The Annals of Applied Probability

Backward stochastic differential equation driven by a marked point process: An elementary approach with an application to optimal control

Fulvia Confortola, Marco Fuhrman, and Jean Jacod

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Abstract

We address a class of backward stochastic differential equations on a bounded interval, where the driving noise is a marked, or multivariate, point process. Assuming that the jump times are totally inaccessible and a technical condition holds (see Assumption (A) below), we prove existence and uniqueness results under Lipschitz conditions on the coefficients. Some counter-examples show that our assumptions are indeed needed. We use a novel approach that allows reduction to a (finite or infinite) system of deterministic differential equations, thus avoiding the use of martingale representation theorems and allowing potential use of standard numerical methods. Finally, we apply the main results to solve an optimal control problem for a marked point process, formulated in a classical way.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 3 (2016), 1743-1773.

Dates
Received: July 2014
Revised: July 2015
First available in Project Euclid: 14 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1465905018

Digital Object Identifier
doi:10.1214/15-AAP1132

Mathematical Reviews number (MathSciNet)
MR3513605

Zentralblatt MATH identifier
1345.60048

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 93E20: Optimal stochastic control

Keywords
Backward stochastic differential equations marked point processes stochastic optimal control

Citation

Confortola, Fulvia; Fuhrman, Marco; Jacod, Jean. Backward stochastic differential equation driven by a marked point process: An elementary approach with an application to optimal control. Ann. Appl. Probab. 26 (2016), no. 3, 1743--1773. doi:10.1214/15-AAP1132. https://projecteuclid.org/euclid.aoap/1465905018


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