热方程的非古典势对称群与不变解

主要研究了热方程与波方程的非古典势对称群生成元及相应的群不变解．研究表明对于守恒形式的偏微分方程,可通过其伴随系统求得的非古典势对称群生成元来构造其显式解．这些显式解不能由方程本身的Lie对称群生成元或Lie-B?cklund对称群生成元构造得到．     

Exact solutions via invariant approach for Black‐Scholes model with time‐dependent parameters

We analyze the Black‐Scholes model with time‐dependent parameters, and it is governed by a parabolic partial differential equation (PDE). First, we compute the Lie symmetries of the Black‐Scholes model with time‐dependent parameters. It admits 6 plus infinite many Lie symmetries, and thus, it can be reduced to the classical heat equation. We use the invariant criteria for a scalar linear (1+1) parabolic PDE and obtain 2 sets of equivalence transformations. With the aid of these equivalence transformations, the Black‐Scholes model with time‐dependent parameters transforms to the classical heat equation. Moreover, the functional forms of the time‐dependent parameters in the PDE are determined via this method. Then we use the equivalence transformations and known solutions of the heat equation to establish a number of exact solutions for the Black‐Scholes model with time‐dependent parameters.     

Lie group symmetries and Riemann function of Klein–Gordon–Fock equation with central symmetry

In the present paper Lie symmetry group method is applied to find new exact invariant solutions for Klein–Gordon–Fock equation with central symmetry. The found invariant solutions are important for testing finite-difference computational schemes of various boundary value problems of Klein–Gordon–Fock equation with central symmetry. The classical admitted symmetries of the equation are found. The infinitesimal symmetries of the equation are used to find the Riemann function constructively.     

Lie symmetry analysis,optimal system,and generalized group invariant solutions of the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work, Lie group theoretic method is used to carry out the similarity reduction and solitary wave solutions of (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation. The equation describes the propagation of nonlinear dispersive waves in inhomogeneous media. Under the invariance property of Lie groups, the infinitesimal generators for the governing equation have been obtained. Thereafter, commutator table, adjoint table, invariant functions, and one-dimensional optimal system of subalgebras are derived by using Lie point symmetries. The symmetry reductions and some group invariant solutions of the DJKM equation are obtained based on some subalgebras. The obtained solutions are new and more general than the rest while known results reported in the literature. In order to show the physical affirmation of the results, the obtained solutions are supplemented through numerical simulation. Thus, the solitary wave, doubly soliton, multi soliton, and dark soliton profiles of the solutions are traced to make this research physically meaningful.     

Investigation of new solutions for an extended (2+ 1)-dimensional Calogero-Bogoyavlenskii-Schif equation

We investigate and concentrate on new infinitesimal generators of Lie symmetries for an extended (2+ 1)-dimensional Calogero-Bogoyavlenskii-Schif (eCBS) equation using the commutator table which results in a system of nonlinear ordinary differential equations (ODEs) which can be manually solved. Through two stages of Lie symmetry reductions, the eCBS equation is reduced to non-solvable nonlinear ODEs using different combinations of optimal Lie vectors. Using the integration method and the Riccati and Bernoulli equation methods, we investigate new analytical solutions to those ODEs. Back substituting to the original variables generates new solutions to the eCBS equation. These results are simulated through three- and two-dimensional plots.     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

An alternative approach to solving the Black–Scholes equation with time-varying parameters

In this note we provide a simple derivation of an explicit formula for the price of an option on a dividend-paying equity when the parameters in the Black–Scholes partial differential equation (PDE) are time dependent. With the aid of general transformations, the option value is expressed as a product of the Black–Scholes price for an option on a non-dividend-paying equity with constant parameters, the ratio of the strike price in the time-varying case to the strike price in the constant-parameter case, and a modified discount factor containing a parametrised time variable.     

Symmetry reduction,exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation

We show that the Benjamin–Bona–Mahoney (BBM) equation with power law nonlinearity can be transformed by a point transformation to the combined KdV–mKdV equation, that is also known as the Gardner equation. We then study the combined KdV–mKdV equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the combined KdV–mKdV equation are derived. We obtain symmetry reduction and a number of exact group-invariant solutions for the underlying equation using the Lie point symmetries of the equation. The conserved densities are also calculated for the BBM equation with dual nonlinearity by using the multiplier approach. Finally, the conserved quantities are computed using the one-soliton solution.     

