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Register a new userAsymptotic analysis of ruin in CEV model
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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 $Ktoinfty$.
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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.
The CEV model is given by the stochastic differential equation $X_t=X_0+int_0^tmu X_sds+int_0^tsigma (X^+_s)^pdW_s$, $frac{1}{2}le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive probability. We show the weak convergence of Euler-Maruyama approximations $X_t^n$ to the process $X_t$, $0le tle T$, in the Skorokhod metric. We give a new approximation by continuous processes which allows to relax some technical conditions in the proof of weak convergence in cite{HZa} done in terms of discrete time martingale problem. We calculate ruin probabilities as an example of such approximation. We establish that the ruin probability evaluated by simulations is not guaranteed to converge to the theoretical one, because the point zero is a discontinuity point of the limiting distribution. To establish such convergence we use the Levy metric, and also confirm the convergence numerically. Although the result is given for the specific model, our method works in a more general case of non-Lipschitz diffusion with absorbtion.
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.
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.
We present formulae for the moments of the ruin time in a Levy risk model. From these we derive the asymptotic behaviour of the moments of the ruin time, as the initial capital tends to infinity. In the perturbed Cramer-Lundberg model with phase-type distributed claims, we explicitely compute the first two moments of the ruin time in terms of roots and derivatives of the corresponding Laplace exponent. In the special case of exponential claims we provide explicit formulae for the first two moments of the ruin time in terms of the model parameters. All our considerations distinguish between the profitable and the unprofitable setting.
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