PDE为连续扩散对及其运行最大值的联合律

Pub Date : 2023-01-06 DOI:10.1017/apr.2022.76

L. Coutin, M. Pontier

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引用次数: 2

摘要

设X是d维扩散，M是其第一个分量的运行上确界。在本文中，我们证明了对于任何$t>0，$对$\big（M_t，X_t\big）$的密度（相对于$（d+1）$维Lebesgue测度）是闭集$\big上的Fokker–Planck偏微分方程的弱解。

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PDE for the joint law of the pair of a continuous diffusion and its running maximum

Let X be a d-dimensional diffusion and M the running supremum of its first component. In this paper, we show that for any

$t>0,$

the density (with respect to the

$(d+1)$

-dimensional Lebesgue measure) of the pair

$\big(M_t,X_t\big)$

is a weak solution of a Fokker–Planck partial differential equation on the closed set

$\big\{(m,x)\in \mathbb{R}^{d+1},\,{m\geq x^1}\big\},$

using an integral expansion of this density.

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