A comparison theorem on constrained viscosity solutions of a HJB equation (Q2744337)

A somewhat generalized comparison theorem on the constrained viscosity solutions of the Hamilton-Jacobi-Bellman (HJB) equation related to the consumption-investment problem is proved, which was first studied in detail by \textit{T. Zariphopoulou} [cf. SIAM J. Control Optim. 32, 59-95 (1994; Zbl 0790.90007)]. The authors prove here that if \(u\) is an upper-semicontinuous concave viscosity solution of the associated HJB equation and \(v\) is a sublinearly growing, uniformly continuous and locally Lipschitz viscosity supersolution of the HJB equation on \(\overline S\), then \(u\leq v\) on \(\overline S\) that is bounded from below, where \(\overline S\) is any open set of \(R\). Note that in the comparison theorem of Zariphopoulou (loc. cit., Theorem 4.1), \(\overline S\) was replaced by \(\overline\Omega=[0,\infty)\).