Pure Mathematicshttp://hdl.handle.net/10413/67702018-10-12T00:05:57Z2018-10-12T00:05:57ZCombined impulse control and optimal stopping in insurance and interest rate theory.http://hdl.handle.net/10413/155462018-10-03T01:01:01Z2015-01-01T00:00:00ZCombined impulse control and optimal stopping in insurance and interest rate theory.
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.
Doctor of Philosophy in Financial Mathematics. University of KwaZulu-Natal, Durban 2015.
2015-01-01T00:00:00ZSome amenability properties on segal algebras.http://hdl.handle.net/10413/155332018-10-03T01:02:34Z2017-01-01T00:00:00ZSome amenability properties on segal algebras.
It has been realized that the definition of amenability given by B. E. Johnson in
his Classical Memoir of American Mathematical Society in 1972 is too restrictive
and does not allow for the development of a rich general theory. For this reason,
by relaxing some of the constraints in the definition of amenability via restricting
the class of bimodules in question or by relaxing the structure of the derivations,
various notions of amenability have been introduced after the pioneering work
of Johnson on amenability in Banach algebras. This dissertation is focused on
six of these notions of amenability in Banach algebras, namely: contractibility,
amenability, weak amenability, generalized amenability, character amenability and
character contractibility. The first five of these notions are studied on arbitrary
Banach algebras and the last two are studied on some classes of Segal algebras.
In particular, results on hereditary properties and several characterizations of
these notions are reviewed and discussed. Indeed, we discussed the equivalent
of these notions with the existence of a bounded approximate diagonal, virtual
diagonal, splitting of exact sequences of Banach bimodules and the existence of a
certain Hahn-Banach extension property. Also, some relations that exist between
these notions of amenability are also established. We show that approximate contractibility
and approximate amenability are equivalent. Some conditions under
which the amenability of the underlying group of a Segal algebra implies the character
amenability of the Segal algebras are also given. Finally, some new results
are obtained which serves as our contribution to knowledge.
Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2017.
2017-01-01T00:00:00ZOn pseudo-amenability of C(X;A) for norm irregular banach algebra A.http://hdl.handle.net/10413/153092018-06-15T01:00:23Z2017-01-01T00:00:00ZOn pseudo-amenability of C(X;A) for norm irregular banach algebra A.
Abstract available in PDF file.
Master of Science in Mathematics. University of KwaZulu-Natal, Durban 2017
2017-01-01T00:00:00ZSome notions of amenability of Banach semigroup algebras.http://hdl.handle.net/10413/153022018-06-15T01:01:43Z2017-01-01T00:00:00ZSome notions of amenability of Banach semigroup algebras.
Abstract available in PDF file.
Master of Science in Mathematics. University of KwaZulu-Natal, Durban 2017.
2017-01-01T00:00:00Z