Uniqueness Problem for the Backward Differential Equation of a Continuous-State Branching Process.

The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup, which is defined by the backward differential equation. We provide a proof of the assertion of Rhyzhov and Skorokhod (<italic>Theory Probab. Appl.</italic>, 1970) on the uniqueness of the solutions to the equation, which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations. [ABSTRACT FROM AUTHOR]