After implementation of Solvency II, insurance companies can use internal risk models. In this paper, we show how to calculate finite-horizon ruin probabilities and prove for them new upper and lower bounds in a risk-switching Sparre Andersen model. Due to its flexibility, the model can be helpful for calculating some regulatory capital requirements. The model generalizes several discrete time- as well as continuous time risk models. A Markov chain is used as a ‘switch’ changing the amount and/or respective wait time distributions of claims while the insurer can adapt the premiums in response. The envelopes of generalized moment generating functions are applied to bound insurer’s ruin probabilities.
In this paper, we investigate deficit distributions at ruin in
a regime-switching Sparre Andersen model. A Markov chain is
assumed to switch the amount and/or respective wait time distributions of
claims while the insurer can adjust the premiums in response.
Special attention is paid to an operator
{\mathbf{L}}
generated by the risk process.
We show that the deficit distributions at ruin during n periods, given the state of the Markov chain at time zero,
form a vector of functions, which is the n-th iteration of
{\mathbf{L}}
on the vector of functions
being identically equal to zero. Moreover, in the case of infinite
horizon, the deficit distributions at ruin are shown to be a fixed point of
{\mathbf{L}}
.
Upper bounds for the vector of deficit distributions at ruin are also proven.
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