In this paper, we discuss the Cramér-Lundberg model with investments, where the price of the invested risk asset follows a geometric Brownian motion with drift $a$ and volatility $σ> 0.$ By assuming there is a cap on the claim sizes, we prove that the probability of ruin has at least an algebraic decay rate if $2a/σ^2 > 1$. More importantly, without this assumption, we show that the probability of ruin is certain for all initial capital $u$, if $2a/σ^2 \le 1$.
Exact solution of the ruin problem in the Cramér--Lundberg model with proportional investment
HSE University（高等经济大学）；Lomonosov Moscow State University（莫斯科国立大学）；Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences（俄罗斯科学院计算机科学与控制联邦研究中心）
Existence of a classical solution to the integro-differential equation arising in the Cramér--Lundberg non-life insurance model with proportional investment
This paper considers a variant of the classical Cramér-Lundberg model that is particularly appropriate in the credit context, with the distinguishing feature that it corresponds to a finite number of obligors. The focus is on computing the ruin probability, i.e., the probability that the initial reserve, increased by the interest received from the obligors and decreased by the losses due to defaults, drops below zero. Besides an exact analysis (in terms of transforms) of this ruin probability, also an asymptotic analysis is performed, including an efficient importance-sampling based simulation…
The recent papers Gajek-Kucinsky(2017) and Avram-Goreac-Li-Wu(2020) investigated the control problem of optimizing dividends when limiting capital injections stopped upon bankruptcy. The first paper works under the spectrally negative Lévy model; the second works under the Cramér-Lundberg model with exponential jumps, where the results are considerably more explicit. The current paper has three purposes. First, it illustrates the fact that quite reasonable approximations of the general problem may be obtained using the particular exponential case studied in Avram-Goreac-Li-Wu(2020). Secondly, …
The classical Cramér-Lundberg risk process models the ruin probability of an insurance company experiencing an incoming cash flow - the premium income, and an outgoing cash flow - the claims. From a system's viewpoint, the web of insurance agents and risk objects can be represented by a bipartite network. In such a bipartite network setting, it has been shown that joint ruin of a group of agents may be avoided even if individual agents would experience ruin in the classical Cramér-Lundberg model. This paper describes and examines a phase transition phenomenon for these ruin probabilities.
We study the asymptotic behavior of ruin probabilities, as the initial reserve goes to infinity, for a reserve process model where claims arrive according to a renewal process, while between the claim times the process has the dynamics of geometric Brownian motion-type Itô processes with time-dependent random coefficients. These coefficients are ``reset" after each claim time, switching to new values independent of the past history of the process. We use the implicit renewal theory to obtain power-function bounds for the eventual ruin probability. In the special case when the random drift and …
We consider a risk model with a counting process whose intensity is a Markovian shot-noise process, to resolve one of the disadvantages of the Cramér-Lundberg model, namely the constant jump intensity of the Poisson process. Due to this structure, we can apply the theory of PDMPs on a multivariate process containing the intensity and the reserve process, which allows us to identify a family of martingales. Eventually, we use change of measure techniques to derive an upper bound for the ruin probability in this model. Exploiting a recurrent structure of the shot-noise process, even the asymptot…
We consider the classical Cramér-Lundberg risk model with claim sizes that are mixtures of phase-type and subexponential variables. Exploiting a specific geometric compound representation, we propose control variate techniques to efficiently simulate the ruin probability in this situation. The resulting estimators perform well for both small and large initial capital. We quantify the variance reduction as well as the efficiency gain of our method over another fast standard technique based on the classical Pollaczek-Khinchine formula. We provide a numerical example to illustrate the performance…
