Optimal Dynamic Credit Investment With Regime-Switching

In this thesis,we propose a class of dynamical models on credit portfolio with regime-switching and default contagion.We study the following optimal credit portfo-lio problems in finance and insurance:(i)optimal credit investment and risk control for an insurer with regime-switching;(ii)risk sensitive credit portfolio optimization with regime-switching;(iii)risk sensitive credit portfolio optimization with regime switching under partial information.Among above problems,default contagion risk is incorpo-rated into the credit portfolio,i.e.,the default events of risky assets may have an impact on the distress state of the surviving assets in the portfolio.Chapter 3 studies an optimal credit investment and risk control problem for an insurer in a regime-switching market.Assume that the regime-switching process has a finite state space.By reinsuring his/her policies,the insurer allocates his/her wealth across multi-name defautable assets and a riskless bond.The aim of the insurer is to maximize the expected utility of the terminal wealth by selecting optimal investment and risk control strategies.We characterize the optimal trading strategy of defaultable stocks and risk control for the insurer.By developing a truncation technique and the method of monotone dynamical system,we analyze the existence and uniqueness of classical solutions to the recursive HJB system.We prove the verification theorem based on the classical solutions of the system.Chapter 4 studies a risk sensitive credit portfolio optimization problem with regime-switching having a countably infinite state space.Returning to the practical implemen-tation in financial markets,the discretization of continuous state financial models can be transferred to Markov chains with countably infinite states.Differently from the ob-jective based on the utility maximization of terminal wealth,we consider the robust in-vestment case,i.e.,the risk preference of an investor can be described as a risk-sensitive parameter.Due to the countably infinite state space of the regime-switching process,the value function of our stochastic control problem satisfies a class of infinite dimensional recursive system of HJB equations.The core of this chapter is to establish the existence and uniqueness of classical solutions of thus system.The scheme used in this chapter is:(a)Using the method introduced in Chapter 3,we prove the existence and uniqueness of classical solutions of finite-dimensional HJB system relating to the finite state space,after which the corresponding verification theorem is proved;(b)we construct a series of auxiliary optimization problems in finite state space,and prove that the value func-tions relating to the auxiliary optimization problems converge to the classical solution of the infinite-dimensional HJB system relating to the original optimization problem.The verification results are also established thanks to the aforementioned classical solution of the infinite-dimensional HJB system.Differently from stochastic control problems based on complete market informa-tion studied in Chapter 3 and Chapter 4,Chapter 5 investigates a risk sensitive credit portfolio optimization problem with regime switching under partial market information.The Markovian regime-switching process is assumed to be unobservable and it might has countable states.The control problem is formulated under partial observations of default events and asset prices before default.By proving an innovative martingale rep-resentation theorem based on incomplete and also phasing out filtration due to sequential defaults,we reduce the partially observed stochastic control problem to fully observed one.We investigate the latter by connecting it to a quadratic BSDE with jumps.To establish the existence of solution to the new quadratic BSDE with jumps,we propose some novel truncation techniques and study the convergence of solutions associated to the truncated BSDEs.On the basis of the solution of the quadratic BSDE with jumps,we are able to characterize the optimal trading strategy and prove the verification theorem.