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Doctor of Philosophy / Department of Mathematics / Ivan Blank / We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in Caffarelli's 1977 paper ``The regularity of free boundaries in higher dimensions." Finally, we show that blowup limits are in general not unique at free boundary points.

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.

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

Optimal stopping and mathematical finance are intimately connected since the value of an American option is given as the solution to an optimal stopping problem. Such a problem can be viewed as a game in which we are trying to maximize an expected reward. The solution involves finding the best possible strategy, or equivalently, an optimal stopping time for the game. Moreover, the reward corresponding to this optimal time should be determined. It is also of interest to know how the solution depends on the model parameters. For example, when pricing and hedging an American option, the volatility needs to be estimated and it is of great practical importance to know how the price and hedging portfolio are affected by a possible misspecification. The first paper of this thesis investigates the performance of the delta hedging strategy for a class of American options with non-convex payoffs. It turns out that an option writer who overestimates the volatility will obtain a superhedge for the option when using the misspecified hedging portfolio. In the second paper we consider the valuation of a so-called stock loan when the lender is allowed to issue a margin call. We show that the price of such an instrument is equivalent to that of an American down-and-out barrier option with a rebate. The value of this option is determined explicitly together with the optimal repayment strategy of the stock loan. The third paper considers the problem of how to optimally stop a Brownian bridge. A finite horizon optimal stopping problem like this can rarely be solved explicitly. However, one expects the value function and the optimal stopping boundary to satisfy a time-dependent free boundary problem. By assuming a special form of the boundary, we are able to transform this problem into one which does not depend on time and solving this we obtain candidates for the value function and the boundary. Using stochastic calculus we then verify that these indeed satisfy our original problem. In the fourth paper we consider an investor wanting to take advantage of a mispricing in the market by purchasing a bull spread, which is liquidated in case of a market downturn. We show that this can be formulated as an optimal stopping problem which we then, using similar techniques as in the third paper, solve explicitly. In the fifth and final paper we study convexity preservation of option prices in a model with jumps. This is done by finding a sufficient condition for the no-crossing property to hold in a jump-diffusion setting.

This thesis consists of three papers devoted to the study of monotonicity formulas and their applications in elliptic and parabolic free boundary problems. The first paper concerns an inhomogeneous parabolic problem. We obtain global and local almost monotonicity formulas and apply one of them to show a regularity result of a problem that arises in connection with continuation of heat potentials.In the second paper, we consider an elliptic two-phase problem with coefficients bellow the Lipschitz threshold. Optimal $C^{1,1}$ regularity of the solution and a regularity result of the free boundary are established.The third and last paper deals with a parabolic free boundary problem with Hölder continuous coefficients. Optimal $C^{1,1}\cap C^{0,1}$ regularity of the solution is proven. / QC20100621

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20153号 / 理博第4238号 / 新制||理||1609(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 坂上 貴之, 教授 上田 哲生, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

On the first part, we consider nonlinear operators I depending on a family of nonlocal linear operators [mathematical equations]. We study the solutions of the Dirichlet initial and boundary value problems [mathematical equations]. We do not assume even symmetry for the kernels. The odd part bring some sort of nonlocal drift term, which in principle competes against the regularization of the solution. Existence and uniqueness is established for viscosity solutions. Several Hölder estimates are established for u and its derivatives under special assumptions. Moreover, the estimates remain uniform as the order of the equation approaches the second order case. This allows to consider our results as an extension of the classical theory of second order fully nonlinear equations. On the second part, we study two phase problems posed over a two dimensional cone generated by a smooth curve [mathematical symbol] on the unit sphere. We show that when [mathematical equation] the free boundary avoids the vertex of the cone. When [mathematical equation]we provide examples of minimizers such that the vertex belongs to the free boundary. / text

We study the regularity of weak solutions for the Stefan and Hele- Shaw problems with Gibbs-Thomson law under special conditions. The main result says that whenever the free boundary is Lipschitz in space and time it becomes (instantaneously) C[superscript 2,alpha] in space and its mean curvature is Hölder continuous. Additionally, a similar model related to the Signorini problem is introduced, in this case it is shown that for large times weak solutions converge to a stationary configuration. / text