论文标题
关于不对称电报过程的最大分布的确切分布
On the Exact Distributions of the Maximum of the Asymmetric Telegraph Process
论文作者
论文摘要
在本文中，我们在任意时间间隔$ [0，t] $的情况下，在初始速度$ v（0）$为$ c_1 $或$ -c_2 $的条件下，以任意时间间隔$ [0，t] $介绍了最大值的分布。对于情况，$ v（0）= -C_2 $最大分布的单数组件显示出意外的循环行为，仅取决于$ C_1 $和$ C_2 $，但不取决于当前时间$ t $。我们还获得了$ v（0）= C_1 $或$ V（0）= -C_2 $的最大值的无条件分布，其表达式具有一系列Bessel功能的形式。我们还表明，此分析中出现的所有条件分布均由广义的Euler-Poisson-Darboux方程管辖。我们将对称电报过程的最大分布恢复为本文的特定情况。我们强调，很少碰巧明确获得过程的最大分布。因此，如《电讯》模型这样的自然过程振荡范围的结果使其对许多应用程序有用。
In this paper we present the distribution of the maximum of the asymmetric telegraph process in an arbitrary time interval $[0,t]$ under the conditions that the initial velocity $V(0)$ is either $c_1$ or $-c_2$ and the number of changes of direction is odd or even. For the case $V(0) = -c_2$ the singular component of the distribution of the maximum displays an unexpected cyclic behavior and depends only on $c_1$ and $c_2$, but not on the current time $t$. We obtain also the unconditional distribution of the maximum for either $V(0) = c_1$ or $V(0) = -c_2$ and its expression has the form of series of Bessel functions. We also show that all the conditional distributions emerging in this analysis are governed by generalized Euler-Poisson-Darboux equations. We recover all the distributions of the maximum of the symmetric telegraph process as particular cases of the present paper. We underline that it rarely happens to obtain explicitly the distribution of the maximum of a process. For this reason the results on the range of oscillations of a natural process like the telegraph model make it useful for many applications.
