http://hdl.handle.net/10386/65 Sat, 18 Apr 2026 14:07:23 GMT 2026-04-18T14:07:23Z http://hdl.handle.net/10386/3432 Malliavin calculus and its applications to mathematical finance Kgomo, Shadrack Makwena In this study,we consider two problems.The first one is the problem of computing hedging portfolios for options that may have discontinuous payoff functions.For this problem we use the Malliavin property called the Clark-Ocone formula and give some examples for diferent types of pay of functions of the options of European type.The second problem is based on the computation of price sensitivities (derivatives of the probabilistic representation of the pay off functions with respect to the underlying parameters of the model) also known as`Greeks' of discontinuous payoff functions and also give some examples.We restrict ourselves to the computation of Delta, Gamma and Vega.For this problem we make use of the properties of the Malliavin calculus like the integration by parts formula and the chain rule.We find the representations of the price sensitivities in terms of the expected value of the random variables that do not involve the actual direct differentiation of the payout function,that is, E[g(XT ) ] where g is a payoff function which depend on the stochasticdic differential equation XT at maturity time T and is the Malliavin weigh tfunction. In general, we study the regularity of the solutions of the stochastic differentia lequations in the sense of Malliavin calculus and explore its applications to Mathematical finance. Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2020 Wed, 01 Jan 2020 00:00:00 GMT http://hdl.handle.net/10386/3432 2020-01-01T00:00:00Z http://hdl.handle.net/10386/3421 Finite element solution of the reaction-diffusion equation Mahlakwana, Richard Kagisho In this study we present the numerical solution o fboundary value problems for the reaction-diffusion equations in 1-d and 2-d that model phenomena such as kinetics and population dynamics.These differential equations are solved nu- merically using the finite element method (FEM).The FEM was chosen due to several desirable properties it possesses and the many advantages it has over other numerical methods.Some of its advantages include its ability to handle complex geometries very well and that it is built on well established Mathemat- ical theory,and that this method solves a wider class of problems than most numerical methods.The Lax-Milgram lemma will be used to prove the existence and uniqueness of the finite element solutions.These solutions are compared with the exact solutions,whenever they exist,in order to examine the accuracy of this method.The adaptive finite element method will be used as a tool for validating the accuracy of theFEM.The convergence of the FEM will be proven only on the real line. Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2020 Wed, 01 Jan 2020 00:00:00 GMT http://hdl.handle.net/10386/3421 2020-01-01T00:00:00Z http://hdl.handle.net/10386/3410 Pathwise functional lto calculus and its applications to the mathematical finance Nkosi, Siboniso Confrence Functional Itˆo calculus is based on an extension of the classical Itˆo calculus to functionals depending on the entire past evolution of the underlying paths and not only on its current value. The calculus builds on F¨ollmer’s deterministic proof of the Itˆo formula Föllmer (1981) and a notion of pathwise functional derivative recently proposed by Dupire (2019). There are no smoothness assumptions required on the functionals, however, they are required to possess certain directional derivatives which may be computed pathwise, see Cont and Fournié (2013); Schied and Voloshchenko (2016a); Cont (2012). In this project we revise the functional Itô calculus together with the notion of quadratic variation. We compute the pathwise change of variable formula utilizing the functional Itô calculus and the quadratic variation notion. We study the martingale representation for the case of weak derivatives, we allow the vertical operator, rX, to operate on continuous functionals on the space of square-integrable Ft-martingales with zero initial value. We approximate the hedging strategy, H, for the case of path-dependent functionals, with Lipschitz continuous coefficients. We study some hedging strategies on the class of discounted market models satisfying the quadratic variation and the non-degeneracy properties. In the classical case of the Black-Scholes, Greeks are an important part of risk-management so we compute Greeks of the price given by path-dependent functionals. Lastly we show that they relate to the classical case in the form of examples. Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2019 Tue, 01 Jan 2019 00:00:00 GMT http://hdl.handle.net/10386/3410 2019-01-01T00:00:00Z http://hdl.handle.net/10386/3354 Properties and calculus on price paths in the model-free approach to the mathematical finance Galane, Lesiba Charles Vovk and Shafer, [41], introduced game-theoretic framework for probability in mathematical finance. This is a new trend in financial mathematics in which no probabilistic assumptions on the space of price paths are made. The only assumption considered is the no-arbitrage opportunity widely accepted by the financial mathematics community. This approach rests on game theory rather than measure theory. We deal with various properties and constructions of quadratic variation for model-free càdlàg price paths and integrals driven by such paths. Quadratic variation plays an important role in the analysis of price paths of financial securities which are modelled by Brownian motion and it is sometimes used as the measure of volatility (i.e. risk). This work considers mainly càdlàg price paths rather than just continuous paths. It turns out that this is a natural settings for processes with jumps. We prove the existence of partition independent quadratic variation. In addition, following assumptions as in Revuz and Yor’s book, the existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by model-free price paths is proven. Thesis (Ph.D. (Applied Mathematics)) -- University of Limpopo, 2021 Fri, 01 Jan 2021 00:00:00 GMT http://hdl.handle.net/10386/3354 2021-01-01T00:00:00Z