Uncertain process is a sequence of uncertain variables indexed by time. This paper presents a series of extreme value theorem of uncertain independent increment process and provides uncertainty distribution of first hitting time. This paper also proposes an insurance risk model with uncertain claims. Finally, a concept of ruin index is defined and a ruin index formula is given.

We consider a class of Markovian risk models in which the insurer collects premiums at rate c1 (c2) whenever the surplus level is below (above) a constant barrier level b. We derive the Laplace­Stieltjes transform (LST) of the distribution of the time to ruin as well as the LST (with respect to time) of the joint distribution of the time to ruin, the surplus prior to ruin, and the deficit at ru...

In the Cramér-Lundberg model and its di usion approximation, it is a classical problem to nd the optimal dividend payment strategy that maximizes the expected value of the discounted dividend payments until ruin. One often raised disadvantage of this approach is the fact that such a strategy does not take the life time of the controlled process into account. In this paper we introduce a value f...

Assume that the surplus of an insurer follows a compound Poisson surplus process. When the surplus is below zero or the insurer is on deficit, the insurer could borrow money at a debit interest rate to pay claims. Meanwhile, the insurer will repay debts from her premium income. The negative surplus may return to a positive level if debts are reasonable. However, when the negative surplus is bel...

We consider the finite time horizon dividend-ruin model where the firm pays out dividends to its shareholders according to a dividend-barrier strategy and becomes ruined when the firm asset value falls below the default threshold. The asset value process is modeled as a restricted Geometric Brownian process with an upper reflecting (dividend) barrier and a lower absorbing (ruin) barrier. Analyt...

It is well-known in ruin theory that the expected present value of penalty at ruin satisfies a defective renewal equation in the Erlang-n renewal risk model. This paper presents a new matrix operator approach to derive a parallel defective renewal equation for the expected present value of total operating costs in a phase-type renewal risk model and hence provides explicit matrix analytic solut...

Consider an insurer who is allowed to make risk-free and risky investments. The price process of the investment portfolio is described as a geometric Lévy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claim-size distribution is of Pareto type, we obtain a simple asymptotic formula which holds uniformly for all time horizons. The same...

We compute ruin probabilities, in both infinite-time and finite-time, for a Gambler’s Ruin problem with both catastrophes and windfalls in addition to the customary win/loss probabilities. For constant transition probabilities, the infinite-time ruin probabilities are derived using difference equations. Finite-time ruin probabilities of a system having constant win/loss probabilities and variab...

Recently, Chen (2011) studied the finite-time ruin probability in a discrete-time risk model in which the insurance and financial risks form a sequence of independent and identically distributed random pairs with common bivariate Farlie–Gumbel–Morgenstern (FGM) distribution. The parameter θ of the FGMdistribution governs the strength of dependence, with a smaller value of θ corresponding to a l...