Transformations of joint distributions of random variables connected with fluctuations of a process with independent increments

E. A. Pechersky ; B. A. Rogozin

Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 3, pp. 431-444

Résumé

Let $\xi(t)$, $t\ge0$, be a homogeneous process with independent increments, $\mathbf M\exp\{\lambda\xi(t)\}=\exp\{t\psi(\lambda)\}$ be its characteristic function. The random variables \begin{gather*} \overline\xi(t)=\sup_{0\le u\le t}\{\xi(u)\},\quad T(t)=\inf\{u\colon\overline\xi(u)=\overline\xi(t)\},\quad\tau(x)=\inf\{u\colon\overline\xi(u)\ge x\}, \\ L(t)=\frac12\int_0^t(1+\operatorname{sign}\xi(u))\,du,\quad\gamma(x)=\overline\xi(\tau(x))-x \end{gather*} being considered, expressions of the following transforms of their distributions \begin{gather*} \int_0^\infty e^{-ut}\mathbf M\exp\{\lambda\overline\xi(t)+\mu(\xi(t)-\overline\xi(t))-\nu T(t)\}\,dt, \\ \int_0^\infty e^{-ut}\mathbf M\exp\{\mu\xi(t)-\nu L(t)\}\,dt,\quad\int_0^\infty e^{\lambda x}\mathbf M\exp\{\lambda\tau(x)+\mu\gamma(x)\}\,dx \end{gather*} are obtained in terms of the components of the infinitely divisible factorization of $uf(u-\psi(\lambda))$.