## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 3, 7 pp.

### Representation of non-Markovian optimal stopping problems by constrained BSDEs with a single jump

Marco Fuhrman, Huyên Pham, and Federica Zeni

#### Abstract

We consider a non-Markovian optimal stopping problem on finite horizon. We prove that the value process can be represented by means of a backward stochastic differential equation (BSDE), defined on an enlarged probability space, containing a stochastic integral having a one-jump point process as integrator and an (unknown) process with a sign constraint as integrand. This provides an alternative representation with respect to the classical one given by a reflected BSDE. The connection between the two BSDEs is also clarified. Finally, we prove that the value of the optimal stopping problem is the same as the value of an auxiliary optimization problem where the intensity of the point process is controlled.

#### Article information

**Source**

Electron. Commun. Probab., Volume 21 (2016), paper no. 3, 7 pp.

**Dates**

Received: 18 February 2015

Accepted: 7 January 2016

First available in Project Euclid: 3 February 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1454514623

**Digital Object Identifier**

doi:10.1214/16-ECP4123

**Mathematical Reviews number (MathSciNet)**

MR3485372

**Zentralblatt MATH identifier**

1338.60118

**Subjects**

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60H10: Stochastic ordinary differential equations [See also 34F05]

**Keywords**

optimal stopping backward stochastic differential equations randomized stopping

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Fuhrman, Marco; Pham, Huyên; Zeni, Federica. Representation of non-Markovian optimal stopping problems by constrained BSDEs with a single jump. Electron. Commun. Probab. 21 (2016), paper no. 3, 7 pp. doi:10.1214/16-ECP4123. https://projecteuclid.org/euclid.ecp/1454514623