Asymptotic results for renewal risk models with risky investments

Albrecher, Hansjörg; Constantinescu, Corina; Thomann, Enrique

doi:10.1016/j.spa.2012.05.017

Cited by 26 publications

(29 citation statements)

References 28 publications

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“…Tang et al (2010) established a result similar to our relation (2.3) for both T < ∞ and T = ∞ for the case in which P is a compound renewal process with regularly varying jumps andR is a general Lévy process. Albrecher et al (2012) also investigated the asymptotic behavior of the infinite-time ruin probability and related quantities for the case in which P is a compound renewal process having light-tailed or heavy-tailed jumps andR is a Brownian motion with drift. Hult and Lindskog (2011) considered a more general case in which P is a Lévy process with regularly varying jumps and R is a semimartingale, and they established an asymptotic formula for the finite-time ruin probability, which holds uniformly for all R with the stochastic exponential eR t fulfilling a certain moment condition.…”

Section: Resultsmentioning

confidence: 99%

Asymptotic Ruin Probabilities for a Bivariate Lévy-Driven Risk Model with Heavy-Tailed Claims and Risky Investments

Hao

Tang

2012

Journal of Applied Probability

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Consider a general bivariate Lévy-driven risk model. The surplus process Y , starting with Y 0 = x > 0, evolves according to dY t = Y t− dR t − dP t for t > 0, where P and R are two independent Lévy processes representing, respectively, a loss process in a world without economic factors and a process describing return on investments in real terms. Motivated by a conjecture of Paulsen, we study the finite-time and infinite-time ruin probabilities for the case in which the loss process P has a Lévy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition. We obtain a simple and unified asymptotic formula as x → ∞, which confirms Paulsen's conjecture.

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Section: Resultsmentioning

confidence: 99%

Asymptotic Ruin Probabilities for a Bivariate Lévy-Driven Risk Model with Heavy-Tailed Claims and Risky Investments

Hao

Tang

2012

Journal of Applied Probability

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“…However, there exist various attempts in relaxing the usual assumptions. For example, the classical Poisson process used to model the number of claims has been replaced with other processes, see, e.g., [3,[5][6][7][8][9][10][11] etc. On the other hand, a dependency of the premium on the size of the surplus has been taken into account in [12].…”

Section: Introductionmentioning

confidence: 99%

On the ruin probability for nonhomogeneous claims and arbitrary inter-claim revenues

Răducan¹,

Vernic

Zbăganu

2015

Journal of Computational and Applied Mathematics

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a b s t r a c tRecently, Raducan et al. (2015) obtained recursive formulas for the ruin probability of a surplus process at or before claim instants under the assumptions that the claim sizes are independent, nonhomogeneous Erlang distributed, and independent of the inter-claim times (i.e., the times between two successive claims), which are assumed to be independent, identically distributed (i.i.d.), following an Erlang or a mixture of exponentials distribution. In this paper, we extend these formulas to the more general case when the inter-claim times are i.i.d. nonnegative random variables following an arbitrary distribution. We also present numerical results based on the new recursions, discuss some computational aspects and state a conjecture that connects the ordering of the claims arrival with the magnitude of the corresponding ruin probabilities.

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“…Among surplus-dependent premiums, risk models with risky investments have been widely analyzed (see e.g., Albrecher et al 2012;Frolova et al 2002;Paulsen 1993;Paulsen and Gjessing 1997). See Paulsen (1998) and Paulsen (2008) for surveys on the topic.…”

Section: Introductionmentioning

confidence: 99%

How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability

Wang

Palmowski

Constantinescu

2021

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In this paper, we generate boundary value problems for ruin probabilities of surplus-dependent premium risk processes, under a renewal case scenario, Erlang (2) claim arrivals, and a hypoexponential claims scenario, Erlang (2) claim sizes. Applying the approximation theory of solutions of linear ordinary differential equations, we derive the asymptotics of the ruin probabilities when the initial reserve tends to infinity. When considering premiums that are linearly dependent on reserves, representing, for instance, returns on risk-free investments of the insurance capital, we firstly derive explicit solutions of the ordinary differential equations under considerations, in terms of special mathematical functions and integrals, from which we can further determine their asymptotics. This allows us to recover the ruin probabilities obtained for general premiums dependent on reserves. We compare them with the asymptotics of the equivalent ruin probabilities when the premium rate is fixed over time, to measure the gain generated by this additional mechanism of binding the premium rates with the amount of reserve owned by the insurance company.

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Asymptotic results for renewal risk models with risky investments

Cited by 26 publications

References 28 publications

Asymptotic Ruin Probabilities for a Bivariate Lévy-Driven Risk Model with Heavy-Tailed Claims and Risky Investments

Asymptotic Ruin Probabilities for a Bivariate Lévy-Driven Risk Model with Heavy-Tailed Claims and Risky Investments

On the ruin probability for nonhomogeneous claims and arbitrary inter-claim revenues

How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability

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