  In this paper, we consider a constrained stochastic linear-quadratic (LQ)
optimal control problem where the control is constrained in a closed cone. The
state process is governed by a controlled SDE with random coefficients.
Moreover, there is a random jump of the state process. In mathematical finance,
the random jump often represents the default of a counter party. Thanks to the
It\^o-Tanaka formula, optimal control and optimal value can be obtained by
solutions of a system of backward stochastic differential equations (BSDEs).
The solvability of the BSDEs is obtained by solving a recursive system of BSDEs
driven by the Brownian motions. We also apply the result to the mean variance
portfolio selection problem in which the stock price can be affected by the
default of a counterparty.
