RUIN PROBABILITIES FOR A MULTIDIMENSIONAL RISK MODEL WITH NON-STATIONARY ARRIVALS AND SUBEXPONENTIAL CLAIMS

Consider a multidimensional risk model, in which an insurer simultaneously confronts m (m ≥ 2) types of claims sharing a common non-stationary and non-renewal arrival process. Assuming that the claims arrival process satisfies a large deviation principle and the claim-size distributions are heavy-tailed, asymptotic estimates for two common types of ruin probabilities for this multidimensional risk model are obtained. As applications, we give two examples of the non-stationary point process: a Hawkes process and a Cox process with shot noise intensity, and asymptotic ruin probabilities are obtained for these two examples.

Optimal Excess-of-Loss Reinsurance for Stochastic Factor Risk Models

We study the optimal excess-of-loss reinsurance problem when both the intensity of the claims arrival process and the claim size distribution are influenced by an exogenous stochastic factor. We assume that the insurer’s surplus is governed by a marked point process with dual-predictable projection affected by an environmental factor and that the insurance company can borrow and invest money at a constant real-valued risk-free interest rate r. Our model allows for stochastic risk premia, which take into account risk fluctuations. Using stochastic control theory based on the Hamilton-Jacobi-Bellman equation, we analyze the optimal reinsurance strategy under the criterion of maximizing the expected exponential utility of the terminal wealth. A verification theorem for the value function in terms of classical solutions of a backward partial differential equation is provided. Finally, some numerical results are discussed.

The time to ruin for a class of Markov additive risk process with two-sided jumps

We consider risk processes that locally behave like Brownian motion with some drift and variance, these both depending on an underlying Markov chain that is also used to generate the claims arrival process. Thus, claims arrive according to a renewal process with waiting times of phase type. Positive claims (downward jumps) are always possible but negative claims (upward jumps) are also allowed. The claims are assumed to form an independent, identically distributed sequence, independent of everything else. As main results, the joint Laplace transform of the time to ruin and the undershoot at ruin, as well as the probability of ruin, are explicitly determined under the assumption that the Laplace transform of the positive claims is a rational function. Both the joint Laplace transform and the ruin probability are decomposed according to the type of ruin: ruin by jump or ruin by continuity. The methods used involve finding certain martingales by first finding partial eigenfunctions for the generator of the Markov process composed of the risk process and the underlying Markov chain. We also use certain results from complex function theory as important tools.

The time to ruin for a class of Markov additive risk process with two-sided jumps

We consider risk processes that locally behave like Brownian motion with some drift and variance, these both depending on an underlying Markov chain that is also used to generate the claims arrival process. Thus, claims arrive according to a renewal process with waiting times of phase type. Positive claims (downward jumps) are always possible but negative claims (upward jumps) are also allowed. The claims are assumed to form an independent, identically distributed sequence, independent of everything else. As main results, the joint Laplace transform of the time to ruin and the undershoot at ruin, as well as the probability of ruin, are explicitly determined under the assumption that the Laplace transform of the positive claims is a rational function. Both the joint Laplace transform and the ruin probability are decomposed according to the type of ruin: ruin by jump or ruin by continuity. The methods used involve finding certain martingales by first finding partial eigenfunctions for the generator of the Markov process composed of the risk process and the underlying Markov chain. We also use certain results from complex function theory as important tools.