A time of ruin constrained optimal dividend problem for spectrally one-sided Lévy processes

Hernández, Camilo; Junca, Mauricio; Moreno-Franco, Harold A.

doi:10.1016/j.insmatheco.2017.12.011

Cited by 14 publications

(21 citation statements)

References 27 publications

(42 reference statements)

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“…The dividend problem, originally considered by de Finetti, asks to maximize the expected net present value of dividend payments over the set of strategies Θ. Now, as studied in [7], we are interested in addressing a modification of this problem by adding a restriction to the dividend process D, which is given by the following constraint…”

Section: 3mentioning

confidence: 99%

“…where, in the case E x [e −qτ D ] > K for all D ∈ Θ, we set V (x; K) = −∞ and call the problem (2.3) infeasible. Proceeding as in [7], we use Lagrange multipliers to reformulate the problem. For Λ ≥ 0 we define the function…”

Section: 3mentioning

confidence: 99%

“…Following the recent work of Hernández et al [7], we study the case in which the longevity feature is added to the problem by considering a constraint on the time of ruin. The longevity aspect of the firm remained as a separate problem; see [19] for a survey on this matter.…”

Section: Introductionmentioning

confidence: 99%

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Optimality of refraction strategies for a constrained dividend problem

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We consider de Finetti's problem for spectrally one-sided Lévy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.

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Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

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Optimality of refraction strategies for a constrained dividend problem

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“…(i) By Proposition 5.5 in [8] we have that for q ≥ 0, the function Z (q) is strictly logconvex on (0, ∞).…”

Section: Preliminariesmentioning

confidence: 98%

Optimal Bail-Out Dividend Problem with Transaction Cost and Capital Injection Constraint

Junca

Moreno-Franco

Pérez

2019

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We consider the bail-out optimal dividend problem under fixed transaction costs for a Lévy risk model with a constraint on the expected net present value of injected capital. In order to solve this problem, we first consider the bail-out optimal dividend problem under transaction costs and capital injection and show the optimality of reflected (c 1 , c 2 )-policies. We then find the optimal Lagrange multiplier, by showing that in the dual Laagrangian problem, the complementary slackness conditions are verified. Finally, we verify our results with some numerical examples.

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“…For the classic de Finetti barrier and for barrier function (88), this has been investigated in several papers. In [HJ16,(15)], the parameter K intervenes as a Lagrange multiplier associated to a time constraint. Assuming complete monotonicity of the Lévy measure , and letting b 0 denote the last maximum of H D (b), they have shown [HJ16,Prop.…”

Section: Optimization Of Dividends For Spectrally Negative Processesmentioning

confidence: 99%

TheW,Zscale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems

2020

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First passage problems for spectrally negative Lévy processes with possible absorbtion or/and reflection at boundaries have been widely applied in mathematical finance, risk, queueing, and inventory/storage theory. Historically, such problems were tackled by taking Laplace transform of the associated Kolmogorov integro-differential equations involving the generator operator.In the last years there appeared an alternative approach based on the solution of two fundamental "two-sided exit" problems from an interval (TSE). A spectrally one-sided process will exit smoothly on one side on an interval, and the solution is simply expressed in terms of a "scale function" W (3) (Bertoin 1997). The non-smooth two-sided exit (or ruin) problem suggests introducing a second scale function Z (4) (Avram, Kyprianou and Pistorius 2004).Since many other problems can be reduced to TSE, researchers produced in the last years a kit of formulas expressed in terms of the "W, Z alphabet" for a great variety of first passage problems.We collect here our favorite recipes from this kit, including a recent one (94) which generalizes the classic De Finetti dividend problem, and whose optimization may be useful for the valuation of financial companies.One interesting use of the kit is for recognizing relationships between apparently unrelated problems -see Lemma 3. Another is expressing results in a standardized form, improving thus the possibility to check when a formula is already known (which is not altogether trivial given that several related strands of literature need to be checked).Last but not least, it turned out recently that once the classic W, Z are replaced with appropriate generalizations, the classic formulas for (absorbed/ reflected) Lévy processes continue to hold for: a) spectrally negative Markov additive processes (Ivanovs and Palmowski 2012), b) spectrally negative Lévy processes with Poissonian Parisian absorbtion or/and reflection (Avram, Perez and Yamazaki 2017), or with Omega killing (Li and Palmowski 2017.This suggests that processes combining two or three of these features could also be handled by appropriate W, Z functions.An implicit question arising from our list is to investigate the existence of similar formulas for more complicated classes of spectrally negative Markovian processes, like for example continuous branching processes with and without immigration (Kawazu and Watanabe 1971), which are characterized by two Laplace exponents. This topic deserves further investigation.

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A time of ruin constrained optimal dividend problem for spectrally one-sided Lévy processes

Cited by 14 publications

References 27 publications

Optimality of refraction strategies for a constrained dividend problem

Optimality of refraction strategies for a constrained dividend problem

Optimal Bail-Out Dividend Problem with Transaction Cost and Capital Injection Constraint

TheW,Zscale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems

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