论文标题
有关电报过程的条件概率的注释
A Note on the Conditional Probabilities of the Telegraph Process
论文作者
论文摘要
我们考虑具有两个速度的电报过程，$ a_1> a_2 \ in \ mathbb {r} $和两个反转速率，$λ_1，λ_2> 0 $。我们研究了有关条件概率度量的某些特征，其中最初的速度和方向变化的数量均已知道。我们通过诱导（有条件的）概率定律以及对时间$ t> 0 $在以前的位置$ 0 <s <t $的位置进行的详细研究来展示新的证明。在对称过程的情况下，我们为在时间$ t> 0 $的关节位置（最大值及其最小值）提出了一些结果。
We consider the telegraph process with two velocities, $a_1>a_2\in\mathbb{R}$, and two rates of reversal, $λ_1,λ_2>0$. We study some of its features with respect to the conditional probability measure where both the initial speed and the number of changes of direction are known. We exhibit a new proof by induction of the (conditional) probability law and a detailed study of the distribution of the motion at time $t>0$ conditioned on its position at a previous time $0<s<t$. In the case of a symmetric process, we present some results on the joint distribution of the position of the motion at time $t>0$, its maximum and its minimum up to that moment.
