arXiv 1208.30782012-08-15math.PR

一维广义漂移随机微分方程的一个注记

A Note on One-dimensional Stochastic Differential Equations with Generalized Drift

Stefan Blei, Hans-Jürgen Engelbert

备注：: 12 pages

摘要：

我们考虑具有广义漂移的一维随机微分方程，其中涉及解过程的局部时间$L^X$：X_t = X_0 + \int_0^t b（X_s）dB_s + \int_\ mathbb{R} L^X（t，y）v（dy），其中b是可测实函数，$B$是Wiener过程，$ν$表示定义在实数线$\mathbb{R}$的有界Borel集上的集函数，使得对于每个$N \in \mathbb{N}$，它是$\mathscr{B}([-N,N])$上的有限符号测度。这类方程通常分别在原子条件$ν({x}) < 1/2$、$ν({x}) > -1/2$和$|ν({x})| < 1$下研究，而与使用右时、左时或对称局部时有关。这个条件允许减少一个方程的广义漂移方程没有漂移和推导条件的存在性和唯一性的解决方案的结果方程没有漂移。本文的主要目的是分别讨论$ν({x}) \geq 1/2$，$ν({x}) \leq -1/2$和$|ν({x})| \geq 1$情形，对于某些$x \in \mathbb{R}$，我们给出了在这些情形下具有广义漂移的方程及其解的特征的完整描述.

英文摘要：

We consider one-dimensional stochastic differential equations with generalized drift which involve the local time $L^X$ of the solution process: X_t = X_0 + \int_0^t b(X_s) dB_s + \int_\mathbb{R} L^X(t,y) ν(dy), where b is a measurable real function, $B$ is a Wiener process and $ν$ denotes a set function which is defined on the bounded Borel sets of the real line $\mathbb{R}$ such that it is a finite signed measure on $\mathscr{B}([-N,N])$ for every $N \in \mathbb{N}$. This kind of equation is, in dependence of using the right, the left or the symmetric local time, usually studied under the atom condition $ν({x}) < 1/2$, $ν({x}) > -1/2$ and $|ν({x})| < 1$, respectively. This condition allows to reduce an equation with generalized drift to an equation without drift and to derive conditions on existence and uniqueness of solutions from results for equations without drift. The main aim of the present note is to treat the cases $ν({x}) \geq 1/2$, $ν({x}) \leq -1/2$ and $|ν({x})| \geq 1$, respectively, for some $x \in \mathbb{R}$, and we give a complete description of the features of equations with generalized drift and their solutions in these cases.

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英文摘要：

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