The main contribution of this thesis is to examine binary options within the British payoff mechanism introduced by Peskir and Samee. This includes British cash-or-nothing put, British asset-or-nothing put, British binary call and American barrier binary options. We assume the geometric Brownian motion model and reduce the optimal stopping problems to free-boundary problems under the Markovian nature of the underlying process. With the help of the local time-space formula on curves, we derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a non-linear integral equation. We begin by investigating the binary options of American-type which are also called `one-touch' binary options. Then we move on to examine the British binary options. Chapter~2 reviews the existing work on all different types of the binary options and sets the background for the British binary options. We price and analyse the American-type (one-touch) binary options using the risk-neutral probability method. In Chapters~3 ~4 and ~5, we present the British binary options where the holder enjoys the early exercise feature of American binary options whereupon his payoff is the `best prediction' of the European binary options payoff under the hypothesis that the true drift equals a contract drift. Based on the observed price movements, if the option holder finds that the true drift of the stock price is unfavourable then he can substitute it with the contract drift and minimise his losses. The key to the British binary option is the protection feature as not only can the option holder exercise at unfavourable stock price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. Chapters~3 and~4 focus on the British binary put options and Chapter~5 on call options. We also analyse the financial meaning of the British binary options and show that with the contract drift properly selected the British binary options become very attractive alternatives to the classic European/American options. Chapter~6 extends the binary options into barrier binary options and discusses the application of the optimal structure without a smooth-fit condition in the option pricing. We first review the existing work for the knock-in options and present the main results from the literature. Then we examine the method in \cite{dai2004knock} in the application to the knock-in binary options. For the American knock-out binary options, the smooth-fit property does not hold when we apply the local time-space formula on curves. We transfer the expectation of the local time term into a computational form under the basic properties of Brownian motion. Using standard arguments based on Markov processes, we analyse the properties of the value function.

This thesis consists of five papers and it mainly addresses the theory and schemes to approximate the quadrature domains, QDs. The first deals with the uniqueness and some qualitative properties of the two QDs. The concept of two phase QDs, is more complicated than its one counterpart and consequently introduces significant and interesting open. We present two numerical schemes to approach the one phase QDs in the paper. The first method is based on the properties of the free boundary the level set techniques. We use shape optimization analysis to construct second method. We illustrate the efficiency of the schemes on a variety of experiments. In the third paper we design two finite difference methods for the approximation of the multi phase QDs. We prove that the second method enjoys monotonicity, consistency and stability and consequently it is a convergent scheme by Barles-Souganidis theorem. We also present various numerical simulations in the case of Dirac measures. We introduce the QDs in a sub domain of and Rn study the existence and uniqueness along with a numerical scheme based on the level set method in the fourth paper. In the last paper we study the tangential touch for a semi-linear problem. We prove that there is just one phase free boundary points on the flat part of the fixed boundary and it is also shown that the free boundary is a uniform C1-graph up to that part. / Denna avhandling består av fem artiklar och behandlar främst teori och numeriska metoder för att approximera "quadrature domians", QDs. Den första artikeln behandlar entydighet och allmänna egenskaper hos tvåfas QDs. Begreppet tvåfas QDs, är mer komplicerat än enafasmotsvarigheten och introducerar därmed intressanta öppna problem. Vi presenterar två numeriska metoder för att approximera enfas QDs i andra artikeln. Den första metoden är baserad på egenskaperna hos den fria randen och nivå mängdmetoden. Vi använder forsoptimeringmanalys för att konstruera den andra metoden. Båda metoderna är testade i olika numeriska simuleringar. I det tredje artikeln vi approximera flerafas QDs med konstruktionen tvåmetoder finita differens. Vi visar att den andra metoden har monotonicitat, konsistens och stabilitet och följaktligen är metoden konvergent tack vare Barles-Souganidis sats. Vi presenterar också olika numeriska simuleringar i fallet med Diracmåt. Vi introducerar QDs i en delmängd av Rn och studerar existens och entydighet jämte en numerisk metod baserad på nivå mängdmetoden i fjärde pappret. I det sista pappret studerar vi den tangentiella touchen för ett semilinjärt problem. Vi visar att det enbart är enafasrandpunkter på den platta delen av den fixerade randen. Vi visar också att den fria randen är en likformig C1-graf upp till den delen av den fixerade randen.

