Hybrid Impulse Control Problem and Related System of Quasi-variational Inequalities

Description

Optimal impulse control problems naturally arise in various applications, when a non-zero intervention cost is incorporated into the objective function. The study of such systems has been regarded as one of the important topics in stochastic control theory. Along another line, in many complex systems, in which continuous dynamics and discrete events coexist, the need of successful controls of such systems in practice leads to an optimization on hybrid models, which has become an emerging new area of the study in stochastic control.In this proposal, we consider the impulse control problems on the hybrid diffusion systems. One of the distinctive features of the underlying system is that the corresponding dynamic programming principle leads to a system of Quasi-Variational-Inequalities (QVIs) of Hamilton-Jacobi-Bellman (HJB) type. The objective of this project is to exploit the properties of the optimal strategy and the value functions in a fairly large class of intervention cost functionals, and to develop a class of efficient numerical algorithms of the control and optimization problems.It is expected that we can characterize the value function of the underlying optimal control problems as the certain unique viscosity solutions of Quasi-variational inequalities. The expected results reveal the connection between underlying stochastic optimization and related PDE formulations. Due to the nonlinearity and switching mechanism, closed-form solutions are virtually impossible. Continuing on our quest, we plan to furtherdevelop e ective computational methods to solve the optimal control problems. Because of the complexity of the hybrid system, the computation cost has in general exponential growth in size of the space of the discrete events. To circumvent the difficulty in dealing with real systems with a huge complexity, we propose to use a two-time-scale approach on hybrid impulse control system to reduce the high dimensionality. In such a way, the limit system of Quasi-variational inequalities with reduced number of discrete event shall preserve the main features of the original control problem asymptotically. For the demonstration purpose, we are planning to implement the numerical scheme with dimensional reduction on various benchmark problems. Furthermore, we consider an applications in the nance market on general transaction costs and liquidity costs, and through the impulse/singular control formulation. We study the relation between the path regularity of the optimal trading strategy and the growth of the cost functionals.