Viscosity Solutions for HJB Equations

Abstract

The theory of viscosity solutions was originated by M.G. Crandall and P.L. Lions in the early 80s for the Hamilton–Jacobi equations and later P.L. Lions developed it for the HJB equations (Lions, J Commun PDE 8:1101–1134, 1983; Acta Math 16:243–278, 1988; Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part II. Optimal control of Zakai equation. In: Da Prato, Tubaro (eds) Stochastic partial differential equations and applications II. Lecture notes in mathematics, vol 1390. Springer, Berlin/Heidelberg, 1989, pp 147–170, 1989; J Funct Anal 86:1–18, 1989). In Chap. 2, we have seen the relation between the value function and the HJB equations. If the value function is smooth, then it provides the classical solution of the HJB equations. Unfortunately, when the diffusion coefficient is degenerate, smoothness does not necessarily hold, even for a simple case, and the HJB equations may in general have no classical equation, either. However, the theory of viscosity solutions gives a powerful tools for studing stochastic control problems. Regarding the viscosity solutions for the HJB equations, we claim only continuity for a solution, not necessarily differentiability. Thus, it has been shown that under mild conditions the value function is the unique viscosity solution of the HJB equation. We will revisit this fact in terms of semigroups in Sect. 3.1.3.This chapter is organized as follows. In Sects. 3.1 and 3.2, we recall some basic results on viscosity solutions for (nonlinear) parabolic equations for later use. In Sect. 3.3 we consider stochastic optimal control-stopping problems in a framework similar to that of finite time horizon controls.