Often Erlang (2) Interest Rate Risk Model

This dissertion mainly study the Erlang(2) risk model with constant interest force, we consider some important distributions and rusults: the non-ruin probability, the surplus immediately before ruin, the deficit at ruin, the joint distribution of the surplus immediately before ruin and the deficit at ruin, the expected discounted penalty at ruin and so on.In chapter one, we introduce the Erlang(2) risk process with constant interest force and give the definition of the probability of ruin, the surplus immediately before ruin, the deficit at ruin, the joint distribution of the surplus immediately before ruin and the deficit at ruin, the expected discounted penalty at ruin respectively.In chapter two, we consider the non-ruin probability . In section one. by adapting the techniques in [5], we obtain the integral expression of non-ruin probabilityIn section two, firstly, we prove the twice continuous differentiability of non-ruin probability, then we obtain the integral-differential equation satisfied by (u)In section three, we introduce the auxiliary function E (u).As u = 0, gives0. By taking L-S transform, we get the differential equation satisfied by the L-Stransform of E (u)We also consider the special case in which 6 = 0 is assumed, and get an explicit expression of L-S transformIn chapter three, we mainly consider the distribution of the surplus immediately before ruin. In section one, we treat the cases u x and u > x separately and obtain the integral expression of F (u,x)In section two, we prove the the twice continuous differentiability of F (u,x) and get its integral-differential equationIn the case of F(x) = 1 - e , x > 0, we obtain an useful differential equationIn section three, by taking L-S transform, the differential equation satisfied by L-S transform is derivedFinally, we consider the case in which 5 = 0 is assumed and get an useful resultIn chapter four and chapter five, we discuss the distribution of the deficit at ruin and the joint distribution of the surplus immediately before ruin and the deficit at ruin. Similarly, we obtain their integral expressions, integral-difference equations and the difference equations satisfied by L-S transformations of their auxiliary functions. Finally, we get explicit expressions of L-S transform when = 0In chapter six, we discuss the expected discounted penalty at ruin (u). In section one, the integral expression is derived. As special cases of our results, we point out that integral expressions of can be obtained easily according to changs of (x,y). In section two, we prove the twice-continuous differentiability of (u) and obtain its integral-differential equation. When = 0, we point out that some results in [6] may be regard as our special cases. In section three, we introduce auxiliary function and take L-S transform, then we get the differential equation satisfied by L-S transform.