Naoyuki Ishimura

REMARKS ON THE BLOW-UP CRITERION FOR THE 3-D BOUSSINESQ EQUATIONS

We consider the problem of blow-up of smooth solutions for the 3-D Boussinesq equations. Owing to the viscosity, we prove that the maximum norm of the gradient of vorticity controls the breakdown of the solutions; the scalar temperature function is shown to be irrelevant to the breakdown.

On the Structure of Steady Solutions for the kinematic Model of Spiral Waves in Excitable Media

Spiral waves are commonly observed in biological and chemical systems. Representing each wave front by a single curve, Brazhnik, Davydov, and Mikhailov introduce a kinematic model equation. The aim of this paper is to provide a detailed analysis for the steady state solutions of these equations. The existence of asymptotically Archimedean solutions is analytically shown.

Curvature evolution of plane curves with prescribed opening angle

We discuss the evolution of plane curves which are described by entire graphs with prescribed opening angle. We show that a solution converges to the unique self-similar solution with the same asymptotics.

Numerical solution of a nonlinear evolution equation for the risk preference

A singular nonlinear partial differential equation (PDE) for the risk preference was derived by the first author in previous publications. The PDE is related to the Arrow-Pratt coefficient of relative risk aversion. In the present paper, we develop a Rothe-Bellman & Kalaba quasilinearization method on quasi-uniform space mesh to numerically investigate such PDE. Numerical experiments are discussed.

Numerical Solution via Transformation Methods of Nonlinear Models in Option Pricing

Based on transformation techniques used in analysis of initial boundary value problems, we propose and discuss two numerical approaches for nonlinear models in option pricing. The first one exploits difference schemes for a degenerate parabolic problem on a finite interval. The second one solves the problem on infinite interval by Rothe’s method. An appropriate substitution reduces the matter to a semilinear integral equation at each time step. Numerical experiments are discussed.

On the eguchi-oki-matsumura equation for phase separation in one space dimension

Eguchi--Oki--Matsumura equations are introduced to describe the dynamics of pattern formation that arises from phase separation in some binary alloys. The model extends the well-known Cahn--Hilliard equation and consists of coupled two functions; one is the local concentration and the other is the local degree of order. We show the existence of a solution, its asymptotic profile, and in part the structure of steady state solutions. Computational studies are also given.

Discrete stochastic calculus and its applications: an expository note

We recast our recent studies on discrete stochastic processes relevant to a discrete analogue of the Ito formula. This analogous formula for discrete environment is introduced by one of the authors, and has a possibility of many applications in the discrete world.We consider the optimal portfolio problem and the pricing of exchange options. The results indicate certain direct connection between the discrete and the continuous processes through the Ito formula.

Stable finite difference scheme for a model equation of phase separation

A stable, conservative, and practical finite difference scheme is presented in order to solve numerically a model equation of phase separation in some binary alloys. The model system is introduced by Eguchi, Oki, and Matsumura in the attempt of theoretically describing such phenomena, and extends the well-known Cahn-Hilliard equation; even numerical solutions are hard to obtain for this dynamics. Our scheme utilizes a Lyapunov functional, that is, the free energy of the system, whose effectiveness is demonstrated by numerical implementations.

Numerical study on the systems of nonlinear ordinary differential equations for default risk model

We deal with the systems of ordinary differential equations (ODEs), which nonlinearly extend a looping default model of defaultable firms. Unknown functions are defined through a weighted integral of the tail distribution functions of the first jump time. We perform numerical study on these systems, especially on the blowing-up behavior of solutions, and consider the meaning of our results in financial economics.

Evolution of Copulas in Discrete Processes with Application to a Numerical Modeling of Dependence Relation Between Exchange Rates

Copulas are known to provide a flexible tool for analyzing the dependence structure between random events. Here we apply the newly introduced notion of evolution of copulas to real data of exchange rates so that we ensure the quality of practically employing our theory. Results show that our algorithm provides a prospective handy method in computational finance.