arXiv 1301.65792013-01-28math.PR

从伪随机游动到伪布朗运动：从单侧或双侧区间的首次退出时间

From pseudo-random walk to pseudo-Brownian motion: first exit time from a one-sided or a two-sided interval

Aimé Lachal

备注：: 69 pages
发表信息：: International Journal of Stochastic Analysis 2014 (2014), Art. ID 520136
DOI：: 10.1155/2014/520136

摘要：

设$N$为正整数，$c$为正常数，$(U_n)_{n\ge 1}$为独立同分布伪随机变量序列.我们假设$U_n$在离散集合$\{-N,-N+1,...,N-1,N\}$中取值，并且它们的公共伪分布由任何$k\in\{-N,-N+1,...,N-1,N\}$的\textit{(positive or negative) real}数\[\mathbb{P}\{U_n=k\}=δ_{k0}+(-1)^{k-1} c\binom{2N}{k+N}\]表征。让我们最后介绍$(S_n)_{n\ge 0}$，由$S_0=0$和用于$n\ge 1$的$S_n=\sum_{j=1}^n U_j$在$\mathbb{Z}$上定义的相关伪随机游动。在本文中，我们展示了$(S_n)_{n\ge 0}$的一些性质。特别是，我们明确地确定了一个给定的阈值$(S_n)_{n\ge 0}$的第一个超调时间的伪分布，以及从一个有界的间隔的第一个退出时间。接下来，通过适当的归一化，我们从伪随机游动传递到由高阶热型方程$\partial/\partial t=(-1)^{N-1} c\;\partial^{2N}/$ $\partial x^{2N}$驱动的伪布朗运动。我们检索伪布朗运动的阈值的第一过冲时间的对应伪分布（Lachal，A.：由方程$\frac{\partial}{\partial t}=\pm \frac{\partial^N}{\partial x^N}$驱动的伪过程的首次命中时间和位置、单极和多极。电子学，J. Probab. 12（2007），300- 353 [MR2299920]）。用同样的方法，我们得到了伪布朗运动在有界区间上第一次离开时的伪分布，这是伪过程的一个新结果。

英文摘要：

Let $N$ be a positive integer, $c$ be a positive constant and $(U_n)_{n\ge 1}$ be a sequence of independent identically distributed pseudo-random variables. We assume that the $U_n$'s take their values in the discrete set $\{-N,-N+1,...,N-1,N\}$ and that their common pseudo-distribution is characterized by the \textit{(positive or negative) real} numbers \[\mathbb{P}\{U_n=k\}=δ_{k0}+(-1)^{k-1} c\binom{2N}{k+N}\] for any $k\in\{-N,-N+1,...,N-1,N\}$. Let us finally introduce $(S_n)_{n\ge 0}$ the associated pseudo-random walk defined on $\mathbb{Z}$ by $S_0=0$ and $S_n=\sum_{j=1}^n U_j$ for $n\ge 1$. In this paper, we exhibit some properties of $(S_n)_{n\ge 0}$. In particular, we explicitly determine the pseudo-distribution of the first overshooting time of a given threshold for $(S_n)_{n\ge 0}$ as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudo-random walk to the pseudo-Brownian motion driven by the high-order heat-type equation $\partial/\partial t=(-1)^{N-1} c\;\partial^{2N}/$ $\partial x^{2N}$. We retrieve the corresponding pseudo-distribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, A.: First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation $\frac{\partial}{\partial t}=\pm \frac{\partial^N}{\partial x^N}$. Electron. J. Probab. 12 (2007), 300--353 [MR2299920]). In the same way, we get the pseudo-distribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudo-process.

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