In this paper we propose and solve an optimal dividend problem with capital

injections over a finite time horizon. The surplus dynamics obeys a linearly controlled drifted

Brownian motion that is reflected at zero, dividends give rise to time-dependent instantaneous

marginal profits, whereas capital injections are subject to time-dependent instantaneous marginal

costs. The aim is to maximize the sum of a liquidation value at terminal time and of the

total expected profits from dividends, net of the total expected costs for capital injections.

Inspired by the study in [13] on reflected follower problems, we relate the optimal dividend

problem with capital injections to an optimal stopping problem for a drifted Brownian motion

that is absorbed at zero. We show that whenever the optimal stopping rule is triggered

by a time-dependent boundary, the value function of the optimal stopping problem gives

the derivative of the value function of the optimal dividend problem. Moreover, the optimal

dividends' distribution strategy is also triggered by the moving boundary of the associated

stopping problem. The properties of this boundary are then investigated in a case study

in which instantaneous marginal profits and costs from dividends and capital injections are

constants discounted at a constant rate.