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SUMMARYA new method for solution of the evolution of plane curves satisfying the geometric equation v=ÿ(x; k; ), where v is the normal velocity, k and are the curvature and tangential angle of a plane curve ⊂ R 2 at the point x ∈ , is proposed. We derive a governing system of partial di erential equations for the curvature, tangential angle, local length and position vector of an evolving family of plane curves and prove local in time existence of a classical solution. These equations include a non-trivial tangential velocity functional governing a uniform redistribution of grid points and thus preventing numerically computed solutions from forming various instabilities. We discretize the governing system of equations in order to ÿnd a numerical solution for 2D anisotropic interface motions and image segmentation problems.

We analyse a model for pricing derivative securities in the presence of both transaction costs as well as the risk from a volatile portfolio. The model is based on the Black-Scholes parabolic PDE in which transaction costs are described following the Hoggard, Whalley, and Wilmott approach. The risk from a volatile portfolio is described by the variance of the synthesized portfolio. Transaction costs as well as the volatile portfolio risk depend on the time lag between two consecutive transactions. Minimizing their sum yields the optimal length of the hedge interval. In this model, prices of vanilla options can be computed from a solution to a fully nonlinear parabolic equation in which a diffusion coefficient representing volatility nonlinearly depends on the solution itself giving rise to explaining the volatility smile analytically. We derive a robust numerical scheme for solving the governing equation and perform extensive numerical testing of the model and compare the results to real option market data. Implied risk and volatility are introduced and computed for large option datasets. We discuss how they can be used in qualitative and quantitative analyses of option market data.

In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear HamiltonJacobi-Bellman equation can be transformed into a quasi-linear parabolic equation whose diffusion function is obtained as the value function of certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence, uniqueness and derive useful bounds of classical Hölder smooth solutions. We furthermore construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit traveling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index as an example of application of the method.2000 Mathematics subject classification: Primary: 35K55, Secondary: 34E05 70H20 91B70 90C15 91B16.

We study evolution of a closed embedded plane curve with the normal velocity depending on the curvature, the orientation and the position of the curve. We propose a new method of tangential redistribution of points by curvature adjusted control in the tangential motion of evolving curves. The tangential velocity may not only distribute grid points uniformly along the curve but also produce a suitable concentration and/or dispersion depending on the curvature. Our study is based on solutions to the governing system of nonlinear parabolic equations for the position vector, tangent angle and curvature of a curve. We furthermore present a semi-implicit numerical discretization scheme based on the flowing finite volume method. Several numerical examples illustrating capability of the new tangential redistribution method are also presented in this paper.Keywords Curvature driven flow of a plane curve · Curvature adjusted tangential velocity · Semi-implicit scheme · Flowing finite volume method · Crystalline curvature flow equation Mathematics Subject Classification (2000)35K65 · 65N40 · 53C80

The purpose of this paper is to analyse the free boundary problem for the Black–Scholes equation for pricing the American call option on stocks paying a continuous dividend. Using the Fourier integral transformation method, we derive and analyse a nonlinear singular integral equation determining the shape of the free boundary. Numerical experiments based on this integral equation are also presented.

We study a flow of closed curves on a given graph surface driven by the geodesic curvature and external force. Using vertical projection of surface curves to the plane we show how the geodesic curvature-driven flow can be reduced to a solution of a fully nonlinear system of parabolic differential equations. We show that the flow of surface curves is gradient-like, i.e. there exists a Lyapunov functional nonincreasing along trajectories. Special attention is placed on the analysis of closed stationary surface curves. We present sufficient conditions for their dynamic stability. Several computational examples of evolution of surface curves driven by the geodesic curvature and external force on various surfaces are presented in this article. We also discuss a link between the geodesic flow and the edge detection problem arising from the image segmentation theory.

In this paper we analyze a nonlinear Black-Scholes model for option pricing under variable transaction costs. The diffusion coefficient of the nonlinear parabolic equation for the price V is assumed to be a function of the underlying asset price and the Gamma of the option. We show that the generalizations of the classical BlackScholes model can be analyzed by means of transformation of the fully nonlinear parabolic equation into a quasilinear parabolic equation for the second derivative of the option price. We show existence of a classical smooth solution and prove useful bounds on the option prices. Furthermore, we construct an effective numerical scheme for approximation of the solution. The solutions are obtained by means of the efficient numerical discretization scheme of the Gamma equation. Several computational examples are presented.