We study a two-dimensional discounted optimal stopping problem related to the pricing of perpetual commodity equities in a model of financial markets in which the behaviour of the underlying asset price follows a generalized geometric Brownian motion and the dynamics of the convenience yield are described by an unobservable continuous-time Markov chain with two states. It is shown that the optimal time of exercise is the first time at which the commodity spot price paid in return to the fixed coupon rate hits a lower stochastic boundary being a monotone function of the running value of the filtering estimate of the state of the chain. We rigorously prove that the optimal stopping boundary is regular for the stopping region relative to the resulting two-dimensional diffusion process and the value function is continuously differentiable with respect to the both variables. It is verified by means of a change-of-variable formula with local time on surfaces that the value function and the boundary are determined as a unique solution of the associated parabolic-type free-boundary problem. We also give a closed-form solution to the optimal stopping problem for the case of an observable Markov chain.

我们研究了与金融市场模型中永续商品股票定价相关的二维贴现最优停止问题，其中基础资产价格的行为遵循广义几何布朗运动，便利收益率的动态由具有两种状态的不可观测的连续时间马尔可夫链。结果表明，最优行权时间是固定票面利率所支付的商品现货价格第一次触及下随机边界时，是链状态过滤估计运行值的单调函数. 我们严格证明，相对于所得二维扩散过程，停止区域的最佳停止边界是规则的，并且值函数对于这两个变量是连续可微的。通过曲面上的局部时间变变量公式验证了值函数和边界被确定为相关抛物线型自由边界问题的唯一解。对于可观察的马尔可夫链，我们还给出了最优停止问题的封闭式解决方案。