Some applications of stochastic control to real options and liquidity risk.

Summary We study some applications of stochastic control to real options and liquidity risk. More precisely, in the first part, we are interested in an optimal portfolio selection problem under a liquidity risk model, then in the second part, in two real options: a regime-switching problem and a coupled singular control and regime-switching problem for a dividend policy with reversible investment, and finally, in the last part, in the existence of an equilibrium in a competitive market under asymmetric information. In solving these problems, especially in the first two parts, stochastic control techniques will be used. The typical approach is to express the principle of dynamic programming related to each problem in order to obtain a PDE characterization of the value functions. By this approach, we show, in the liquidity risk problem and the two real options, that the corresponding value functions are the unique solution of the associated system of variational HJB inequalities. In each problem of the first two parts, the solutions, in particular the optimal controls, can be obtained either in an explicit way or by an iterative method.