Equilibrium for Regular-Singular Control under Mean-Variance Criterion

This paper studies a class of mixed regular-singular control problems under mean-variance criteria.

We seek time-consistent equilibrium strategies in an intrapersonal game setting and propose a novel equilibrium notion.

Under which, we derive a verification theorem and necessary conditions providing a full mathematical characterization of the equilibrium.

To illustrate the theory, we construct explicit coupled equilibrium solutions for a reinsurance problem, where the regular control depends on the singular control state, and the free boundary of the singular control switches dynamically in accordance with the variation of the regular control expression, yielding nontrivial coupling.

In the degenerate case $\alpha_2=0$, the coupled solution reduces to the combination of two independent single-control equilibria and coincides with the limit as the parameter tends to zero.