Abstract
AbstractWe revisit the dividend payment problem in the dual model of Avanzi et al. ([2–4]). Using the fluctuation theory of spectrally positive Lévy processes, we give a short exposition in which we show the optimality of barrier strategies for all such Lévy processes. Moreover, we characterize the optimal barrier using the functional inverse of a scale function. We also consider the capital injection problem of [4] and show that its value function has a very similar form to the one in which the horizon is the time of ruin.
Highlights
In the so-called dual model, the surplus of a company is modeled by a Levy process with positive jumps; see [2,3,4, 7]
In [2], Avanzi and Gerber consider the dividend payment problem when the Levy process is assumed to be the sum of an independent Brownian motion and a compound Poisson process with i.i.d. positive hyper-exponential jumps; they determine the optimal strategy among the set of barrier strategies. (The special case in which the jumps are exponentially distributed was obtained by [7].) The optimality over all admissible strategies is later shown by [4] using the veri cation approach in [7]
We give a short proof showing that for a general spectrally positive Levy process, barrier strategies are optimal for both problems, and we give a simple characterization of the optimal barriers in terms of the scale functions; see (2.13) and (3.4)
Summary
In the so-called dual model, the surplus of a company is modeled by a Levy process with positive jumps (spectrally positive Levy processes); see [2,3,4, 7]. Our goal is to determine the optimal dividend strategy until the time of ruin for all spectrally positive Levy processes. In [2], Avanzi and Gerber consider the dividend payment problem when the Levy process is assumed to be the sum of an independent Brownian motion and a compound Poisson process with i.i.d. positive hyper-exponential jumps; they determine the optimal strategy among the set of barrier strategies. Where ν is a Levy measure with the support (0, ∞) that satis es the integrability condition (0,∞)(1 ∧ z2)ν( dz) < ∞ It has paths of bounded variation if and only if σ = 0 and (0,1) z ν( dz) < ∞; in this case, we write (1.1) as ψ(s) = ds +. We assume that μ < ∞ (and |ψ (0+)| < ∞) to ensure that the problem is nontrivial
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