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Register a new userHow much we gain by surplus-dependent premiums -- asymptotic analysis of ruin probability
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In this paper, we build on the techniques developed in Albrecher et al. (2013), to generate initial-boundary value problems for ruin probabilities of surplus-dependent premium risk processes, under a renewal case scenario, Erlang (2) claim arrivals, and an exponential claims scenario, Erlang (2) claim sizes. Applying the approximation theory of solutions of linear ordinary differential equations developed in Fedoryuk (1993), we derive the asymptotics of the ruin probabilities when the initial reserve tends to infinity. When considering premiums that are {it linearly} dependent on reserves, representing for instance returns on risk-free investments of the insurance capital, we firstly derive explicit formulas for the ruin probabilities, from which we can easily determine their asymptotics, only to match the ones 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 own by the insurance company.
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We give asymptotic analysis for probability of absorbtion $mathsf{P}(tau_0le T)$ on the interval $[0,T]$, where $ tau_0=inf{t:X_t=0}$ and $X_t$ is a nonnegative diffusion process relative to Brownian motion $B_t$, dX_t&=mu X_tdt+sigma X^gamma_tdB_t. X_0&=K>0 Diffusion parameter $sigma x^gamma$, $gammain [{1/2},1)$ is not Lipschitz continuous and assures $mathsf{P}(tau_0>T)>0$. Our main result: $$ limlimits_{Ktoinfty} frac{1}{K^{2(1-gamma)}}logmathsf{P}(tau_{0}le T) =-frac{1}{2E M^2_T}, $$ where $ M_T=int_0^Tsigma(1-gamma)e^{-(1-gamma)mu s}dB_s $. Moreover we describe the most likely path to absorbtion of the normed process $frac{X_t}{K}$ for $Ktoinfty$.
Based on a discrete version of the Pollaczeck-Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.
The study deals with the ruin problem when an insurance company having two business branches, life insurance and non-life insurance, invests its reserve into a risky asset with the price dynamics given by a geometric Brownian motion. We prove a result on smoothness of the ruin probability as a function of the initial capital and obtain for it an integro-differential equation understood in the classical sense. For the case of exponentially distributed jumps we show that the survival probability is a solution of an ordinary differential equation of the 4th order. Asymptotic analysis of the latter leads to the conclusion that the ruin probability decays to zero in the same way as in the already studied cases of models with one-side jumps.
We consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ($i=1,2,dots$) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature, that is, if the surplus process just before the $i$th arrival is at level $u$, then for $a>0$ the capital jumps up to the level $(1+a)u+C_i$. The ruin probability and the distribution of the time to ruin are determined. We furthermore identify the value of discounted cumulative dividend payments, for the case of a Poisson arrival process of proportional gains. In the dividend calculations, we also consider a random perturbation of our basic risk process modeled by an independent Brownian motion with drift.
In this paper, we are concerned with the numerical solution of one type integro-differential equation by a probability method based on the fundamental martingale of mixed Gaussian processes. As an application, we will try to simulate the estimation of ruin probability with an unknown parameter driven not by the classical Levy process but by the mixed fractional Brownian motion.
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