F. John's stability conditions vs. A. Carasso's SECB constraint for backward parabolic problems

In order to solve backward parabolic problems F. John [{\it Comm. Pure. Appl. Math.} (1960)] introduced the two constraints "$\|u(T)\|\le M$" and $\|u(0) - g \| \le \delta$ where $u(t)$ satisfies the backward heat equation for $t\in(0,T)$ with the initial data $u(0).$ The {\it slow-evolution-from-the-continuation-boundary} (SECB) constraint has been introduced by A. Carasso in [{\it SIAM J. Numer. Anal.} (1994)] to attain continuous dependence on data for backward parabolic problems even at the continuation boundary $t=T$. The additional "SECB constraint" guarantees a significant improvement in stability up to $t=T.$ In this paper we prove that the same type of stability can be obtained by using only two constraints among the three. More precisely, we show that the a priori boundedness condition $\|u(T)\|\le M$ is redundant. This implies that the Carasso's SECB condition can be used to replace the a priori boundedness condition of F. John with an improved stability estimate. Also a new class of regularized solutions is introduced for backward parabolic problems with an SECB constraint. The new regularized solutions are optimally stable and we also provide a constructive scheme to compute. Finally numerical examples are provided.Comment: 15 pages. To appear in Inverse Problem

Numerical methods for option pricing under jump-diffusion models.

Wu, Tao.Thesis (M.Phil.)--Chinese University of Hong Kong, 2010.Includes bibliographical references (leaves 56-61).Abstracts in English and Chinese.Chapter 1 --- Background and Organization --- p.7Chapter 2 --- Parallel Talbot method for solving partial integro- differential equations --- p.9Chapter 2.1 --- Introduction --- p.9Chapter 2.2 --- Initial-boundary value problem --- p.11Chapter 2.3 --- Spatial discretization and semidiscrete problem --- p.12Chapter 2.4 --- Parallel Talbot method --- p.15Chapter 2.4.1 --- Φ-functions and Talbot quadrature --- p.15Chapter 2.4.2 --- Control on nonnormality and feasibility con- straints --- p.18Chapter 2.4.3 --- Optimal parameterization of parabolic Talbot contour --- p.22Chapter 2.5 --- Numerical experiments --- p.26Chapter 2.6 --- Conclusion --- p.32Chapter 3 --- Memory-reduction Monte Carlo method for pricing American options --- p.37Chapter 3.1 --- Introduction --- p.37Chapter 3.2 --- Exponential Levy processes and the full-storage method --- p.39Chapter 3.3 --- Random number generators --- p.41Chapter 3.4 --- The memory-reduction method --- p.43Chapter 3.5 --- Numerical examples --- p.45Chapter 3.5.1 --- Black-Scholes model --- p.46Chapter 3.5.2 --- Merton's jump-diffusion model --- p.48Chapter 3.5.3 --- Variance gamma model --- p.50Chapter 3.5.4 --- Remarks on the efficiency of the memory-reduction method --- p.52Chapter 3.6 --- Conclusion --- p.53Chapter 3.7 --- Appendix --- p.5

Laplace Transform Method and Its Applications for Weather Derivatives

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2013. 8. 신동우.In this thesis we deal with the most efficient methods for numerical Laplace inversion and analyze the effect of roundoff errors. There are three issues in the control of numerical Laplace inversion: the choice of contour, its parameterization and numerical quadrature. We extend roundoff error control to the case of numerical inversion for hyperbolic contour. Also in order to examine the effect of roundoff error, computation is carried out both in double-precision and multi-precision, the latter which provides better understanding of the numerical Laplace inversion algorithms. We analyze temperature data for Seoul based on a well defined daily average temperature and consider related weather derivatives. The temperature data exhibit some quite distinctive features, compared to other cities that have been considered before. Due to these characteristics, seasonal variance and oscillation in Seoul is more apparent in winter and less evident in summer than in the other cities. We construct a deterministic model for the average temperature and then simulate future weather patterns, before pricing various weather derivative options and calculating the market price of risk. And Laplace transform method is applicable for solving the partial differential equation of weather derivatives.Chapter 1 Introduction Chapter 2 Laplace Transform Methods 2.1 Introduction 2.2 A unified framework to several numerical Laplace inversion schemes 2.2.1 Contours and their parameterization 2.2.2 Infinity-to-finite interval maps and quadrature rule 2.3 Roundoff error control on numerical Laplace inversion 2.3.1 Review of error estimation 2.3.2 Roundoff error control for hyperbolic contour 2.4 Numerical examples Chapter 3 Weather Derivatives 3.1 Introduction 3.2 Modelling of Seoul temperature 3.3 Temperature Derivatives 3.3.1 Option pricing for temperature derivatives 1: HDD and CDD 3.3.2 Option pricing for temperature derivatives 2: CAT 3.4 Estimating the Market Price of Risk (MPR) Chapter 4 Pricing Weather Derivatives using Laplace Transform Methods 4.1 Pricing option for weather sensitive asset 4.2 Pricing weather option using weather swapsDocto