# The beta product distribution with complex parametersby

*Stochastic Processes Seminar*

Joint Seminar Series on Stochastic Processes and Financial Mathematics

*by Daniel Dufresne*

*Institution:*Centre for Actuarial Studies, Department of Economics, Uni of Melb

*Date: Thu 15th November 2007*

*Time: 1:15 PM*

*Location: Room 213, Richard Berry Buiding, The University of Melbourne*

*Abstract*: The talk will describe how the distribution of the product of two independent

beta variables, which has 2+2=4 parameters, actually extends to cases where some of those parameters are complex or negative. This has apparently not been noticed previously. An actuarial application is given:

in the classical risk theoretic model with Poisson claim arrivals,

consider the discounted value of claims nos. 3, 6,..., each claim having

an exponential distribution; it turns out that this discounted value has

the same distribution as the product of two independent variables, one a

complex-parameter beta product and the other a gamma. (To risk theory

specialists, considering every third claim in a

Poisson arrival process is the same as assuming waiting times are Erlang(3) in a Sparre-Andersen model.)

*For More Information:* Prof Daniel Dufresne dufresne@unimelb.edu.au or Dr Aihua Xia A.Xia@ms.unimelb.edu.au