Necessary conditions for constant solutions of coupled Riccati equations in Nash games

The asymptotic behaviour of solutions as t → ∞ for coupled matrix Riccati equations occurring in open-loop linear-quadratic Nash games is studied in this paper. A general formula representing all possible solutions is given. Necessary conditions for constant real solutions are derived and an estimate for the rate of convergence is obtained. Two examples illustrate these results.     

A class of non-zero-sum stochastic differential investment and reinsurance games

In this article, we provide a systematic study on the non-zero-sum stochastic differential investment and reinsurance game between two insurance companies. Each insurance company’s surplus process consists of a proportional reinsurance protection and an investment in risky and risk-free assets. Each insurance company is assumed to maximize his utility of the difference between his terminal surplus and that of his competitor. The surplus process of each insurance company is modeled by a mixed regime-switching Cramer–Lundberg diffusion approximation process, i.e. the coefficients of the diffusion risk processes are modulated by a continuous-time Markov chain and an independent market-index process. Correlation between the two surplus processes, independent of the risky asset process, is allowed. Despite the complex structure, we manage to solve the resulting non-zero sum game problem by applying the dynamic programming principle. The Nash equilibrium, the optimal reinsurance/investment, and the resulting value processes of the insurance companies are obtained in closed forms, together with sound economic interpretations, for the case of an exponential utility function.     

A Chebyshev pseudospectral multidomain method for the soliton solution of coupled nonlinear Schrödinger equations

The collision of solitary waves is an important problem in both physics and applied mathematics. In this paper, we study the solution of coupled nonlinear Schrödinger equations based on pseudospectral collocation method with domain decomposition algorithm for approximating the spatial variable. The problem is converted to a system of nonlinear ordinary differential equations which will be integrated in time by explicit Runge–Kutta method of order four. The multidomain scheme has much better stability properties than the single domain. Thus this permits using much larger step size for the time integration which fulfills stability restrictions. The proposed scheme reduces the effects of round-of-error for the Chebyshev collocation and also uses less memory without sacrificing the accuracy. The numerical experiments are presented which show the multidomain pseudospectral method has excellent long-time numerical behavior and preserves energy conservation property.     

Dynamic policy in linear models with rational expectations of future events: A computer package

A computer package is presented, called POREM, for policy optimisation of linear dynamic, continuous-time models with constant coefficients and rational expectations of future events, based on infinite horizons and quadratic preferences. It is possible to calculate cooperative, decentralised Nash and decentralised Stackelberg outcomes and for each outcome it is possible to allow for pre-commitment and for lack of pre-commitment vis-à-vis private sector agents. It is possible to allow for hierarchical games, that is to allow for a group of Stackelberg leaders and a group of Stackelberg followers. The input of the model is very user-friendly and can be done with the aid of mnemonics. The package is programmed in PORTRAN77 and a single-precision version is available for personal computers