Option pricing formulas based on uncertain fractional differential equation

Uncertain fractional differential equations have been playing an important role in modelling complex dynamic systems. Early researchers have presented the extreme value theorems and time integral theorem on uncertain fractional differential equation. As applications of these theorems, this paper investigates the pricing problems of American option and Asian option under uncertain financial markets based on uncertain fractional differential equations. Then the analytical solutions and numerical solutions of these option prices are derived, respectively. Finally, some numerical experiments are performed to verify the effectiveness of our results.

     

Riccati方程的可积性条件

得到了R iccati方程的一些可积的充分条件及其在这些条件下的通积分,此结果包含了已有文献中的一些结论,最后给出定理应用的例子.     

Boundary Integral Equation Method in the Steady State Oscillation Problems for Anisotropic Bodies

The three-dimensional steady state oscillation problems of the elasticity theory for homogeneous anisotropic bodies are studied. By means of the limiting absortion principle the fundamental matrices maximally decaying at infinity are constructed and the generalized Sommerfeld–Kupradze type radiation conditions are formulated. Special functional spaces are introduced in which the basic and mixed exterior boundary value problems of the steady state oscillation theory have unique solutions for arbitrary values of the oscillation parameter. Existence theorems are proved by reduction of the original boundary value problems to equivalent boundary integral (pseudodifferential) equations. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Boundary value problems in the theory of thermoporoelasticity for materials with double porosity

In this paper the linear quasi-static theory of thermoelasticity for solids with double porosity is considered. The system of equations of this theory is based on the equilibrium equations for solids with double porosity, conservation of fluid mass, constitutive equations, Darcy's law for materials with double porosity and Fourier's law for heat conduction. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for classical solutions of the above mentioned BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Study of the Solvability of Some Volterra-Type Integral and Integro-Differential Equations of Third Kind

For Volterra integral equations of the third kind and for Volterra-type integrodifferential equations of the third kind, theorems on the existence of solutions in Sobolev spaces (i.e., regular solutions) are proved. The proofs are based on the theory of boundary value problems for degenerate ordinary differential equations and on the theory of boundary value problems for parabolic equations with a changing evolution direction.     

Analytical description of the radiative-conductive heat transfer in a gray medium contained between two diffuse parallel plates

Explicit analytical solutions for the temperature and heat flux in a gray medium contained between two diffuse parallel plates are derived for both pure thermal radiation and coupled conduction-radiation heat transfer. This is achieved by combining the integral equations for the heat flux and temperature predicted by the radiative transfer equation with the corresponding predictions of the discrete ordinates method. The algebraic formulation of this well-known method is used to derive analytical results that agree with their corresponding numerical ones with an accuracy greater than 99.9%, for a large interval of optical thicknesses and conduction-to-radiation factors. The explicit and original solutions, for both pure radiation and radiative-conductive heat transfer, therefore solve the problem of one dimensional steady-state heat transfer in gray cavities.     

Schwarz problem for higher‐order complex partial differential equations in the upper half plane

Linear and nonlinear elliptic complex partial differential equations of higher‐order are considered under Schwarz conditions in the upper‐half plane. Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schwarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro‐differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively.     

Existence of Solutions for Fractional Integro-differential Equations with Impulsive and Integral Conditions

This paper presents the existence of solutions for a class of Cauchy problems with integral condition for impulsive fractional integro-differential equations. Based on definition of solution for impulsive fractional integro- differential equations, the existence theorems of solutions of fractional differ- ential equation are obtained by applying fixed point methods. Finally, three examples are given to demonstrate the feasibility of the obtained results.     

Boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations

In the present paper we study the unique solvability of two non-local boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations. The uniqueness of the solutions of the considered problems are proven by the “abc” method. Existence theorems for the solutions of these problems are proven by the method of integral equations. The obtained results can be used for studying local and non-local boundary-value problems for mixed-hyperbolic type equations with two and three lines of changing type.      

