HJB Equations with Gradient Constraint Associated with Controlled Jump-Diffusion Processes

Abstract

In this paper, we guarantee the existence and uniqueness (in the almost everywhere sense) of the solution to a Hamilton-Jacobi-Bellman (HJB) equation with gradient constraint and a partial integro-differential operator whose Lévy measure has bounded variation. This type of equation arises in a singular control problem, where the state process is a multidimensional jump-diffusion with jumps of finite variation and infinite activity. We verify, by means of ε-penalized controls, that the value function associated with this problem satisfies the aforementioned HJB equation.

Variational inequalities on unbounded domains for zero-sum singular-controller vs. stopper games

“…Proof. This proof refines and extends arguments from [37, Lemma A.2] (see also [25,Lemma 2.8]). For simplicity we denote u = u ε,δ m .…”

“…We find bounds on the Sobolev norm of the solution of the penalised (semilinear) PDE problem, uniformly with respect to the penalisation parameters, thanks to analytical techniques rooted in early work by Evans [12] and new probabilistic tricks developed ad-hoc in our framework. Indeed, it turns out that the co-existence of two hard constraints in (1.2), the 'min-max' structure of the problem, its parabolic nature and unboundedness of the domain make the use of purely analytical ideas as in Evans [12] not sufficient to provide the necessary bounds (see also the references given in the previous paragraph and more recent work by Hynd [22] and Kelbert and Moreno-Franco [25], for comparison). In the process of obtaining our main result (Theorem 3.3) we also contribute a detailed proof of the existence and uniqueness of the solution for the penalised problem (Theorem 4.11 for bounded domain and Theorem 5.3+Proposition 5.4 for unbounded domain), which hopefully will serve as a useful reference for future work in the field.…”

On a mixed singular/switching control problem with multiples regimes

“…Later, Yamada [16] used a NPDS similar to (3.1), to study the HJB equation (2.8). This type of equations appears also in some stochastic singular control problems; see [10] and the references therein.…”

“…Singular and switching stochastic control problems have been of great research interest in control theory due to their applicability in diverse problems of finance, economy, biology and other fields; see, e.g., [10,14] and the reference therein. For that reason, new techniques and problems are continuously developed.…”

On a mixed singular/switching control problem with multiple regimes

“…Singular and switching stochastic control problems have been of great research interest in control theory owing to their applicability to diverse problems of finance, economy, biology, and other fields; see, e.g., [12,16] and the references therein. For that reason, new techniques and problems are continuously being developed.…”