AN OPTION PRICING PROBLEM WITH THEUNDERLYING STOCK PAY1NG DIVIDENDS~

In this paper, a pricing problem of European call options is considered, wbete the underlying stock generates dividends d, at some fixed future dates T, before the expiration date T .without the inappropriate assumption made in that the dlvkdeMs being payed continously.The arbitrage free pricing of the option is determined via a series of partial differential equations.which is derived at the view point of backward s‘tochasric differential ertuation (BBDE). It isshowed how the dividends affect the fair price of the call options. Some simulating results are alsogiven to illust rate the respective in fluence of parameters a.T.r,K.di and F1 on the option pricing.     

A dynamic boundary value problem arising in the ecology of mangroves

We consider an evolution model describing the vertical movement of water and salt in a domain splitted in two parts: a water reservoir and a saturated porous medium below it, in which a continuous extraction of fresh water takes place (by the roots of mangroves). The problem is formulated in terms of a coupled system of partial differential equations for the salt concentration and the water flow in the porous medium, with a dynamic boundary condition which connects both subdomains.We study the existence and uniqueness of solutions, the stability of the trivial steady state solution, and the conditions for the root zone to reach, in finite time, the threshold value of salt concentration under which mangroves may live.     

Partial differential integral equation model for pricing American option under multi state regime switching with jumps

In this paper, we consider a two dimensional partial differential integral equation (PDIE) model for pricing American option. A nonlinear rationality parameter function for two asset problems is introduced to deal with the free boundary. The rationality parameter function is added in the PDIEs used for pricing American option problems under multi-state regime switching with jumps. The resulting two dimensional nonlinear system of PDIE is then numerically solved. Based on real poles rational approximation, a strongly stable highly efficient and reliable method is developed to solve such complicated systems of PIDEs. The method is build in a predictor corrector style which makes it linearly implicit, therefore, avoids solving nonlinear systems of equations at each time step in all regimes. The method is seen to maintain the stability and convergence for large jump sizes and high volatility in each regime. The impact of regime switching on option prices corresponding to different values interest rate, volatility, and rationality parameter is computed, illustrated by graphs and given in the tables. Convergence results in each regime are presented and time evolution graphs are given to show the effectiveness and reliability of the method.     

On the asymptotic free boundary for the American put option problem

In practical work with American put options, it is important to be able to know when to exercise the option, and when not to do so. In computer simulation based on the standard theory of geometric Brownian motion for simulating stock price movements, this problem is fairly easy to handle for options with a short lifespan, by analyzing binomial trees. It is considerably more challenging to make the decision for American put options with long lifespan. In order to provide a satisfactory analysis, we look at the corresponding free boundary problem, and show that the free boundary—which is the curve that separates the two decisions, to exercise or not to—has an asymptotic expansion, where the coefficient of the main term is expressed as an integral in terms of the free boundary. This raises the perspective that one could use numerical simulation to approximate the integral and thus get an effective way to make correct decisions for long life options.     

Analysis of a two-phase field model for the solidification of an alloy

In this paper we present some theoretical results for a system of nonlinear partial differential equations that provide a phase field model for the solidification/melting of a metallic alloy. It is assumed that two different kinds of crystallization are possible. Consequently, the unknowns are the temperature τ and the phase field functions u and v. The time derivatives ut and vt appear in the equation for τ (the heat equation). On the other hand, the equations for u and v contain nonlinear terms where we find τ.     

期权定价公式的注

根据建立在连续支付红利且利率变动的股票上的期权的到期特点,利用两个数字式期权构造的投资组合收益来复制期权从而导出欧氏看跌期权的定价公式,避开了通过求解B-S方程来得到期权价格的困难.运用同样的方法也获得了期货期权公式.     

Uniqueness for an inverse source problem of determining a space dependent source in a time-fractional diffusion equation

We study uniqueness of a solution for an inverse source problem arising in linear time-fractional diffusion equations with time dependent coefficients. New uniqueness results are formulated in Theorem 3.1. We also show optimality of the conditions under which uniqueness holds by explicitly constructing counterexamples, that is by constructing more than one solution in the case when the conditions for uniqueness are violated.     

A new method of solving the coefficient inverse problem

This paper is concerned with the new method for solving the coefficient inverse problem in the reproducing kernel space. It is different from the previous studies. This method gives accurate results and shows that it is valid by the numerical example.     

A recursive pricing formula for a path-dependent option under the constant elasticity of variance diffusion

In this paper, we consider a path-dependent option in finance under the constant elasticity of variance diffusion. We use a perturbation argument and the probabilistic representation (the Feynman–Kac theorem) of a partial differential equation to obtain a complete asymptotic expansion of the option price in a recursive manner based on the Black–Scholes formula and prove rigorously the existence of the expansion with a convergence error.     

