## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 26, Number 5 (2016), 3154-3177.

### The inverse first-passage problem and optimal stopping

Erik Ekström and Svante Janson

#### Abstract

Given a survival distribution on the positive half-axis and a Brownian motion, a solution of the inverse first-passage problem consists of a boundary so that the first passage time over the boundary has the given distribution. We show that the solution of the inverse first-passage problem coincides with the solution of a related optimal stopping problem. Consequently, methods from optimal stopping theory may be applied in the study of the inverse first-passage problem. We illustrate this with a study of the associated integral equation for the boundary.

#### Article information

**Source**

Ann. Appl. Probab. Volume 26, Number 5 (2016), 3154-3177.

**Dates**

Received: August 2015

First available in Project Euclid: 19 October 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1476884314

**Digital Object Identifier**

doi:10.1214/16-AAP1172

**Mathematical Reviews number (MathSciNet)**

MR3563204

**Zentralblatt MATH identifier**

1351.60110

**Subjects**

Primary: 60J65: Brownian motion [See also 58J65]

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

**Keywords**

Inverse first-passage problem optimal stopping nonlinear integral equation

#### Citation

Ekström, Erik; Janson, Svante. The inverse first-passage problem and optimal stopping. Ann. Appl. Probab. 26 (2016), no. 5, 3154--3177. doi:10.1214/16-AAP1172. https://projecteuclid.org/euclid.aoap/1476884314