Initial-Data-Parameterized linear quadratic stochastic optimal control problems with random jumps

A stochastic controlproblem is formulated and we get the explicit form of the optimal controlfor initial-data-parameterized linear quadratic stochastic optimalcontrol problems with random jumps. The optimal controlcan be proved to be unique. A stochastic Riccati equationis rigorous derived from the stochastic Hamilton system, which provides an optimal feedback control. This completes the the interrelationship between the stochastic Riccati equationand stochastic Hamilton system as two different but equivalent tools for the stochastic linear quadraticproblem.