Present value distributions with applications to ruin theory and stochastic equations. (English) Zbl 0943.60098

Gjessing, Håkon K.; Paulsen, Jostein

Stochastic Processes Appl. 71, No. 1, 123-144 (1997).

Summary: We study the distribution of the stochastic integral \(\int ^{\infty }_0 e^{-R_t} dP_t\) where \(P\) and \(R\) are independent Lévy processes with a finite number of jumps on finite time intervals. The exact distribution is obtained in many special cases, and we derive asymptotic properties of the tails of the distributions in the general case. These results are applied to give two new examples of exact solutions of the probability of eventual ruin of an insurance portfolio where return on investments are stochastic. Finally we use the results to give new examples of exact solutions of the stochastic equations \(Z \overset \text{d} = AZ + B\) and \(Z \overset \text{d} = A(Z+C)\) where \(Z\) and \((A,B)\) (or \((A,C)\)) are independent.

MSC:

60K30	Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

present value distribution; ruin probability; stochastic equation; integro-differential equation; characteristic function; Laplace transform

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