abstract: In a 2004 paper, John Nagy raised the possibility of the existence of a hypertumor \emph{i.e.}, a focus of aggressively reproducing parenchyma cells that invade part or all of a tumor. His model used a system of nonlinear ordinary differential equations to find a suitable set of conditions for which these hypertumors exist. Here that model is expanded by transforming it into a system of nonlinear partial differential equations with diffusion, advection, and a free boundary condition to represent a radially symmetric tumor growth. Two strains of parenchymal cells are incorporated; one forming almost the entirety of the tumor while the much more aggressive strain appears in a smaller region inside of the tumor. Simulations show that if the aggressive strain focuses its efforts on proliferating and does not contribute to angiogenesis signaling when in a hypoxic state, a hypertumor will form. More importantly, this resultant aggressive tumor is paradoxically prone to extinction and hypothesize is the cause of necrosis in many vascularized tumors. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2014

Philosophiae Doctor - PhD / Conventional partial differential equations under the classical Black-Scholes approach have been extensively explored over the past few decades in solving option pricing problems. However, the underlying Efficient Market Hypothesis (EMH) of classical economic theory neglects the effects of memory in asset return series, though memory has long been observed in a number financial data. With advancements in computational methodologies, it has now become possible to model different real life physical phenomenons using complex approaches such as, fractional differential equations (FDEs). Fractional models are generalised models which based on literature have been found appropriate for explaining memory effects observed in a number of financial markets including the stock market. The use of fractional model has thus recently taken over the context of academic literatures and debates on financial modelling. / 2023-12-02

Most models for the spread of an invasive species into a new environment are based on Fisher's reaction-diffusion equation. They assume that habitat quality is independent of the presence or absence of the invading population. Ecosystem engineers are species that modify their environment to make it (more) suitable for them. A potentially more appropriate modeling approach for such an invasive species is to adapt the well-known Stefan problem of melting ice. Ahead of the front, the habitat is unsuitable for the species (the ice); behind the front, the habitat is suitable (the open water). The engineering action of the population moves the boundary ahead (the melting). This approach leads to a free boundary problem. In this thesis, we mathematically analyze a novel free-boundary model for the spread of ecosystem engineers that was recently derived from an individual random walk model. The Stefan condition for the moving boundary is replaced by a biologically derived two-sided condition that models the movement behavior of individuals at the boundary as well as the process by which the population moves the boundary to expand their territory. We first consider the model with logistic growth and study its well-posedness. We assign a convex functional to this problem so that the evolution system governed by this convex potential is exactly the system of evolution equations describing the above model. We then apply variational and fixed-point methods to deal with this free boundary problem and prove the existence of local in-time solutions. We next study traveling wave solutions of the model with the strong Allee growth function. We use phase plane analysis to find traveling wave solutions of different types and their corresponding existence range of speed for the model with an imposed speed of the moving boundary. We then find the speeds in those ranges at which the corresponding traveling wave follows the speed of the free boundary.

The Mullins-Sekerka problem, also called two-sided Hele-Shaw flow, arises in modeling a binary material with two stable concentration phases. A coarsening process occurs, and large particles grow while smaller particles eventually dissolve. Single particles become spherical. This process is described by evolving harmonic functions within the two phases with the moving interface driven by the jump in the normal derivatives of the harmonic functions at the interface. The harmonic functions are continuous across the interface, taking on values equal to the mean curvature of the interface. This dissertation reformulates the three-dimensional problem as one on the two-dimensional interface by using boundary integrals. A semi-implicit scheme to solve the free boundary problem numerically is implemented. Numerical analysis tasks include discretizing surfaces, overcoming node bunching, and dealing with topology change in a toroidal particle. A particle (node)-cluster technique is developed with the aim of alleviating excessive run time caused by filling the dense matrix used in solving a system of linear equations.

