School Seminars and Colloquia

Exit and dividend problems of a two-dimensional risk process

Stochastic Processes and Financial Mathematics

by Zbigniew Palmowski

Institution: Mathematical Institute, University of Wroclaw
Date: Wed 15th February 2012
Time: 2:15 PM
Location: Richard Berry Building, room 215

Abstract: In this talk we will consider two insurance companies (or two branches of the same
company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according
to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process we obtain a closed form expression for the ultimate ruin probability. We focus on the case of exponential claims. In the general case we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cram\'er light-tail assumption on the claim size distribution.
Finally, we will also analyze the dividend problems for this two-dimensional risk process.
This talk is based on [1-4].

[1] Avram, F., Palmowski, Z. and Pistorius, M. (2008) Exit problem of a two-dimensional risk process from a cone: exact and asymptotic results. Annals of Applied Probability 18(6), 2421-2449.

[2] Avram, F., Palmowski, Z. and Pistorius, M. (2008) A two-dimensional ruin problem on the positive quadrant. Insurance: Mathematics and Economics} 42(1), 227-234.

[3] Palmowski, Z. and Pistorius, M. (2009) Cram\'{e}r asymptotics for finite time first passage probabilities for general L\'{e}vy processes. Statistics and Probability Letters 79(16), 1752-1758.

[4] Czarna, I. and Palmowski (2011) De Finetti's dividend problem and impulse control for a two-dimensional insurance risk process. Stochastic Models 27, 1-31.