M.Sc. / The innovative work of Black and Scholes [1, 2] extended the mathematical understanding of the options pricing model, beginning the deliberate study of the theory of option pricing. Its impact on the nancial markets was immediate and unprecedented and is arguably one of the most important discoveries within nance theory to date. By just inserting a few variables, which include the stock price, risk-free rate of return, option's strike price, expiration date, and an estimate of the volatility of the stock's price, the option-pricing formula is easily used by nancial investors. It allows them to price various derivatives ( nancial instrument whose price and value are derived from the value of assets underlying them), including options on commodities, nancial assets and even pricing of employee stock options. Hence, European1 and American2 call or put options on a non-dividend-paying stock can be valued using the Black-Scholes model. All further advances in option pricing since the Black-Scholes analysis have been re nements, generalisations and expansions of the original idea presented by them.

The modelling power of Itˆo integrals has a far reaching impact on a spectrum of diverse fields. For example, in mathematics of finance, its use has given insights into the relationship between call options and their non-deterministic underlying stock prices; in the study of blood clotting dynamics, its utility has helped provide an understanding of the behaviour of platelets in the blood stream; and in the investigation of experimental psychology, it has been used to build random fluctuations into deterministic models which model the dynamics of repetitive movements in humans. Finding the quadrature for these integrals using continuous groups or Lie groups has to take families of time indexed random variables, known as Wiener processes, into consideration. Adaptations of Sophus Lie’s work to stochastic ordinary differential equations (SODEs) have been done by Gaeta and Quintero [1], Wafo Soh and Mahomed [2], ¨Unal [3], Meleshko et al. [4], Fredericks and Mahomed [5], and Fredericks and Mahomed [6]. The seminal work [1] was extended in Gaeta [7]; the differential methodology of [2] and [3] were reconciled in [5]; and the integral methodology of [4] was corrected and reconciled in [5] via [6]. Symmetries of SODEs are analysed. This work focuses on maintaining the properties of the Weiner processes after the application of infinitesimal transformations. The determining equations for first-order SODEs are derived in an Itˆo calculus context. These determining equations are non-stochastic. Many methods of deriving Lie point-symmetries for Itˆo SODEs have surfaced. In the Itˆo calculus context both the formal and intuitive understanding of how to construct these symmetries has led to seemingly disparate results. The impact of Lie point-symmetries on the stock market, population growth and weather SODE models, for example, will not be understood until these different results are reconciled as has been attempted here. Extending the symmetry generator to include the infinitesimal transformation of the Wiener process for Itˆo stochastic differential equations (SDEs), has successfully been done in this thesis. The impact of this work leads to an intuitive understanding of the random time change formulae in the context of Lie point symmetries without having to consult much of the intense Itˆo calculus theory needed to derive it formerly (see Øksendal [8, 9]). Symmetries of nth-order SODEs are studied. The determining equations of these SODEs are derived in an Itˆo calculus context. These determining equations are not stochastic in nature. SODEs of this nature are normally used to model nature (e.g. earthquakes) or for testing the safety and reliability of models in construction engineering when looking at the impact of random perturbations. The symmetries of high-order multi-dimensional SODEs are found using form invariance arguments on both the instantaneous drift and diffusion properties of the SODEs. We then apply this to a generalised approximation analysis algorithm. The determining equations of SODEs are derived in an It¨o calculus context. A methodology for constructing conserved quantities with Lie symmetry infinitesimals in an Itˆo integral context is pursued as well. The basis of this construction relies on Lie bracket relations on both the instantaneous drift and diffusion operators.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy, 2017 / Killing vectors are widely used to study conservation laws admitted by spacetime metrics or to determine exact solutions of Einstein field equations (EFE) via Killing’s equation. Its solutions on a manifold are in one-to-one correspondence with continuous symmetries of the metric on that manifold. Two well known spherically symmetric static spacetime metrics in Relativity that admit maximal symmetry are given by Minkowski and de-Sitter metrics. Some other spherically symmetric metrics forming interesting solutions of the EFE are known as Schwarzschild, Kerr, Bertotti-Robinson and Einstein metrics. We study the symmetry properties and conservation laws of the geodesic equations following these metrics as well as the wave and Klein-Gordon (KG) type equations constructed using the covariant d’Alembertian operator on these manifolds. As expected, properties of reduction procedures using symmetries are more involved than on the well known flat (Minkowski) manifold. / XL2017

Xie Chunjing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 65-71). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Introduction --- p.3 / Chapter 1 --- Stability of Shock Waves in Viscous Conservation Laws --- p.10 / Chapter 1.1 --- Cauchy Problem for Scalar Viscous Conservation Laws and Viscous Shock Profiles --- p.10 / Chapter 1.2 --- Stability of Shock Waves by Energy Method --- p.15 / Chapter 1.3 --- Nonlinear Stability of Shock Waves by Spectrum Anal- ysis --- p.20 / Chapter 1.4 --- L1 Stability of Shock Waves in Scalar Viscous Con- servation Laws --- p.26 / Chapter 2 --- Propagation of a Viscous Shock in Bounded Domain and Half Space --- p.35 / Chapter 2.1 --- Slow Motion of a Viscous Shock in Bounded Domain --- p.36 / Chapter 2.1.1 --- Steady Problem and Projection Method --- p.36 / Chapter 2.1.2 --- Projection Method for Time-Dependent Prob- lem --- p.40 / Chapter 2.1.3 --- Super-Sensitivity of Boundary Conditions --- p.43 / Chapter 2.1.4 --- WKB Transformation Method --- p.45 / Chapter 2.2 --- Propagation of a Stationary Shock in Half Space --- p.50 / Chapter 2.2.1 --- Asymptotic Analysis --- p.50 / Chapter 2.2.2 --- Pointwise Estimate --- p.51 / Chapter 3 --- Nonlinear Stability of Viscous Transonic Flow Through a Nozzle --- p.58 / Chapter 3.1 --- Matched Asymptotic Analysis --- p.58 / Bibliography --- p.65

