  In this paper, we study an optimal mean-variance investment-reinsurance
problem for an insurer (she) under a Cram\'er-Lundberg model with random
coefficients. At any time, the insurer can purchase reinsurance or acquire new
business and invest her surplus in a security market consisting of a risk-free
asset and multiple risky assets, subject to a general convex cone investment
constraint. We reduce the problem to a constrained stochastic linear-quadratic
control problem with jumps whose solution is related to a system of partially
coupled stochastic Riccati equations (SREs). Then we devote ourselves to
establishing the existence and uniqueness of solutions to the SREs by pure
backward stochastic differential equation (BSDE) techniques. We achieve this
with the help of approximation procedure, comparison theorems for BSDEs with
jumps, log transformation and BMO martingales. The efficient
investment-reinsurance strategy and efficient mean-variance frontier are
explicitly given through the solutions of the SREs, which are shown to be a
linear feedback form of the wealth process and a half-line, respectively.
