Μελέτη της ολοκληρώσιμης προεξοφλημένης συνάρτησης ποινής στη θεωρία χρεοκοπίας
The operation of an insurance company and the decisions taken are largely determined by the reserve that has the given time. One of the main problems in the ruin theory is the calculation of the probability of ruin, i.e. the probability of the possibility the reserve to be some time negative. Apart from the time of ruin T, two significant quantities are the random variables that describe the surlpus before the ruin occurs U(T—) and the deficit at the time of ruin |U(T)|. Gerber-Shiu (1998) introduced a function called Gerber-Shiu expected discounted penalty function, also known as Gerber-Shiu function. Specifically it is defined as the following expectation ms(u) = E[β-δτ • w(U(T—), |U(T)|) • I(T < to)|U(0) = u] , where : • u : the initial surplus. • w(x, y) : a penalty function. • δ > 0 : a dicount function. • I(T < TO) : an indicator function, equal to 1 when ruin occurs and equal to 0 otherwise. In this dissertation we introduce the integrated mδ (u) and we study specific cases for various choices of δ and w(x,y). Furthermore, we consider the distribution of the deficit and the integrated form.