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

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.

Published at http://dx.doi.org/10.1214/15-AAP1132 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples
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• Approximate and Approximate Null-Controllability of a Class of Piecewise Linear Markov Switch Systems
• Optimal control of semi-Markov processes with a backward stochastic differential equations approach
• Reflections on BSDEs
• Optimal control and zero-sum games for Markov chains of mean-field type
• Optimal switching problem for marked point process and systems of reflected BSDE
• Generalized BSDEs with random time horizon in a progressively enlarged filtration
• Nonequivalence of Controllability Properties for Piecewise Linear Markov Switch Processes
• Controllability Metrics on Networks with Linear Decision Process-type Interactions and Multiplicative Noise