This work is motivated by numerical solutions to Hamilton-Jacobi-Bellman
quasi-variational inequalities (HJBQVIs) associated with combined stochastic
and impulse control problems. In particular, we consider (i) direct control,
(ii) penalized, and (iii) semi-Lagrangian discretization schemes applied to the
HJBQVI problem. Scheme (i) takes the form of a Bellman problem involving an
operator which is not necessarily contractive. We consider the well-posedness
of the Bellman problem and give sufficient conditions for convergence of the
corresponding policy iteration. To do so, we use weakly chained diagonally
dominant matrices, which give a graph-theoretic characterization of weakly
diagonally dominant M-matrices. We compare schemes (i)--(iii) under the
following examples: (a) optimal control of the exchange rate, (b) optimal
consumption with fixed and proportional transaction costs, and (c) pricing
guaranteed minimum withdrawal benefits in variable annuities. We find that one
should abstain from using scheme (i).

本研究探讨了与随机控制和脉冲控制相关的 Hamilton-Jacobi-Bellman 准变分不等式（HJBQVIs）的数字解法，提出了三种离散化方案并比较了其在不同实例中的表现。其中，直接控制的方案的表现较差，不推荐使用。

弱链式矩阵、策略迭代和脉冲控制

Weakly chained matrices, policy iteration, and impulse control

We consider impulse control problems in finite horizon for diffusions with
decision lag and execution delay. The new feature is that our general framework
deals with the important case when several consecutive orders may be decided
before the effective execution of the first one. This is motivated by financial
applications in the trading of illiquid assets such as hedge funds. We show
that the value functions for such control problems satisfy a suitable version
of dynamic programming principle in finite dimension, which takes into account
the past dependence of state process through the pending orders. The
corresponding Bellman partial differential equations (PDE) system is derived,
and exhibit some peculiarities on the coupled equations, domains and boundary
conditions. We prove a unique characterization of the value functions to this
nonstandard PDE system by means of viscosity solutions. We then provide an
algorithm to find the value functions and the optimal control. This easily
implementable algorithm involves backward and forward iterations on the domains
and the value functions, which appear in turn as original arguments in the
proofs for the boundary conditions and uniqueness results.

本文研究了带决策滞后和执行延迟的扩散情况下的脉冲控制问题。我们在动态规划原理的适当版本下，通过挂起的订单考虑状态过程的过去依赖关系，导出对应的 Bellman 偏微分方程系统，并通过粘性解的方式得出了对该非标准 PDE 系统的价值函数唯一刻画和计算最优控制的算法。