The Tax Identity For Markov Additive Risk Processes

Albrecher, Hansjörg; Avram, Florin; Constantinescu, Corina; Ivanovs, Jevgeņijs

doi:10.1007/s11009-012-9310-y

Cited by 10 publications

(7 citation statements)

References 22 publications

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“…Before presenting our main theorem, we emphasise that our results hold true for a large class of stochastic processes for X that includes, amongst others, spectrally negative Lévy processes, spectrally negative Markov additive processes (see [4]), diffusion processes (see [11]) and fractional Brownian motion. However, practically, (1) and (2) may not in all cases be the right way to define a taxed process.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

The equivalence of two tax processes

Ghanim

Loeffen

Watson

2020

Insurance: Mathematics and Economics

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We introduce two models of taxation, the latent and natural tax processes, which have both been used to represent loss-carry-forward taxation on the capital of an insurance company. In the natural tax process, the tax rate is a function of the current level of capital, whereas in the latent tax process, the tax rate is a function of the capital that would have resulted if no tax had been paid. Whereas up to now these two types of tax processes have been treated separately, we show that, in fact, they are essentially equivalent. This allows a unified treatment, translating results from one model to the other. Significantly, we solve the question of existence and uniqueness for the natural tax process, which is defined via an integral equation. Our results clarify the existing literature on processes with tax.

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Section: Introduction and Main Resultsmentioning

confidence: 93%

The equivalence of two tax processes

Ghanim

Loeffen

Watson

2020

Insurance: Mathematics and Economics

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“…In this case, putting φ γ (u) = φ γ,ξ,a (u) = E u [e −δτ + a,γ ; τ + a,γ < σ ξ,γ ], Thm 1.1 is equivalent to for v γ (u, a) = γE u τ + a,γ ∧σ ξ,γ 0 e −δt dX(t), where the derivative is with respect to u. The equations (112), (113) can be derived by elementary considerations in the Cramér-Lundberg case -see [AACI14,AH07] for similar computations in the particular case of stopping at ruin. Or, giving up rigor, one may just recognize them as the Kolmogorov equation for a) the survival probability and b) γ× the expected time until death for the excised taxed process with drawdown killing.…”

Section: De Finetti's Problem For Taxed Process With Affine Drawdown mentioning

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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“…The loss-carry-forward taxation model is first proposed by Albrecher and Hipp [4] under the compound Poisson model. It has been extended to the spectrally negative Lévy model by Albrecher et al [6], the time-homogeneous diffusion model by Li et al [19], and the Markov additive model by Albrecher [2].…”

Section: Extension To the General Loss-carry-forward Taxation Modelmentioning

confidence: 99%

“…where now the newly defined scale function naturally depends on the two variables x, u. Several control problems for (X, D) are known to reduce to the study of the process X t with all its negative excursions excised, which turns out to be a deterministic process, killed at a random time [2,3]-see Figure 1 below. This supports the parallel fundamental idea of [18] to base the study of (X, D) on the existence of two differential parameters.…”

Section: Introductionmentioning

confidence: 99%

General drawdown of general tax model in a time-homogeneous Markov framework

Avram¹,

Li²,

Li³

2018

Preprint

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Drawdown/regret times feature prominently in optimal stopping problems, in statistics (CUSUM procedure) and in mathematical finance (Russian options). Recently it was discovered that a first passage theory with general drawdown times, which generalize classic ruin times, may be explicitly developed for spectrally negative Lévy processes -see Avram, Vu, Zhou(2017), Li, Vu, Zhou(2017. In this paper, we further examine general drawdown related quantities for taxed time-homogeneous Markov processes, using the pathwise connection between general drawdown and tax process.

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The Tax Identity For Markov Additive Risk Processes

Cited by 10 publications

References 22 publications

The equivalence of two tax processes

The equivalence of two tax processes

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

General drawdown of general tax model in a time-homogeneous Markov framework

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