In a classical problem for the stopping of a diffusion process $(X_t)_{t \geq 0}$, where the goal is to maximise the expected discounted value of a function of the stopped process ${\mathbb E}^x[e^{-\beta \tau}g(X_\tau)]$, maximisation takes place over all stopping times $\tau$. In a Poisson optimal stopping problem, stopping is restricted to event times of an independent Poisson process. In this article we consider whether the resulting value function $V_\theta(x) = \sup_{\tau \in {\mathcal T}({\mathbb T}^\theta)}{\mathbb E}^x[e^{-\beta \tau}g(X_\tau)]$ (where the supremum is taken over stopping times taking values in the event times of an inhomogeneous Poisson process with rate $\theta = (\theta(X_t))_{t \geq 0}$) inherits monotonicity and convexity properties from $g$. It turns out that monotonicity (respectively convexity) of $V_\theta$ in $x$ depends on the monotonicity (respectively convexity) of the quantity $\frac{\theta(x) g(x)}{\theta(x) + \beta}$ rather than $g$. Our main technique is stochastic coupling.

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Poisson最优停止下价值函数的形状

在停止扩散过程 $(X_t)_{t \geq 0}$ 的经典问题中，目标是最大化停止过程 ${\mathbb E}^x[ e^{-\beta \tau}g(X_\tau)]$，最大化发生在所有停止时间 $\tau$。在泊松最优停止问题中，停止仅限于独立泊松过程的事件时间。在本文中我们考虑结果值函数是否 $V_\theta(x) = \sup_{\tau \in {\mathcal T}({\mathbb T}^\theta)}{\mathbb E}^x[e ^{-\beta \tau}g(X_\tau)]$（其中最高值是在停止时间上取值的非齐次泊松过程的事件时间，速率为 $\theta = (\theta(X_t))_ {t \geq 0}$) 从 $g$ 继承单调性和凸性属性。事实证明，$x$ 中 $V_\theta$ 的单调性（分别为凸性）取决于数量 $\frac{\theta(x) g(x)}{\theta(x) 的单调性（分别为凸性） + \beta}$ 而不是 $g$。我们的主要技术是随机耦合。