论文标题
在泊松最佳停止下的价值函数的形状
The shape of the value function under Poisson optimal stopping
论文作者
论文摘要
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 ${\mathbb E}^x[e^{-βτ}g(X_τ)]$, maximisation takes place over all stopping times $τ$.在泊松最佳停止问题中，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$v_θ（x）= \ sup_ {τ\ in {\ mathcal t}（{\ sathbb t}^θ）}} {\ mathbb e}^x [e^e^{e^{ - βτ} g（x____°带速率$θ=（θ（x_t））_ {t \ geq 0} $的泊松过程_ {t \ geq 0} $）从$ g $继承单调性和凸性属性。事实证明，$ x $中的$v_θ$的单调性（分别凸）取决于数量$ \ frac {θ（x）g（x）g（x）} {θ（x）{θ（x） +β} $而不是$ g $而不是$ g $的单调性（分别凸）。我们的主要技术是随机耦合。
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 ${\mathbb E}^x[e^{-βτ}g(X_τ)]$, maximisation takes place over all stopping times $τ$. 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 $V_θ(x) = \sup_{τ\in {\mathcal T}({\mathbb T}^θ)}{\mathbb E}^x[e^{-βτ}g(X_τ)]$ (where the supremum is taken over stopping times taking values in the event times of an inhomogeneous Poisson process with rate $θ= (θ(X_t))_{t \geq 0}$) inherits monotonicity and convexity properties from $g$. It turns out that monotonicity (respectively convexity) of $V_θ$ in $x$ depends on the monotonicity (respectively convexity) of the quantity $\frac{θ(x) g(x)}{θ(x) + β}$ rather than $g$. Our main technique is stochastic coupling.
