Browsing by Author "Gideon, Frednard"

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    A Classification of Fuzzy Subgroups of Finite Abelian Groups
    (University of Namibia, 2013) Gideon, Frednard
    The knowledge of fuzzy sets and systems has become a considerable aspect to apply in various mathematical systems. In this paper, we apply a knowledge of fuzzy sets to group structures. We consider a fuzzy subgroups of finite abelian groups, denoted by G = Zpn +Zqm , where Z is an integer, p and q are distinct primes and m;n are natural numbers. The fuzzy subgroups are classified using the notion of equivalence classes. In essence the equivalence relations of fuzzy subsets X is extended to equivalence relations of fuzzy subgroups of a group G. We then use the notion of flags and keychains as tools to enumerate fuzzy subgroups of G. In this way, we characterized the properties of the fuzzy subgroups of G. Finally, we use maximal chains to construct a fuzzy subgroups-lattice diagram for these groups of G.
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    An efficient numerical method for pricing double-barrieroptions on an underlying stock governed by a fractal stochastic process
    (MDPI, 2023) Nuugulu, Samuel Megameno; Gideon, Frednard; Patidar, Kailash C.
    After the discovery of the fractal structures of financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes to solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a numerical scheme for pricing double-barrier options, written on an underlying stock whose dynamics are governed by a non-standard fractal stochastic process. The resultant model is time-fractional and is herein referred to as a time-fractional Black-Scholes model. The presence of the time-fractional derivative helps to capture the time-decaying effects of the underlying stock while capturing the globalized change in underlying prices and barriers. In this paper, we present the construction of the proposed scheme, analyse it in terms of its stability and convergence, and present two numerical examples of pricing double knock-in barrier-option problems. The results suggest that the proposed scheme is unconditionally stable and convergent with order O(h2 + k2).