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Doctoral Degrees (Pure Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10413/7120

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Browsing Doctoral Degrees (Pure Mathematics) by Author "Govinder, Keshlan Sathasiva."

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    Extensions and generalisations of Lie analysis.
    (1995) Govinder, Keshlan Sathasiva.; Leach, Peter Gavin Lawrence.
    The Lie theory of extended groups applied to differential equations is arguably one of the most successful methods in the solution of differential equations. In fact, the theory unifies a number of previously unrelated methods into a single algorithm. However, as with all theories, there are instances in which it provides no useful information. Thus extensions and generalisations of the method (which classically employs only point and contact transformations) are necessary to broaden the class of equations solvable by this method. The most obvious extension is to generalised (or Lie-Backlund) symmetries. While a subset of these, called contact symmetries, were considered by Lie and Backlund they have been thought to be curiosities. We show that contact transformations have an important role to play in the solution of differential equations. In particular we linearise the Kummer-Schwarz equation (which is not linearisable via a point transformation) via a contact transformation. We also determine the full contact symmetry Lie algebra of the third order equation with maximal symmetry (y'''= 0), viz sp(4). We also undertake an investigation of nonlocal symmetries which have been shown to be the origin of so-called hidden symmetries. A new procedure for the determination of these symmetries is presented and applied to some examples. The impact of nonlocal symmetries is further demonstrated in the solution of equations devoid of point symmetries. As a result we present new classes of second order equations solvable by group theoretic means. A brief foray into Painleve analysis is undertaken and then applied to some physical examples (together with a Lie analysis thereof). The close relationship between these two areas of analysis is investigated. We conclude by noting that our view of the world of symmetry has been clouded. A more broad-minded approach to the concept of symmetry is imperative to successfully realise Sophus Lie's dream of a single unified theory to solve differential equations.
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    Lie group analysis of exotic options.
    (2013) Okelola, Michael.; Govinder, Keshlan Sathasiva.; O'Hara, John Gerard.
    Exotic options are derivatives which have features that makes them more complex than vanilla traded products. Thus, finding their fair value is not always an easy task. We look at a particular example of the exotic options - the power option - whose payoffs are nonlinear functions of the underlying asset price. Previous analyses of the power option have only obtained solutions using probability methods for the case of the constant stock volatility and interest rate. Using Lie symmetry analysis we obtain an optimal system of the Lie point symmetries of the power option PDE and demonstrate an algorithmic method for finding solutions to the equation. In addition, we find a new analytical solution to the asymmetric type of the power option. We also focus on the more practical and realistic case of time dependent parameters: volatility and interest rate. Utilizing Lie symmetries, we are able to provide a new exact solution for the terminal pay off case. We also consider the power parameter of the option as a time dependent factor. A new solution is once again obtained for this scenario.