Oscillation for systems of nonlinear neutral type parabolic partial functional differential equations

This paper discusses the oscillation of solutions for systems of nonlinear neutral type parabolic partial fuctional differential equations of the form     

An inverse problem for a higher-order parabolic equation

We prove existence and uniqueness theorems for the inverse problem of finding the right-hand side of a higher-order parabolic equation with two independent variables and an additional condition in the form of integral overdetermination. The results obtained are used to study the passage to the limit in a sequence of such inverse problems with weakly convergent coefficients. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 680–691, November, 1998.     

Analysis of a semigroup approach in the inverse problem of identifying an unknown parameters

This article presents a semigroup approach to the mathematical analysis of the inverse parameter problems of identifying the unknown parameters p(t) and q in the linear parabolic equation u_t(x, t)  = u_xx + qu_x(x, t) + p(t)u(x, t), with Dirichlet boundary conditions u(0, t) = ψ₀, u(1, t) = ψ₁. The main purpose of this paper is to investigate the distinguishability of the input-output mapping Φ[·]:P→H^1,2[0,T], via semigroup theory. In this paper, it is shown that if the nullspace of the semigroup T(t) consists of only zero function, then the input-output mapping Φ[·] has the distinguishability property. It is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of the mapping. Moreover, under the light of the measured output data u_x(0, t) = f(t) the unknown parameter p(t) at (x, t) = (0, 0) and the unknown coefficient q are determined via the input data. Furthermore, it is shown that measured output data f(t) can be determined analytically by an integral representation. Hence the input-output mapping Φ[·]:P→H^1,2[0,T] is given explicitly interms of the semigroup.     

Boundary data smoothness for solutions of nonlocal boundary value problems for second order differential equations

Under certain conditions, solutions of the boundary value problem, y^″=f(x,y,y^′), y(x₁)=y₁, and , are differentiated with respect to boundary conditions, where a<x₁<η₁<?<ηm<x₂<b, r₁,…,rm∈R, and y₁,y₂∈R.     

A new formulation of the water wave problem for Stokes waves of constant vorticity

We consider steady symmetric gravity water waves on finite depth with constant vorticity and a monotone surface profile between crests and troughs. The problem is transformed into one concerning the vertical velocity. A representation formula for the stream function in terms of the surface and the vorticity is presented, and we show that the surface can be determined from the vertical velocity.     

无限时滞中立型随机泛函微分方程解的存在唯一性

有限时滞随机泛函微分方程的存在唯一性已经得到较多的研究,但对于无限时滞随机泛函微分方程的性质极少.本文在不需要线性增长条件,在一致Lipschitz条件下证明了无限时滞中立型随机泛函微分方程的存在唯一性,给出了精确解和近似解的误差估计,最后给出了解的矩估计.     

An inverse problem of Bilayer textile thickness determination in dynamic heat and moisture transfer

An inverse problem of bilayer textile thickness determination in dynamic heat and moisture transfer is presented satisfying the heat–moisture comfort level of human body. Heat and mass transfer law in bilayer textiles is displayed by proving the existence and uniqueness of solution to the coupled partial differential equations with initial-boundary value conditions. The finite difference method is employed to derive the numerical solution to partial differential equations. The regularized solution of the inverse problem is reformulated into solving function minimum problem through the Tikhonov regularization method. The golden section method is applied to solve the direct search problem and achieve the optimal solution to the inverse problem. Numerical algorithm and its numerical results provide theoretical explanation for textile materials research and development.     

On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition

We study the unique solvability of the inverse problem of determining the righthand side of a parabolic equation whose leading coefficient depends on both the time and the spatial variable under an integral overdetermination condition with respect to time. We obtain two types of condition sufficient for the local solvability of the inverse problem as well as study the so-called Fredholm solvability of the inverse problem under consideration.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 522–534.Original Russian Text Copyright © 2005 by V. L. Kamynin.This revised version was published online in April 2005 with a corrected issue number.     

Method of lines solutions of the parabolic inverse problem with an overspecification at a point

The present work is motivated by the desire to obtain numerical solution to a quasilinear parabolic inverse problem. The solution is presented by means of the method of lines. Method of lines is an alternative computational approach which involves making an approximation to the space derivatives and reducing the problem to a system of ordinary differential equations in the variable time, then a proper initial value problem solver can be used to solve this ordinary differential equations system. Some numerical examples and also comparison with finite difference methods will be investigated to confirm the efficiency of this procedure.     

Necessary conditions for the existence of positive solutions to fractional boundary value problems at resonance

In this article, we obtain some properties of the positive solutions to a class of fractional differential equation multipoint boundary value problems at resonance. Necessary conditions for the existence of positive solutions are established by means of the spectral theory of linear operator. As application, necessary and sufficient conditions for the existence of positive solutions are given for the case that the nonlinearity has a fixed sign. Some examples are presented to illustrate our results.