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In this paper we study the joint ruin problem for two insurance companiesthat divide between them both claims and premia in some specified proportions(modeling two branches of the same insurance company or an insurance andre-insurance company). Modeling the risk processes of the insurance companiesby Cram\'{e}r-Lundberg processes we obtain the Laplace transform in space ofthe probability that either of the insurance companies is ruined in finitetime. Subsequently, for exponentially distributed claims, we derive an explicitanalytical expression for this joint ruin probability by explicitly invertingthis Laplace transform. We also provide a characterization of the Laplacetransform of the joint ruin time.

A two-dimensional ruin problem on the positive quadrant