Browsing by Author "Okelola, Michael."

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Group analysis of equations arising in embedding theory.
(2010) Okelola, Michael.; Govinder, Keshlan Sathasiva.
Embedding theories are concerned with the embedding of a lower dimensional manifold (dim = n, say) into a higher dimensional one (usually dim = n+1, but not necessarily so). We are concerned with the particular case of embedding 4D spherically symmetric equations into 5D Einstein spaces. This scenario is of particular relevance to contemporary cosmology and astrophysics. Essentially, they are 5D vacuum field equations with initial data given on a 4D spacetime hypersurface. The equations that arise in this framework are highly nonlinear systems of ordinary differential equations and they have been particularly resistant to solution techniques over the past few years. As a matter of fact, to date, despite theoretical results for the existence of solutions for embedding classes of 4D space times, no general solutions to the local embedding equations are known. The Lie theory of extended groups applied to differential equations has proved to be very successful since its inception in the nineteenth century. More recently, it has been successfully utilized in relativity and has provided solutions where none were previously found, as well as explaining the existence of ad hoc methods. In our work, we utilize this method in an attempt to find solutions to the embedding equations. It is hoped that we can place the analysis of these equations onto a firm theoretical basis and thus provide valuable insight into embedding theories.
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.