For pricing American options, %after suitable discretization in space and time, a sequence of discrete linear complementarity problems (LCPs) or equivalently Hamilton-Jacobi-Bellman (HJB) equations need to be solved in a sequential time-stepping manner. In each time step, the policy iteration or its penalty variant is often applied due to their fast convergence rates. In this paper, we aim to solve for all time steps simultaneously, by applying the policy iteration to an ``all-at-once form" of the HJB equations, where two different parallel-in-time preconditioners are proposed to accelerate the solution of the linear systems within the policy iteration. Our proposed methods are generally applicable for such all-at-once forms of the HJB equation, arising from option pricing problems with optimal stopping and nontrivial underlying asset models. Numerical examples are presented to show the feasibility and robust convergence behavior of the proposed methodology.

翻译：对于美式期权定价，在空间和时间上进行适当离散化后，需要以顺序时间步进方式求解一系列离散线性互补问题（LCP）或等价的Hamilton-Jacobi-Bellman（HJB）方程。在每个时间步中，由于策略迭代及其罚函数变体具有快速收敛特性，因此常被采用。本文旨在通过将策略迭代应用于HJB方程的"全耦合形式"，同时求解所有时间步，为此提出了两种不同的时间并行预处理器，用于加速策略迭代中线性系统的求解。所提方法普遍适用于由最优停时和非平凡标的资产模型衍生的期权定价问题中的HJB方程全耦合形式。数值算例展示了所提方法的可行性和稳健收敛行为。