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

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