We derive formulas for the moments of ruin time in a L\'evy risk model and use these to determine asymptotic behavior as initial capital tends infinity. In special case perturbed Cram\'er-Lundberg with phase-type or exponentially distributed claims, we explicitly compute first two time. All our considerations distinguish between profitable unprofitable setting.

For general risk processes, we introduce and study the expected time-integrated negative part of the process on a fixed time interval. Differentiation theorems are stated and proved. They make it possible to derive the expected value of this risk measure, and to link it with the average total time below 0, studied by Dos Reis, and the probability of ruin. We carry out differentiation of other f...

We study the continuous-time portfolio optimization problem of an insurer. The wealth of the insurer is given by a classical risk process plus gains from trading in a risky asset, modelled by a geometric Brownian motion. The insurer is not only interested in maximizing the expected utility of wealth but is also concerned about the ruin probability. We thus investigate the problem of optimizing ...

The ruin probability of the reserve of an insurance company, in finite and infinite horizon, when there is the possibility to invest in a risky asset, has recently received a lot of attention. It is well known that for the classical Cramér-Lundberg process (where there is no investment and the claims have exponential moments), the ruin probability decreases exponentially with respect to the ini...

Abstract: We consider three closely related problems in optimal control: (1) minimizing the probability of lifetime ruin when the rate of consumption is stochastic and when the individual can invest in a Black-Scholes financial market; (2) minimizing the probability of lifetime ruin when the rate of consumption is constant but the individual can invest in two risky correlated assets; and (3) a ...

We establish a functional large deviation principle and a functional moderate deviation principle for Markov-modulated risk models with reinsurance by constructing an exponential martingale approach. Lundberg’s estimate of the ruin time is also presented.