How to measure and manage insurance
and finance risks is a practically important and
theoretically interesting problem. This project
studies some actuarial and finance models with
dependent risks. Our main focus is on a
popular risk measure in actuarial science: ruin
probability. Ruin probability refers to the
probability of a company going to bankruptcy. It is
well-known that when the model includes dependent
structure, the problem becomes very difficult.
We modeled the premium and claims
using models which depend on historical information.
Markov processes and time series have been used to
model the dependent structure. In order to use
information of both claim and premium series to
predict the future value of one series, we proposed
to model this by using Granger's causal model. That
model includes the possibility of having causality
between premium and claim processes in Granger's
sense. The method we used is martingale which is a
very powerful mathematical tool. Another class of
dependent insurance risk models we have investigated
is the threshold insurance risk model. We assume
that the claim size of an insurance business
depends on the claim time and the solution is to
solve the delayed integro-differential equations
satisfied by the ruin probability.
By using martingale arguments, we
obtained the upper bounds for the ruin probability.
The upper bound for the ruin probability can serve
as a conservative estimation
of the ruin probability. By
solving the delayed integro-differential equations
satisfied by the ruin probability, Lundberg type
upper bound for the ruin probability has been
obtained. We have also inves-tigated the insurance
risk models with investment income; both the ruin
probability and the absolute ruin probability have
been studied. By solving certain type of
integro-differential equations, we have obtained
closed form solutions for the ruin probability and
the absolute ruin probability in the case of claim
size following certain class of distributions.
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