Zhang Hedan. / Thesis submitted in: September 2004. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 78-81). / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Convex Scalar Conservation Laws --- p.9 / Chapter 2.1 --- Cauchy Problems and Weak Solutions --- p.9 / Chapter 2.2 --- Rankine-Hugoniot Condition --- p.11 / Chapter 2.3 --- Entropy Condition --- p.13 / Chapter 2.4 --- Uniqueness of Weak Solution --- p.15 / Chapter 2.5 --- Riemann Problems --- p.17 / Chapter 3 --- General Scalar Conservation Laws --- p.21 / Chapter 3.1 --- Entropy-Entropy Flux Pairs --- p.21 / Chapter 3.2 --- Admissibility Conditions --- p.22 / Chapter 3.3 --- Kruzkov Theory --- p.23 / Chapter 4 --- Elementary waves and Riemann Problems for Nonconvex Scalar Conservation Laws --- p.35 / Chapter 4.1 --- Basic Facts --- p.35 / Chapter 4.2 --- Riemann Solutions --- p.36 / Chapter 5 --- Asymptotic Behavior --- p.46 / Chapter 5.1 --- Periodic Asymptotic Behavior --- p.46 / Chapter 5.2 --- Asymptotic Behavior of Convex Conservation Law --- p.49 / Chapter 5.3 --- Asymptotic Behavior of Non-convex case --- p.52 / Chapter 5.3.1 --- L∞ Behavior --- p.53 / Chapter 5.3.2 --- Wave-Interactions and Asymptotic Behavior Toward Shock Waves --- p.55 / Bibliography --- p.78

Ye Mao. / "May 2002." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (p. 116-123). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Wang Hui Ying. / "June 2002." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (p. 67-72). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Weng, Shangkun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (p. 73-78). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Existence and Uniqueness results of transonic shock solution to full Euler system in a large variable nozzle --- p.11 / Chapter 2.1 --- The mathematical description of the transonic shock problem and main results --- p.11 / Chapter 2.2 --- The reformulation on problem (2.1.1) with (2.1.5)-(2.1.9) --- p.18 / Chapter 2.3 --- An Iteration Scheme --- p.30 / Chapter 2.4 --- A priori estimates and proofs of Theorem 2.2.1 and Theorem 2.1.1 --- p.39 / Chapter 3 --- A monotonic theorem on the shock position with respect to the exit pressure --- p.50 / Chapter 4 --- Discussions and Future work --- p.64 / Chapter 5 --- Appendix --- p.66 / Chapter 5.1 --- Appendix A: Background solution --- p.66 / Chapter 5.2 --- Appendix B: An outline of the proof of Theorem 2.1.2 --- p.67

O objetivo deste projeto é o estudo de técnicas numéricas robustas para aproximação da solução de leis de conservação hiperbólicas escalares unidimensionais e bidimensionais e de sistemas de leis de conservação hiperbólicas. Para alcançar tal objetivo, estudamos esquemas conservativos com propriedades especiais, tais como, esquemas upwind, TVD, Godunov, limitante de fluxo e limitante de inclinação. A solução de um sistema de leis de conservação pode exibir descontinuidades do tipo choque, rarefação ou de contato. Assim, o desenvolvimento de técnicas numéricas capazes de reproduzir e tratar esses comportamentos é desejável. Além de representar corretamente a descontinuidade os esquemas numéricos têm ainda uma tarefa mais árdua; aquela de escolher a solução singular correta, a chamada solução entrópica. Os métodos de Godunov, limitantes de fluxo e limitantes de inclinação são técnicas numéricas que possuem as características apropriadas para aproximar a solução entrópica de uma lei de conservação. / The aim of this work is the study of robust numerical techniques for approximating the solution of scalar and systems of hyperbolic conservation laws. To achieve this, we studied conservative schemes with special properties, such as, schemes upwind, TVD, Godunov, flux limiters and slope limiters. The solution of a system of conservation laws can present discontinuities, like shocks, rarefaction or contact. Therefore, the development of numerical techniques capable of reproducing such featurs are highly desirable. Furthermore, besides resolving singularities, it is required that the numerical method chooses the correct weak solution, that is, the entropic solution. Godunov, flux limiters and slope limiters are techniques that show the appropriate behaviour when applied to conservation laws.

This thesis examines the Euler equations of gas dynamics and develops a new adaption indicator, which is based on the weak local residual measured for the non- conservative pressure variable. We demonstrate that the proposed indicator is capable of automatically detecting discontinuities and distinguishing between the shock and contact waves when they are isolated from each other. We use the developed indi- cator to design a scheme adaption algorithm, according to which nonlinear limiters are used only in the vicinity of shocks. The new adaption algorithm is realized using a second-order limited scheme and a high-order nonlimited central-upwind scheme. Robustness and high resolution of the designed method is shown on a number of one- and two-dimensional numerical examples. / acase@tulane.edu