The dividend problem with a finite horizon

Angelis, Tiziano De; Ekström, Erik

doi:10.1214/17-aap1286

Cited by 29 publications

(62 citation statements)

References 31 publications

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“…A striking difference with the problem studied in [16] is the much more involved dynamics underlying the OSP and the behaviour of the gain process. In [16], the state dynamics in the control problem is of the form (t,X D t ), withX D as in (1.1) but with deterministic constant drift.…”

Section: Mathematical Background and Overview Of Main Resultsmentioning

confidence: 92%

“…In the absence of capital injection, even just assuming a finite time-horizon for the dividend problem, i.e., taking γ D ∧ T for some deterministic T > 0, introduces major technical difficulties. The latter were addressed first by Grandits in [29] and [30] with PDE methods, and then by De Angelis and Ekström [16] with probabilistic methods. Interestingly, the finite time-horizon is more easily tractable in the presence of capital injection, as shown in Ferrari and Schuhmann [26] using ideas originally contained in El Karoui and Karatzas [25].…”

Section: Mathematical Background and Overview Of Main Resultsmentioning

confidence: 99%

“…Here we take the approach suggested in [16], but as we explain below, we substantially expand the results therein. First we link our dividend problem to a suitable optimal stopping one.…”

Section: Mathematical Background and Overview Of Main Resultsmentioning

confidence: 99%

“…As in [16], the presence of an absorbing point for the process X D 'destroys' the standard link between optimal stopping and singular control. Such a link has been studied by many authors.…”

Section: Mathematical Background and Overview Of Main Resultsmentioning

confidence: 99%

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Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

Angelis

2019

Finance Stoch

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We study the optimal dividend problem for a firm's manager who has partial information on the profitability of the firm. The problem is formulated as one of singular stochastic control with partial information on the drift of the underlying process and with absorption. In the Markovian formulation, we have a two-dimensional degenerate diffusion whose first component is singularly controlled. Moreover, the process is absorbed when its first component hits zero. The free boundary problem (FBP) associated to the value function of the control problem is challenging from the analytical point of view due to the interplay of degeneracy and absorption. We find a probabilistic way to show that the value function of the dividend problem is a smooth solution of the FBP and to construct an optimal dividend strategy. Our approach establishes a new link between multidimensional singular stochastic control problems with absorption and problems of optimal stopping with 'creation'. One key feature of the stopping problem is that creation occurs at a state-dependent rate of the 'local time' of an auxiliary two-dimensional reflecting diffusion.

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Section: Mathematical Background and Overview Of Main Resultsmentioning

confidence: 92%

Section: Mathematical Background and Overview Of Main Resultsmentioning

confidence: 99%

“…Here we take the approach suggested in [16], but as we explain below, we substantially expand the results therein. First we link our dividend problem to a suitable optimal stopping one.…”

Section: Mathematical Background and Overview Of Main Resultsmentioning

confidence: 99%

Section: Mathematical Background and Overview Of Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

Angelis

2019

Finance Stoch

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“…Next we show left-continuity for all t ∈ (0, T ) and for this we adapt to our setting ideas as those in the proof of Proposition 4.2 in De Angelis and Ekström [10]. Suppose that b makes a jump at some t ∈ (0, T ).…”

Section: Verifying Assumption 31: a Case Study With Discounted Constmentioning

confidence: 99%

An Optimal Dividend Problem with Capital Injections over a Finite Horizon

Ferrari¹,

Schuhmann²

2019

SIAM J. Control Optim.

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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 the origin, 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 of El Karoui and Karatzas [14] 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 the origin. 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 dividend 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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Optimal dividends and capital injection under dividend restrictions

Lindensjö

Lindskog

2020

Math Meth Oper Res

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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.

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The dividend problem with a finite horizon

Cited by 29 publications

References 31 publications

Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

Optimal dividends with partial information and stopping of a degenerate reflecting diffusion

An Optimal Dividend Problem with Capital Injections over a Finite Horizon

Optimal dividends and capital injection under dividend restrictions

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