable approach for modeling the defaultable market is the intensity-based reduced-form.That is,we can use a nonnegative diffusion process to construct the conditional survival process of the default stopping time(see Appendix A for more details).However the intensity-based reduced-form is different from another method called structural approach for modeling the defaultable market.On the other hand,among the literature for discussing the optimal portfolio with the defaultable risk,Jang and Bielecki and Jang proposed a dynamics for the price process of a defaultable bond with the constant default risk premium and default intensity and further established an optimal asset allocation for maximizing the expected HARA utility of the terminal wealth.Under the assumption of the constant default risk premium and default intensity,the optimal asset allocation and the corresponding value function admit the closed-form.However in practice,the default intensity and risk premium generally fluctuate over the time(see Pham for the default-free case).Based on this insight,we consider the optimal portfolio problem with the log and non-log utility when the default risk premium and default intensity depend on a common stochastic factor which is described as a diffusion process in Section 2.1 and 2.2,respectively.Comparing with the results in Jang and Bielecki and Jang,we can also deduce the closed form of the optimal allocation in the case. Although the HJB equation does not admit an explicit soltion,we obtain upper and lower function bounds for the classic solution to the HJB equaion via an approach to the sub-super solutions for uniformly elliptic equations.These bounds can also help us to prove the verification theorem.If the assumptions are released,then the HJB equation may be not possess the classic solution.A flexible approach is to define the visocity solution to the HJB equation.We can prove that the value function is a visocity solution.However the visocity solution is a weaker solution form for the HJB equation.The uniqueness of the solution is difficult to prove.In Section 2.3, we consider the optimal portfolio problem similarly as in Section 2.1 and 2.2 but with the weaker assumptions by using the visocity solution approach.Where we apply the price formulae in Belanger,Shreve and Wong to prove the price dynamics of the defaultable zero-coupon bond under the physical probability measure.Note that the state(wealth) equation reflects at point zero and posseses the jumps.This motivates us to combine the methods from Atar,Budhiraja and Williams and Benth,Karlsen and Reikvam to prove the uniqueness of the visocity solution to our HJB equation in Section 2.3.In Chapter 3,we focus on two class of higher-order parabolic SPDEs with non-Gaussian noises perturbation:Cahn-Hilliard and Kuramoto-Sivashinsky SPDEs.The deterministic Cahn-Hilliard equation describes some important qualitative features of two-phase systems,in particular,the spinodal decomposition,i.e.a rapid separation of phases when the material is cooled down sufficiently.Currently this equation has become a very important subject in material science.However,the evolution of the phase systems are generally stochastic,hence Cahn-Hilliard equation deriven by random noises can describe the evolution better.Da Prato and Debussche,and Cardon-Weber were first to explore Cahn-Hilliard equations with Gaussian noises.However we here study the non-Gaussian noises perturbated Cahn-Hilliard equation.The existence and uniqueness of local mild solutions to Cahn-Hilliard equations with Levy space-time white noises are proved in Section 3.1.A difficulty to overcome is that the traditional Burkholder-Davis-Gundy inequality is not be applied to the Cahn-Hilliard equation with jumps.Hence we introduce a new version of Burkholder-Davis-Gundy inequality.In Section 3.2,we continue to consider Cahn-Hilliard equations driven by fractional noises.A weak convergence approach is used to prove the existence and uniqueness of the global mild solution,due to the nonlinearity of the equation.To apply the approach,we have to prove the tightness of the truncated solutions.This makes us to deduce a class of new estimations associated with the Green kernel and which are demonstrated in Appendix B.Combining with the estimates in Appendix B, we prove the Stroock-Varadhan characterization support theorem for the distribution of the solution to Cahn-Hilliard equation in Section 3.3.Due to the nonlinearity of the equation,we need to introduce a localization framework.Regularization properties of the convolution with several functions connected to the Green kernel are deduced. In Section 3.4,we consider the fourth order stochastic Anderson models with spacecorrelated Gaussian noises and fractional(which can be viewed as a time-correlated) noises in It(?) and Skorokhod senses,respectively.We present the regularization property and the estimation of Lyapunov exponent to the solution.Finally we show the existence and uniqueness of the global weak solution to the nonlocal stochastic Kuramoto-Sivashinsky equations driven by compensated Poisson random measures.Furthermore, we prove the Fellerian property of the semi-group associated with the solition and show the existence and support property of the semi-group by employing Krylov-Bogoliubov theorem.In parallel with the content of Chapter 3,we discuss stochastic wave equations in Chapter 4.At present,the deterministic wave equation with damping has aroused much attentions in the literature.An important theme is to study the explosive property of the solution.However,it seems difficult to study the theme for stochastic version of hyperbolic equations.In Section 4.1,we give sufficient conditions such that the local solutions of(strong) damped stochastic wave equations with Q-Wiener process pertubartion are blowup with positive probability or explosive in L~2 sense via appropriate energy inequalities.In the last two sections,we concentrate on stochastic hyperbolic equations with pure jumps.To the best of my knowledge,it seems that it is the first time to consider the case.A semi-linear damped stochastic wave equation driven by a compensated Poisson random measure is considered in Section 4.2.The global weak and strong solutions of the equation are established,respectively.Furthermore,the Markov property and Fellerian property are demonstrated for the global weak solution, and the existence and support property of an invariant measure associated with the transition semigroup are proved.We continue to study a wave equation but with non-Gaussian L'evy noise.As for the proof of the weak solution,we are mainly based on the following property associated with the jumps of Levy processes:the L'evy process with only small jumps is a martingale and which admits any order moment,the big jumps of L'evy process can be arranged from small to big size.Since the higher order moments do not exist generally for the non-Gaussian Levy process(e.g.α-stable process),we make the drift and jump coefficients to restrict in some function class in the proof of the invariant measure.Thus we can show the existence and uniqueness of the invariant measure under some 'stable' conditions.In the last chapter,we propose a discontinuous Galerkin numerical scheme for a class of elliptic SPDEs driven by space white noise with Dirichlet boundary condition in 2 and 3 space dimension.L~2 error estimates are established and a numerical test as d=2 shows that the discoutinuous scheme is effective.On the other hand,there has been much interest in the application of discontinuous Galerkin methods for deterministic nonlinear hyperbolic and convection dominated problems.In particular,it was shown to be of use in solving pure elliptic problems.The local discontinuous Galerkin method which we consider in this chapter was the one of variants of discontinuous Galerkin methods introduced by Cockburn and Shu.To the best of my knowledge, it seems that it is the first time to apply the local discontinuous Galerkin method to the stochastic elliptic equation. At the end of the dissertation,I present several research themes which we consider now.They are concentrated on the study of the exchange rate target zone modeling by using RSDEs.The exchange rate target zone modeling by employing reflected Brownian motion fundamental are owed to Paul Krugman who just won the Nobel Prize for Economics 2008.Being different from reflected Brownian motions,we present a reflcted O-U process with two-sided barriers to model the exchange rate target zone. The parameters associated with the model are estimated by a simple yet fexible semiparametric approach.The study of Monte Carlo simulation shows that the scheme is well suited to the model.In addition,we incorporate the realignment risk into the exchange rate in a target zone.The realignment is modeled as a continuous time two states Markov chain.The states denote upward(appreciates) and downward(depreciates) realignment,respectively.Under the stationary setting of the Markov chain,a general pricing formulae for the realignment exchange rate derivative is derived in the presence of the realignment risk premium and the closed-form solution is presented for a Jacob diffusion model.