一类非线性算子方程的多重正解及其应用

本文利用不动点指数理论对一类非线性算子方程建立了多重正解的存在性定理，并将所获结果应用到Ｈａｍｍｅｒｓｔｅｉｎ积分方程和Ｓｔｕｒｍ Ｌｉｏｕｖｉｌｅ两点边值问题，得到了新的结论，本质上改进和推广了［２，４，５，６，８－１２］的工作．     

Direct and inverse dynamic problems for SH-waves in porous media

Direct and inverse dynamic problems for the equation of SH-waves in porous media are considered. A singular solution of the direct dynamic problem is constructed. A system of nonlinear Volterra integral equations of the second kind is obtained for the dynamic inverse problems in question. Theorems of uniqueness and theorems of existence in the small for the considered inverse problems are proved. Also, theorems of continuous dependence of solutions of inverse dynamic problems on input data are proved.     

一般Lidstone边值问题的解的存在性

考察了含有各阶导数的一般Lidstone边值问题的存在性．利用Leray-Schauder不动 点定理、等价范数及积分方程组技巧证明了解或正解的两个存在定理．这些定理在应用上非常 方便．换句话说,通过考察非线性项在某个有界子集上的“高度”就能决定解的存在性．     

Rank criteria for the controllability of a boundary-value problem for a linear system of integro-differential equations with pulse influence

We determine necessary and sufficient conditions for the solvability of boundary-value problems for a linear system of integro-differential equations with pulse influence. We prove theorems on the existence and integral representation of solutions of linear second order integral-sum Volterra equations and linear systems of integro-differential equations with pulse influence at fixed times. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 723–730, June, 2000.     

随机积分和微分方程在弱拓扑下解的存在性和比较结果*

在本文中,我们对非线性随机Volterra积分方程在Banach空间的弱拓扑下的随机解证明了几个存在定理.然后作为应用,我们得到了随机微分方程的弱随机解的存在定理.还得到了这些随机方程的极值随机解的存在性和随机比较定理.我们的定理改进和推广了[4,5,10,11,12]中的相应结果.     

Generalized solutions of mixed problems for first-order partial functional differential equations

A theorem on the existence of solutions and their continuous dependence upon initial boundary conditions is proved. The method of bicharacteristics is used to transform the mixed problem into a system of integral functional equations of the Volterra type. The existence of solutions of this system is proved by the method of successive approximations using theorems on integral inequalities. Classical solutions of integral functional equations lead to generalized solutions of the original problem. Differential equations with deviated variables and differential integral problems can be obtained from the general model by specializing given operators. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 804–828, June, 2006.     

Existence of non-trivial solution for superlinear system of integral equations and its applications

1.IntroductionTherearealotofresultsaboutHammersteinintegralequations(see[1]),butsupen-linearproblemsaredifficult;thereareonlyafewresultsaboutit.In[2,3]theauthorsstudysuperliniearintegralequations;in[4]theauthorsstudysystemofsuperlinearintegralequatio...     

Some boundary value problems for pseudodifferential equations with degeneration

Boundary value problems for a new class of degenerate pseudodifferential equations containing a variable-symbol degenerate pseudodifferential operator based on a special integral transform and the first derivative with respect to one of the variables are studied. Existence theorems for these problems are proved. A priori estimates for their solutions are obtained in special weighted spaces similar to Sobolev ones.     

Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions

In this paper, we consider the properties of Green’s function for a class of nonlinear Caputo fractional differential equations with integral boundary conditions by constructing an available integral operator. By means of well-known fixed point theorems and lower and upper solutions method, some new existence and nonexistence criteria of single or multiple positive solutions for fractional differential equation boundary value problems are established. As applications, some interesting examples are presented to illustrate the main results.     

Boundary value problems in the theory of binary mixtures

In this paper, the boundary value problems of steady oscillation (vibration) of the linear theory of thermoelasticity for binary mixtures are investigated by means of the boundary integral equation method (potential method). The uniqueness and existence theorems of solutions of the exterior boundary value problems by means potential method and multidimensional singular integral equations are proved. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一般Lidstone边值问题的解的存在性

考察了含有各阶导数的一般Lidstone边值问题的存在性. 利用Leray-Schauder不动点定理、等价范数及积分方程组技巧证明了解或正解的两个存在定理. 这些定理在应用上非常方便. 换句话说, 通过考察非线性项在某个有界子集上的“高度”就能决定解的存在性.