Browsing by Author "Ebobisse Bille, Francois"

Now showing 1 - 4 of 4
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    Mean-variance hedging in an illiquid market
    (2015) Mavuso, Melusi Manqoba; Ebobisse Bille, Francois
    Consider a market consisting of two correlated assets: one liquidly traded asset and one illiquid asset that can only be traded at time 0. For a European derivative written on the illiquid asset, we find a hedging strategy consisting of a constant (time 0) holding in the illiquid asset and dynamic trading strategies in the liquid asset and a riskless bank account that minimizes the expected square replication error at maturity. This mean-variance optimal strategy is first found when the liquidly traded asset is a local martingale under the real world probability measure through an application of the Kunita-Watanabe projection onto the space of attainable claims. The result is then extended to the case where the liquidly traded asset is a continuous square integrable semimartingale, and we again use the Kunita-Watanabe decomposition, now under the variance optimal martingale measure, to find the mean-variance optimal strategy in feedback form. In an example, we consider the case where the two assets are driven by correlated Brownian motions and the derivative is a call option on the illiquid asset. We use this example to compare the terminal hedging profit and loss of the optimal strategy to a corresponding strategy that does not use the static hedge in the illiquid asset and conclude that the use of the static hedge reduces the expected square replication error significantly (by up to 90% in some cases). We also give closed form expressions for the expected square replication error in terms of integrals of well-known special functions.
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    Monotone and pseudomonotone operators with applications to variational problems
    (2015) Alexander, Byron Joseph; Ebobisse Bille, Francois
    This work is primarily concerned with investigating how monotone and pseudomonotone operators between Banach spaces are used to prove the existence of solutions to nonlinear elliptic boundary value problems. A well-known approach to solving nonlinear elliptic boundary value problems is to reformulate them as equations of the form A (u) = f, where A is a monotone or pseudomonotone operator from a Sobolev space to its dual. We seek to study the abstract theory which underpins this approach and proves the existence of a solution to the equation A (u) = f, implying the existence of a weak solution to the elliptic boundary value problem. Further, we examine properties of monotone and pseudomonotone operators, with an emphasis on a characterization, which involves the latter, and establishes a connection between the operator and the principal part of a partial differential equation. In addition, results relating monotone and pseudomonotone operators with variational inequalities are explored.
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    The required ansatz to construct Lie point transformations and the symmetries of a first-order stochastic differential equation
    (2011) Masike, Kakanyo Knowledge; Fredericks, Ebrahim; Ebobisse Bille, Francois
    In this thesis we demonstrate how to obtain the required ansatz to determine Lie point transformations of evolution-type equations from the contact transformation approach. We indicate that the Lie point transformations of the Fokker-Planck equation (FPE), which is a second-order linear parabolic partial differential equation (PDE), are projectable by using the ansatz. We further obtain the symmetries of a stochastic ordinary differential equation (SODE) which corresponds to those of the FPE. This is possible because there exists a relationship between an SODE and the associated (deterministic) FPE. The study of SODEs is an interesting and applicable concept in the real world and one of the building factors to this study is an Ito integral. These Ito integrals are of much use, for instance, in the field of mathematical finance whereby its use has shown the relationship between call options and their non-deterministic underlying stock prices. Wiener processes must be considered in finding an approximation of these integrals. Acclimatization of Sophus Lie's work to SODEs has been done by (Gaeta and Quintero [2]; Wafo Soh and Mahomed [41]; Unal [42]; Fredericks and Mahomed [43]). The determining equations for the first-order SODEs are derived in an Ito calculus context and are non-stochastic. Consequently, symmetries of an SODE are obtained without the consultation of its corresponding FPE.
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    Semilinear elliptic partial differential equations with the critical Sobolev exponent
    (2017) Mavuso, Melusi Manqoba; Ebobisse Bille, Francois
    We present how variational methods and results from linear and non-linear functional analysis are applied to solving certain types of semilinear elliptic partial differential equations (PDEs). The ultimate goal is to prove results on the existence and non-existence of solutions to the Semilinear Elliptic PDEs with the Critical Sobolev Exponent. To this end, we first recall some useful results from functional analysis, including the Sobolev spaces, which provide a natural setting for the idea of weak or generalised solutions. We then present linear PDE theory, including eigenvalues of the Dirichlet Laplacian operator. We discuss the Direct Methods of Calculus of Variations and Critical Point Theory, together with examples of how these techniques are applied to solving PDEs. We show how the existence of solutions to semilinear elliptic equations depends on the exponent of the growth of the non-linear term. This then naturally leads to the discussion of the critical Sobolev exponent, where we present both positive and negative results.