Convergence of bounded-drift Brownian control values to the initial BCP

Prove that the optimal value function V_a(z) of the modified Brownian control problem with common drift-rate upper bound a converges downward to the optimal value function V(z) of the initial Brownian control problem as a tends to infinity, namely, V_a(z) \downarrow V(z) as a \uparrow \infty.

Background

The paper introduces an initial Brownian control problem (BCP) whose deviation controls may have unbounded variation and can therefore produce instantaneous state displacements. It then proposes a modified BCP in which the drift-rate controls are absolutely continuous and uniformly bounded, while a prescribed boundary-control mechanism handles reflection at the boundary of the nonnegative orthant.

Because every admissible control for the modified BCP is also admissible for the initial BCP, the modified problem has an objective value no smaller than that of the initial problem. The authors conjecture that increasing the common drift-rate bound eliminates the practical distinction between finite-rate and instantaneous displacement control, even though the modified BCP retains a fixed, non-optimized boundary policy. Establishing this convergence would justify the modified BCP as an approximation to the initial BCP when the control bounds are large.