Zhuo Jin

Numerical solutions of optimal risk control and dividend optimization policies under a generalized singular control formulation

This paper develops numerical methods for finding optimal dividend pay-out and reinsurance policies. A generalized singular control formulation of surplus and discounted payoff function is introduced, where the surplus is modeled by a regime-switching process subject to both regular and singular controls. To approximate the value function and optimal controls, Markov chain approximation techniques are used to construct a discrete-time controlled Markov chain. The proofs of the convergence of the approximation sequence to the surplus process and the value function are given. Examples of proportional and excess-of-loss reinsurance are presented to illustrate the applicability of numerical methods.

Numerical methods for portfolio selection with bounded constraints

This work develops an approximation procedure for portfolio selection with bounded constraints. Based on the Markov chain approximation techniques, numerical procedures are constructed for the utility optimization task. Under simple conditions, the convergence of the approximation sequences to the wealth process and the optimal utility function is established. Numerical examples are provided to illustrate the performance of the algorithms.

Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections

This work focuses on numerical methods for finding optimal investment, dividend payment, and capital injection policies to maximize the present value of the difference between the cumulative dividend payment and the possible capital injections. The surplus is modeled by a regime-switching jump diffusion process subject to both regular and singular controls. Using the dynamic programming principle, the value function is a solution of the coupled system of nonlinear integro-differential quasi-variational inequalities. In this paper, the state constraint of the impulsive control gives rise to a capital injection region with free boundary, which makes the problem even more difficult to analyze. Together with the regular control and regime-switching, the closed-form solutions are virtually impossible to obtain. We use Markov chain approximation techniques to construct a discrete-time controlled Markov chain to approximate the value function and optimal controls. Convergence of the approximation algorithms is proved. Examples are presented to illustrate the applicability of the numerical methods.

ALMOST SURE AND pTH-MOMENT STABILITY AND STABILIZATION OF REGIME-SWITCHING JUMP DIFFUSION SYSTEMS ∗

This work focuses on regime-switching jump diffusions, which include three classes of random processes, Brownian motions, Poisson processes, and Markov chains. First, a scalar linear system is treated as a benchmark model. Then stabilization of systems with one-sided linear growth is considered. Next, nonlinear systems that have a finite explosion time are treated, in which regularization (explosion suppression) and stabilization are achieved by introducing appropriate diffusions together with Poisson and Markov chain perturbations. This work reveals the impact of various random effects on the underlying systems for almost sure and

Numerical Methods for Optimal Dividend Payment and Investment Strategies of Markov-Modulated Jump Diffusion Models with Regular and Singular Controls

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Numerical solutions of quantile hedging for guaranteed minimum death benefits under a regime-switching jump-diffusion formulation

th-moment stability and provides insight on stability and stabilization of switching jump diffusion systems.

Local risk minimization for vulnerable European contingent claims on nontradable assets under regime switching models

This work focuses on numerical methods for finding optimal dividend payment and investment policies to maximize the present value of the cumulative dividend payment until ruin; the surplus is modeled by a regime-switching jump diffusion process subject to both regular and singular controls. Using the dynamic programming principle, the optimal value function obeys a coupled system of nonlinear integro-differential quasi-variational inequalities. Since the closed-form solutions are virtually impossible to obtain, we use Markov chain approximation techniques to approximate the value function and optimal controls. Convergence of the approximation algorithms are proved. Examples are presented to illustrate the applicability of the numerical methods.

Optimal Debt Ratio and Consumption Strategies in Financial Crisis

This work develops numerical approximation methods for quantile hedging involving mortality components for contingent claims in incomplete markets, in which guaranteed minimum death benefits (GMDBs) could not be perfectly hedged. A regime-switching jump-diffusion model is used to delineate the dynamic system and the hedging function for GMDBs, where the switching is represented by a continuous-time Markov chain. Using Markov chain approximation techniques, a discrete-time controlled Markov chain with two component is constructed. Under simple conditions, the convergence of the approximation to the value function is established. Examples of quantile hedging model for guaranteed minimum death benefits under linear jumps and general jumps are also presented.

A numerical method for annuity-purchasing decision making to minimize the probability of financial ruin for regime-switching wealth models

ABSTRACT This article focuses on an optimal hedging problem of the vulnerable European contingent claims. The underlying asset of the vulnerable European contingent claims is assumed to be nontradable. The interest rate, the appreciation rate and the volatility of risky assets are modulated by a finite-state continuous-time Markov chain. By using the local risk minimization method, we obtain an explicit closed-form solution for the optimal hedging strategies of the vulnerable European contingent claims. Further, we consider a problem of hedging for a vulnerable European call option. Optimal hedging strategies are obtained. Finally, a numerical example for the optimal hedging strategies of the vulnerable European call option in a two-regime case is provided to illustrate the sensitivities of the hedging strategies.

Pricing dynamic fund protections with regime switching

This paper derives the optimal debt ratio and consumption strategies for an economy during the financial crisis. Taking into account the impact of labor market condition during the financial crisis, the production rate function is stochastic and affected by the government fiscal policy and unanticipated shocks. The objective is to maximize the total expected discounted utility of consumption in the infinite time horizon. Using dynamic programming principle, the value function is a solution of Hamilton–Jacobi–Bellman (HJB) equation. The subsolution-supersolution method is used to verify the existence of classical solutions of the HJB equation. The explicit solution of the value function is derived, and the corresponding optimal debt ratio and consumption strategies are obtained. An example is provided to illustrate the methodologies and some interesting economic insights.