## The Annals of Probability

- Ann. Probab.
- Volume 3, Number 2 (1975), 215-233.

### First Passage Distributions of Processes With Independent Increments

#### Abstract

Let $\{X_t, t \geqq 0\}$ be a process with stationary independent increments taking values in $d$-dimensional Euclidean space. Let $S$ be a set in $R^d$, and let $T = \inf\{t > 0: X_t \not\in S\}$. For a reasonably wide class of processes and sets $S$, criteria are given for deciding when $P\{X_T \in B\} > 0$ and when $P\{X_T \in B\} = 0$, where $B \subset \partial S$.

#### Article information

**Source**

Ann. Probab. Volume 3, Number 2 (1975), 215-233.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996394

**Digital Object Identifier**

doi:10.1214/aop/1176996394

**Mathematical Reviews number (MathSciNet)**

MR368177

**Zentralblatt MATH identifier**

0318.60063

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J30

Secondary: 60G17: Sample path properties 60G10: Stationary processes 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J25: Continuous-time Markov processes on general state spaces 60J40: Right processes

**Keywords**

Stochastic processes Markov process stationary independent increments Levy measure first passage distribution local growth sample function behavior

#### Citation

Millar, P. W. First Passage Distributions of Processes With Independent Increments. Ann. Probab. 3 (1975), no. 2, 215--233. doi:10.1214/aop/1176996394. https://projecteuclid.org/euclid.aop/1176996394