arXiv 1102.31722011-02-15math.PR

马氏过程的随机导数和广义h-变换

Stochastic derivatives and generalized h-transforms of Markov processes

Christian Léonard

摘要：

设$R$是时间区间$[0,1]$上的连续时间马尔可夫过程，其值在某个状态空间$X$中。我们将这个参考过程$R$转化为$P:=f(X_0)\exp (-\int_0^1 V_t(X_t) dt) g(X_1)\,R$，其中$f,g$是X上的非负可测函数，V是$[0,1]\times X$上的某个可测函数。很容易看出，$P$也是马尔可夫的。本文的目的是确定的马尔可夫生成器的$P$的马尔可夫生成器的$R$和额外的成分：$f,g$和$V$在没有规律性的假设$f,g$和$V.$作为第一步，我们表明，扩展的生成器的马尔可夫过程基本上是它的随机衍生物。然后，我们计算的随机导数的$P$，以确定其生成器，在有限的熵条件下。抽象的结果示出连续扩散过程的$\mathbb{R}^d$和Metropolis算法的离散空间。

英文摘要：

Let $R$ be a continuous-time Markov process on the time interval $[0,1]$ with values in some state space $X$. We transform this reference process $R$ into $P:=f(X_0)\exp (-\int_0^1 V_t(X_t) dt) g(X_1)\,R$ where $f,g$ are nonnegative measurable functions on X and V is some measurable function on $[0,1]\times X$. It is easily seen that $P$ is also Markov. The aim of this paper is to identify the Markov generator of $P$ in terms of the Markov generator of $R$ and of the additional ingredients: $f,g$ and $V$ in absence of regularity assumptions on $f,g$ and $V.$ As a first step, we show that the extended generator of a Markov process is essentially its stochastic derivative. Then, we compute the stochastic derivative of $P$ to identify its generator, under a finite entropy condition. The abstract results are illustrated with continuous diffusion processes on $\mathbb{R}^d$ and Metropolis algorithms on a discrete space.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文