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

Abstract

The main goal of this paper is to establish existence, regularity and uniqueness results for the solution of a Hamilton-Jacobi-Bellman (HJB) equation, whose operator is an elliptic integro-differential operator. The HJB equation studied in this work arises in singular stochastic control problems where the state process is a controlled d-dimensional Lévy process.

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

“…Remark 1.6. Previously to the paper by Moreno-Franco [27] and this paper, the singular stochastic control problem described above has been studied extensively in the onedimensional case when the state process includes the continuous part only; see, e.g., [3,6,14,19,20]. Several articles focused on the multidimensional case when the state process is a multidimensional SBM [9,21,26,31], a diffusion process [11,16,17], or a multidimensional SBM with jumps process, whose Lévy measure ν satisfies Ê d * |z| p ν(dz) < ∞, for all p ≥ 2 [25].…”

“…Although the NPIDD problem (1.8) is a tool to guarantee the existence of the solution to the HJB equation (1.1), this turns out to be a problem of interest itself because, previously to this paper, we find few references related to this class of problems. Say, paper [27] analyses the NPIDD problem (1.8) when the Lévy measure ν is finite on Ê d * , and [28] studies a degenerate Neumann problem for quasi-linear elliptic integro-differential operators when the Lévy measure ν has unbounded variation, i.e., Ê d * [|z| 2 ∧ 1]ν(dz) < ∞, and s satisfies s(x, z) = 0, for (x, z) ∈ O × Ê d * such that x + z / ∈ O. This type of problem can also be related to an absolutely continuous optimal control problem when the controlled process is a jump-diffusion with jump measure of finite variation; see Section 4.…”

“…Recently, Moreno-Franco [27] analysed the HJB equation (1.1) when the domain set is a ball B R (0) ⊂ Ê d , the coefficients of the partial integro-differential operator Γ are constant, s = g = 1, c = q, with q being a positive constant large enough, and the Lévy measure ν has a density κ ∈ C 0,α ′ (Ê d * ) with respect to the Lebesgue measure dz such that ν(Ê d * ) < ∞.…”

“…Notice that the HJB equation (1.1) with the operator Γ defined as in (1.2) is more general than in [27]. The Lévy measure ν can satisfy ν(Ê d * ) = ∞ and it is not required that ν has a density κ with respect to the Lebesgue measure dz.…”

“…For each p ∈ (d, ∞), there exists a unique non-negative solution u to the HJB equation (1.1) in the space C 0,1 (O) ∩ W 2,p loc (O). The solution u to the HJB equation (1.1) is established in the almost everywhere sense in line with [27]. To prove Theorem 1.1; see Section 3, we will employ a penalization technique, which has been used by [9,15,16,17,18,27,31,32], when the operator Γ has only the elliptic differential part L or when the Lévy measure of its integral part I is finite.…”