Combined impulse control and optimal stopping in insurance and interest rate theory.
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Date
2015
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Abstract
In this thesis, we consider the problem of portfolio optimization for an
insurance company with transactional costs. Our aim is to examine the
interplay between insurance and interest rate. We consider a corporation,
such as an insurance firm, which pays dividends to shareholders.
We assume that at any time t the financial reserves of the insurance company
evolve according to a generalized stochastic differential equation. We
also consider that these liquid assets of the firm earn interest at a constant
rate. We consider that when dividends are paid out, transaction costs are
incurred. Due to the presence of transactions costs in the proposed model,
the mathematical problem becomes a combined impulse and stochastic control
problem.
This thesis is an extension of the work by Zhang and Song [69]. Their paper
considered dividend control for a financial corporation that also takes
reinsurance to reduce risk with surplus earning interest at the constant
force p > 0.
We will extend their model by incorporating jump diffusions into the market
with dividend payout and reinsurance policies. Jump-diffusion models,
as compared to their diffusion counterpart, are a more realistic mathematical
representation of real-life processes in finance.
The extension of Zhang and Song [69] model to the jump case will require
us to reduce the analytical part of the problem to Hamilton-Jacobi-Bellman
Qausi-Variation Inequalities for combined impulse control in the presence
of jump diffusion. This will assist us to find the optimal strategy for the
proposed jump diffusion model while keeping the financial corporation in
the solvency region. We will then compare our results in the jump-diffusion
case to those obtained by Zhang and Song [69] in the no jump case.
We will then consider models with stochastic volatility and uncertainty as
a means of extending the current theory of modeling insurance reserves.
Description
Doctor of Philosophy in Financial Mathematics. University of KwaZulu-Natal, Durban 2015.
Keywords
Theses--Applied Mathematics.