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This paper is concerned with a linear quadratic (LQ, for short) optimalcontrol problem for mean-field backward stochastic differential equations(MF-BSDE, for short) driven by a Poisson random martingale measure and aBrownian motion. Firstly, by the classic convex variation principle, theexistence and uniqueness of the optimal control is established. Secondly, theoptimal control is characterized by the stochastic Hamilton system which turnsout to be a linear fully coupled mean-field forward-backward stochasticdifferential equation with jumps by the duality method. Thirdly, in terms of adecoupling technique, the stochastic Hamilton system is decoupled byintroducing two Riccati equations and a MF-BSDE with jumps. Then an explicitrepresentation for the optimal control is obtained.

Linear-Quadratic Optimal Control Problems for Mean-Field Backward Stochastic Differential Equations with Jumps