This thesis is in the field of non-linear partial differential equations (PDE), focusing on problems which show some type of phase-transition. A single phase Hele-Shaw flow models a Newtoninan fluid which is being injected in the space between two narrowly separated parallel planes. The time evolution of the space that the fluid occupies can be modelled by a semi-linear PDE. This is a problem within the field of free boundary problems. In the multi-phase problem we consider the time-evolution of a system of phases which interact according to the principle that the joint boundary which emerges when two phases meet is fixed for all future times. The problem is handled by introducing a parameterized equation which is regularized and penalized. The penalization is non-local in time and tracks the history of the system, penalizing the joint support of two different phases in space-time. The main result in the first paper is the existence theory of a weak solution to the parameterized equations in a Bochner space using the implicit function theorem. The family of solutions to the parameterized problem is uniformly bounded allowing us to extract a weakly convergent subsequence for the case when the penalization tends to infinity. The second problem deals with a parameterized highly oscillatory quasi-linear elliptic equation in divergence form. As the regularization parameter tends to zero the equation gets a jump in the conductivity which occur at the level set of a locally periodic function, the obstacle. As the oscillations in the problem data increases the solution to the equation experiences high frequency jumps in the conductivity, resulting in the corresponding solutions showing an effective global behaviour. The global behavior is related to the so called homogenized solution. We show that the parameterized equation has a weak solution in a Sobolev space and derive bounds on the solutions used in the analysis for the case when the regularization is lost. Surprisingly, the limiting problem in this case includes an extra term describing the interaction between the solution and the obstacle, not appearing in the case when obstacle is the zero level-set. The oscillatory nature of the problem makes standard numerical algorithms computationally expensive, since the global domain needs to be resolved on the micro scale. We develop a multi scale method for this problem based on the heterogeneous multiscale method (HMM) framework and using a finite element (FE) approach to capture the macroscopic variations of the solutions at a significantly lower cost. We numerically investigate the effect of the obstacle on the homogenized solution, finding empirical proof that certain choices of obstacles make the limiting problem have a form structurally different from that of the parameterized problem. / <p>QC 20180222</p>

The main contribution of this thesis is to derive the properties and present a closed from solution of the exotic options under some specific types of Levy processes, such as American put options, American call options, British put options, British call options and American knock-out put options under either double exponential jump-diffusion processes or one-sided exponential jump-diffusion processes. Compared to the geometric Brownian motion, exponential jump-diffusion processes can better incorporate the asymmetric leptokurtic features and the volatility smile observed from the market. Pricing the option with early exercise feature is the optimal stopping problem to determine the optimal stopping time to maximize the expected options payoff. Due to the Markovian structure of the underlying process, the optimal stopping problem is related to the free-boundary problem consisting of an integral differential equation and suitable boundary conditions. By the local time-space formula for semi-martingales, the closed form solution for the options value can be derived from the free-boundary problem and we characterize the optimal stopping boundary as the unique solution to a nonlinear integral equation arising from the early exercise premium (EEP) representation. Chapter 2 and Chapter 3 discuss American put options and American call options respectively. When pricing options with early exercise feature under the double exponential jump-diffusion processes, a non-local integral term will be found in the infinitesimal generator of the underlying process. By the local time-space formula for semi-martingales, we show that the value function and the optimal stopping boundary are the unique solution pair to the system of two integral equations. The significant contributions of these two chapters are to prove the uniqueness of the value function and the optimal stopping boundary under less restrictive assumptions compared to previous literatures. In the degenerate case with only one-sided jumps, we find that the results are in line with the geometric Brownian motion models, which extends the analytical tractability of the Black-Scholes analysis to alternative models with jumps. In Chapter 4 and Chapter 5, we examine the British payoff mechanism under one-sided exponential jump-diffusion processes, which is the first analysis of British options for process with jumps. We show that the optimal stopping boundaries of British put options with only negative jumps or British call options with only positive jumps can also be characterized as the unique solution to a nonlinear integral equation arising from the early exercise premium representation. Chapter 6 provides the study of American knock-out put options under negative exponential jump-diffusion processes. The conditional memoryless property of the exponential distribution enables us to obtain an analytical form of the arbitrage-free price for American knock-out put options, which is usually more difficult for many other jump-diffusion models.

