The aim of this paper is to investigate the finite element methods for pricing the American put option on bonds. Based on a new variational inequality equation for the option pricing problems, both semidiscrete and fully discretized finite element approximation schemesare established. It is proved that the finite element methods are stable and convergent under L2 and HI norms.

In this paper, we investigate the finite element methods for a second type variational inequality problem. The linear finite element approximation scheme including its numerical integration modification form is proposed by a new approach. We show the unique existence and stability of finite element solutions. In particular, some abstract error estimates in H1 and L2 norms are established which imply the optimal convergence rates in order and regularity. Finally, we also give an error estimate in L1 norm.

In this paper, we present a general error analysis framework for the finite volume element (FVE) approximation to the Ritz-Volterra projection, the Sobolev equations and parabolic integro-differential equations. The main idea in our paper is to consider the FVE methods as perturbations of standard finite element methods which enables us to derive the optimal L2 and H1 norm error estimates, and the L1 and W1 1 norm error estimates by means of the time dependent Green functions. Our discussions also include elliptic and parabolic problems as the special cases.

A highly accurate derivative recovery formula is presented for the k-order finite element approximations to the two-point boundary value problems. This formula possesses the O (hk+1) order of superconvergence on the whole domain in L1 norm and O (h2k) order of ultraconvergence at the mesh points, and also the lowest regularity requirement for the exact solutions. Numerical experiments are given to verify the high accuracy of our formula.

The object of this paper is to investigate the uperconvergence properties of finite element approximations to parabolic and hyperbolic integral-differential equations. The quasi projection technique introduced earlier by Douglas, etc. is developed to derive the O (h2r) order knot superconvergence in a single pace variable, and to show the optimal order negative norm estimates in case of several space variables.

In this paper, the linear finite element approximation to the positive and symmetric, linear hyperbolic systems is analyzed and an O (h2) order error estimate is established under the conditions of strongly regular triangulation and the H3-regularity for the exact solutions. The convergence analysis is based on some superclose estimates derived in this paper. Our method and result here are also applicable to general hyperbolic problems. Finally, we discuss the linearized shallow water system of equations.

The objective of this article is to investigate the pricing problem for the lookback options with American type constrains. Based on the di®erential linear complementary formula associated with the pricing problem, an implicit di®erence scheme is constructed and analyzed. We show that the difference solution is uniquely existent and unconditionally stable. Using the notion of viscosity solutions, we also prove that the finite di®erence solution converges uniformly to the viscosity solution of the continuous problem. Furthermore, by means of the variational inequality analysis method, the O (Δt+h2) order error estimate is derived in the discrete L2-norm provided that the continuous problem is suffciently regular.

The object of this paper is to investigate the convergence of semidiscrete-nite element pproximations to the parabolic and hyperbolic integral-di erential equations, Sobolev equations and visco-elasticity equations. The Ritz-Volterra projection will be used to unity much of the analysis for the di erent types of problems. Optimal order error esti-mates are obtained in Lp and W1 spaces for 2≤p≤∞

对目前普遍使用的期权定价二叉树模型的缺陷进行了分析，利用随机误差校止方法构造出新型的二叉树参数模型。新的模型避免了负的概率并且具有很高的精确度，因而可应用于计算各种期权的价格。

本文研究美式股票看跌期权定价问题的数值方法。通过将问题转化为等价的变分不等式方程，分别建立了半离散和全离散有限元逼近格式，并给出了有限元解的收敛性和稳定性分析。数值实验表明本文算法是一个高效和收敛的算法。