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arXiv 1106.2245math.PR
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具有大高度过程的Lévy过程

Lévy processes conditioned on having a large height process

Mathieu Richard

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34 pages, 2 figures
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摘要：

在目前的工作中，我们认为频谱积极的Lévy过程$(X_t,t\geq0)$不漂移到$+\infty$，我们有兴趣在条件这些过程达到任意大的高度（在与$X$相关的高度过程的意义上）在击中0之前。这样，我们得到了一个新的条件Lévy过程保持积极的。这个条件过程的（诚实）律$\pfl$被定义为通过鞅的Doob $h$ -变换。对于具有无限变差路径的Lévy过程，对于某些$α$，该鞅是$(\int\tilde\rt(\mathrm{d}z)e^{αz}+I_t)\2{t\leq T_0}$，其中$(I_t,t\geq0)$是PDMATHAMO的过去下确界过程，$(\tilde\rt,t\geq0)$是Duquesne，2002中定义的所谓\emph{exploration process}，其中$T_0$是PDMATHAMO为0的命中时间。在$\pfl$下，我们还得到了$X$在其最小值处的路径分解，这使得我们能够证明$\pfl$的收敛性为$x\to0$。当过程$X$是补偿复合Poisson过程时，前一鞅通过$X$的未来下确界过程的跳来定义。在这种情况下，计算更容易，因为$X$可以被视为（亚）临界\emph{splitting tree}的轮廓过程。我们也可以在脊椎分解的静脉中给出我们的条件过程的另一种表征。

英文摘要：

In the present work, we consider spectrally positive Lévy processes $(X_t,t\geq0)$ not drifting to $+\infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$) before hitting 0. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law $\pfl$ of this conditioned process is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths, this martingale is $(\int\tilde\rt(\mathrm{d}z)e^{αz}+I_t)\2{t\leq T_0}$ for some $α$ and where $(I_t,t\geq0)$ is the past infimum process of $X$, where $(\tilde\rt,t\geq0)$ is the so-called \emph{exploration process} defined in Duquesne, 2002, and where $T_0$ is the hitting time of 0 for $X$. Under $\pfl$, we also obtain a path decomposition of $X$ at its minimum, which enables us to prove the convergence of $\pfl$ as $x\to0$. When the process $X$ is a compensated compound Poisson process, the previous martingale is defined through the jumps of the future infimum process of $X$. The computations are easier in this case because $X$ can be viewed as the contour process of a (sub)critical \emph{splitting tree}. We also can give an alternative characterization of our conditioned process in the vein of spine decompositions.

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