In the first part of this dissertation, we produce and study a generalized mathematical model of solid combustion. Our generalized model encompasses two special cases from the literature: a case of negligible heat diffusion in the product, for example, when the burnt product is a foam-like substance; and another case in which diffusivities in the reactant and product are assumed equal. In addition to that, our model pinpoints the dynamics in a range of settings, in which the diffusivity ratio between the burned and unburned materials varies between 0 and 1. The dynamics of temperature distribution and interfacial front propagation in this generalized solid combustion model are studied through both asymptotic and numerical analyses. For asymptotic analysis, we first analyze the linear instability of a basic solution to the generalized model. We then focus on the weakly nonlinear case where a small perturbation of a neutrally stable parameter is taken so that the linearized problem is marginally unstable. Multiple scale expansion method is used to obtain an asymptotic solution for large time by modulating the most linearly unstable mode. On the other hand, we integrate numerically the exact problem by the Crank-Nicolson method. Since the numerical solutions are very sensitive to the derivative interfacial jump condition, we integrate the partial differential equation to obtain an integral-differential equation as an alternative condition. The result system of nonlinear algebraic equations is then solved by the Newton’s method, taking advantage of the sparse structure of the Jacobian matrix. By a comparison of our asymptotic and numerical solutions, we show that our asymptotic solution captures the marginally unstable behaviors of the solution for a range of model parameters. Using the numerical solutions, we also delineate the role of the diffusivity ratio between the burned and unburned materials. We find that for a representative set of this parameter values, the solution is stabilized by increasing the temperature ratio between the temperature of the fresh mixture and the adiabatic temperature of the combustion products. This trend is quite linear when a parameter related to the activation energy is close to the stability threshold. Farther from this threshold, the behavior is more nonlinear as expected. Finally, for small values of the temperature ratio, we find that the solution is stabilized by increasing the diffusivity ratio. This stabilizing effect does not persist as the temperature ratio increases. Competing effects produce a “cross-over” phenomenon when the temperature ratio increases beyond about 0.2. In the second part, we study the existence and decay rate of a transmission problem for the plate vibration equation with a memory condition on one part of the boundary. From the physical point of view, the memory effect described by our integral boundary condition can be caused by the interaction of our domain with another viscoelastic element on one part of the boundary. In fact, the three different boundary conditions in our problem formulation imply that our domain is composed of two different materials with one condition imposed on the interface and two other conditions on the inner and outer boundaries, respectively. These transmission problems are interesting not only from the point of view of PDE general theory, but also due to their application in mechanics. For our mathematical analysis, we first prove the global existence of weak solution by using Faedo-Galerkin’s method and compactness arguments. Then, without imposing zero initial conditions on one part of the boundary, two explicit decay rate results are established under two different assumptions of the resolvent kernels. Both of these decay results allow a wider class of relaxation functions and initial data, and thus generalize some previous results existing in the literature.

Doctor of Philosophy (PhD) / For many practitioners and market participants, the valuation of financial derivatives is considered of very high importance as its uses range from a risk management tool, to a speculative investment strategy or capital enhancement. A developing market requires efficient but accurate methods for valuing financial derivatives such as American options. A closed form analytical solution for American options has been very difficult to obtain due to the different boundary conditions imposed on the valuation problem. Following the method of solving the American option as a free boundary problem in the spirit of the "no-arbitrage" pricing framework of Black-Scholes, the option price and hedging parameters can be represented as an integral equation consisting of the European option value and an early exercise value dependent upon the optimal free boundary. Such methods exist in the literature and along with risk-neutral pricing methods have been implemented in practice. Yet existing methods are accurate but inefficient, or accuracy has been compensated for computational speed. A new numerical approach to the valuation of American options by cubic splines is proposed which is proven to be accurate and efficient when compared to existing option pricing methods. Further comparison is made to the behaviour of the American option's early exercise boundary with other pricing models.