Browsing by Author "Mgobhozi, Sivuyile Wiseman."

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Combined impulse control and optimal stopping in insurance and interest rate theory.
(2015) Mgobhozi, Sivuyile Wiseman.; Chikodza, Eriyoti.; Mukwembi, Simon.
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.
Completion of an incomplete market by quadratic variation assets.
(2011) Mgobhozi, Sivuyile Wiseman.; Mataramvura, Sure.
It is well known that the general geometric L´evy market models are incomplete, except for the geometric Brownian and the geometric Poissonian, but such a market can be completed by enlarging it with power-jump assets as Corcuera and Nualart [12] did on their paper. With the knowledge that an incomplete market due to jumps can be completed, we look at other cases of incompleteness. We will consider incompleteness due to more sources of randomness than tradable assets, transactions costs and stochastic volatility. We will show that such markets are incomplete and propose a way to complete them. By doing this we show that such markets can be completed. In the case of incompleteness due to more randomness than tradable assets, we will enlarge the market using the market’s underlying quadratic variation assets. By doing this we show that the market can be completed. Looking at a market paying transactional costs, which is also an incomplete market model due to indifference between the buyers and sellers price, we will show that a market paying transactional costs as the one given by, Cvitanic and Karatzas [13] can be completed. Empirical findings have shown that the Black and Scholes assumption of constant volatility is inaccurate (see Tompkins [40] for empirical evidence). Volatility is in some sense stochastic, and is divided into two broad classes. The first class being single-factor models, which have only one source of randomness, and are complete markets models. The other class being the multi-factor models in which other random elements are introduced, hence are an incomplete markets models. In this project we look at some commonly used multi-factor models and attempt to complete one of them.