Optimal dividends and capital injection under dividend restrictions

Abstract

We study a singular stochastic control problem faced by the owner of an insurance company that dynamically pays dividends and raises capital in the presence of the restriction that the surplus process must be above a given dividend payout barrier in order for dividend payments to be allowed. Bankruptcy occurs if the surplus process becomes negative and there are proportional costs for capital injection. We show that one of the following strategies is optimal: (i) Pay dividends and inject capital in order to reflect the surplus process at an upper barrier and at 0, implying bankruptcy never occurs. (ii) Pay dividends in order to reflect the surplus process at an upper barrier and never inject capital-corresponding to absorption at 0-implying bankruptcy occurs the first time the surplus reaches zero. We show that if the costs of capital injection are low, then a sufficiently high dividend payout barrier will change the optimal strategy from type (i) (without bankruptcy) to type (ii) (with bankruptcy). Moreover, if the costs are high, then the optimal strategy is of type (ii) regardless of the dividend payout barrier. We also consider the possibility for the owner to choose a stopping time at which the insurance company is liquidated and the owner obtains a liquidation value. The uncontrolled surplus process is a Wiener process with drift.

On the surplus management of funds with assets and liabilities in presence of solvency requirements

“…With proportional transaction costs, it turns out (unsurprisingly) that if such measures are warranted (have positive expected value), these will be made only at level α 0 to avoid bankruptcy. A recent treatment of optimal dividends and capital injections with transaction costs is Lindensjö and Lindskog (2020), although they considered a pure diffusion rather than a ratio of two correlated geometric Brownian motions as in this paper.…”

Moment-constrained optimal dividends: precommitment and consistent planning

“…Optimal dividends under ruin probability constraints are studied in [27,30], while optimal dividends under a constraint for the ruin time is studied in [29]. A dividend problem under the constraint that the surplus process must be above a given fixed level in order for dividend payments to be admissible is studied in [40]; see also [36], where this problem is studied in a model which allows for capital injection. Optimal stopping under expectation constraints is studied in [4,10], while stochastic control under expectation constraints is studied in [20].…”

Moment constrained optimal dividends: precommitment \& consistent planning

“…Optimal dividends under ruin probability constraints are studied in [23,26], while optimal dividends under a constraint for the ruin time is studied in [25]. A dividend problem under the constraint that the surplus process must be above a given fixed level in order for dividend payments to be admissible is studied in [36]; see also [32] where this problem is studied in a model which allows for capital injection. Optimal stopping under expectation constraints is studied in [3,9] while stochastic control under expectation constraints is studied in [44].…”

“…There is a vast literature on many different versions of the optimal dividend problem, see e.g. the literature reviews [1,4] and the more recent surveys included in [22,24,32].…”