An Optimal Dividend Problem with Capital Injections over a Finite Horizon
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.
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Center for Mathematical Economics
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