弱Hörmander条件下退化扩散方程Varadhan估计的Yosida参数化方法
A Yosida's parametrix approach to Varadhan's estimates for a degenerate diffusion under the weak Hörmander condition
摘要：
我们适应和扩展Yosida的参数化方法，最初介绍的黎曼流形上的抛物型算子的基本解的建设，推导Varadhan型渐近估计的退化扩散下的弱Hörmander条件下的转移密度。这个扩散过程，被Yor在一系列论文中广泛研究，发现直接应用于一类称为亚式期权的路径依赖金融衍生品的研究。我们得到Varadhan公式\begin{equation} \frac{-2 \log p(t,x;T,y) } { Ψ(t,x;T,y) } \to 1, \qquad \text{as } \quad T-t \to 0^+, \end{equation}其中$p$表示转移密度和$Ψ$表示最优费用函数的确定性控制问题的扩散。我们提供了这个公式的部分证明，并提出了数值证据来支持一个中间不等式的有效性，需要完成的证明。我们还推导出一个渐近展开的费用函数$Ψ$，表示在初等函数，这是有用的，以设计有效的近似公式的过渡密度。
英文摘要：
We adapt and extend Yosida's parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under the weak Hörmander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of path-dependent financial derivatives known as Asian options. We obtain the Varadhan formula \begin{equation} \frac{-2 \log p(t,x;T,y) } { Ψ(t,x;T,y) } \to 1, \qquad \text{as } \quad T-t \to 0^+, \end{equation} where $p$ denotes the transition density and $Ψ$ denotes the optimal cost function of a deterministic control problem associated to the diffusion. We provide a partial proof of this formula, and present numerical evidence to support the validity of an intermediate inequality that is required to complete the proof. We also derive an asymptotic expansion of the cost function $Ψ$, expressed in terms of elementary functions, which is useful in order to design efficient approximation formulas for the transition density.
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