Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control

Abstract

We study the partial differential equationwhere u is the unknown function, L is a second-order elliptic operator, f is a given smooth function and H is a convex function. This is a model equation for HamiltonJacobi-Bellman equations arising in stochastic singular control. We establish the existence of a unique viscosity solution of the Dirichlet problem that has a Hölder continuous gradient. We also show that if H is uniformly convex, the gradient of this solution is Lipschitz continuous.

On Hamilton–Jacobi–Bellman equations with convex gradient constraints

“…This work also generalizes the first author's previous work [14], which he considered equation (1.4) with F (M, x) = −a(x) · M. In [14], an a priori W 2,∞ loc (Ω) estimate was derived on solutions under the assumption (1.11). However, an a priori W 2,p loc (Ω) estimate was claimed to be obtained for arbitrary convex gradient constraint functions H. Upon further review, we now believe that the asserted W 2,p loc (Ω) estimate requires a uniform convexity hypothesis similar to (1.11).…”

Solution to HJB Equations with an Elliptic Integro-Differential Operator and Gradient Constraint

“…The method used in (1.7) is usually called penalty method and was introduced by L. C. Evans to establish existence and regularity of solutions to second order elliptic equations with gradient constraints [12]. This method has also been used in other works like [20,37,18,19].…”

“…Previous to this work, the equations (1.1) and (1.7) have mainly been studied in the case that Γ is an elliptic differential operator; see, e.g. [12,20,37,30,21,19]. The closest to our work is the the paper of Menaldi and Robin [29].…”

“…We will establish Theorems 1.1 and 1.2, by probabilistic, integro-differential and PDE classical methods, inspired by Evans [12], Lenhart [25], Gimbert and Lions [17], Soner and Shreve [37], Garroni and Menaldi [14] and Hynd [19]. Since ν(Ê * ) < ∞, we have that the HJB equation (1.1) can be written as…”

A two-dimensional control problem arising from dynamic contracting theory

“…Pham [17] shows that the value functions of a class of optimal switching problems are differentiable by means of viscosity solutions. Regarding multi-dimensional control problems, Soner and Shreve [21], Hynd [12], [13] and Hynd and Mawi [14] prove regularity results for elliptic PDEs arising from singular control problems. A common feature of these studies is the uniform ellipticity assumption about the second order differential operator that defines the elliptic PDE.…”