Symmetry analysis of a model for the exercise of a barrier option

A barrier option takes into account the possibility of an unacceptable change in the price of the underlying stock. Such a change could carry considerable financial loss. We examine one model based upon the Black–Scholes–Merton Equation and determine the functional forms of the barrier function and rebate function which are consistent with a solution of the underlying evolution partial differential equation using the Lie Theory of Extended Groups. The solution is consistent with the possibility of no rebate and the barrier function is very similar to one adopted on an heuristic basis.     

Explicit solution of Black–Scholes option pricing mathematical models with an impulsive payoff function

This paper deals with the construction of explicit solutions of the Black–Scholes equation with a weak payoff function. By using the Mellin transform of a class of weak functions a candidate integral formula for the solution is first obtained and then it is proved that it is a rigorous solution of the problem. Well known solutions of option pricing value problems are obtained as particular cases of the solution proposed here.     

Nonhypoellipticity and comparison principle for partial differential equations of Black–Scholes type

This paper studies some less known properties of the Black–Scholes equation and of its nonlinear modifications arising in Finance. In particular, the nonhypoellipticity of the linear Black–Scholes equation is shown; a comparison principle is formulated for a class of nonlinear degenerate parabolic equations which incorporates the most relevant financial applications; finally, some comments on the properties of the viscosity solutions are given.     

On analytical solutions of the Black–Scholes equation

This work presents a theoretical analysis for the Black–Scholes equation. Given a terminal condition, the analytical solution of the Black–Scholes equation is obtained by using the Adomian approximate decomposition technique. The mathematical technique employed in this work also has significance in studying some other problems in finance theory.     

Isovector fields and similarity solutions of Einstein vacuum equations for rotating fields

The isovector fields (infinitesimal generators of Lie groups) of Einstein vacuum equations for stationary axially symmetric rotating fields, in conventional form, that is a coupled system of nonlinear partial differential equations (PDEs) of second order are derived using the geometric prolongation technique. Some symmetry transformations and similarity (exact) solutions of Einstein vacuum equations are obtained.     

Derivation of conservation laws for a nonlinear wave equation modelling melt migration using Lie point symmetry generators

The derivation of conservation laws for a nonlinear wave equation modelling the migration of melt through the Earth’s mantle is considered. New conserved vectors which depend explicitly on the spatial coordinate are generated using the Lie point symmetry generators of the equation and known conserved vectors. It is demonstrated how conserved vectors that are conformally associated with a Lie point symmetry generator can be derived more simply than by the direct method by imposing the symmetry condition on the conservation law equation.     

Algebraic properties of evolution partial differential equations modelling prices of commodities

Schwartz (J. Finance 1997; 52 :923–973) presented three models for the pricing of a commodity. The simplest was a variation on the Black–Scholes equation. The second allowed for a stochastic convenience yield on the commodity and the third added a stochastic variation in the underlying interest rate. We apply the techniques of Lie group analysis to resolve these equations, discuss their peculiar algebraic properties and indicate the route to the addition of other stochastic influences. Copyright © 2007 John Wiley & Sons, Ltd.     

High-order exponential spline method for pricing European options

ABSTRACT

The key purpose of the present work is to constitute an analysis of a numerical method for a degenerate partial differential equation, called the Black–Scholes equation, governing European option pricing. The method is based on exponential spline spatial discretization and an explicit finite-difference time-stepping technique. We establish the convergence and an error bound for the solutions of the fully discretized system. The numerical and graphical results elucidate that the suggested approach is very straightforward and accurate.     

Professor V. Lakshmikantham,Editor Stochastic Analysis and Applications (1983–2012)

We use the stochastic calculus of variations for the fractional Brownian motion to derive formulas for the replicating portfolios for a class of contingent claims in a Bachelier and a Black–Scholes markets modulated by fractional Brownian motion. An example of such a model is the Black–Scholes process whose volatility solves a stochastic differential equation driven by a fractional Brownian motion that may depend on the underlying Brownian motion.     

The method of noncommutative integration for linear differential equations. Functional algebras and noncommutative dimensional reduction

The method of noncommutative integration for linear partial differential equations [1] is extended to the case of the so-called functional algebras for which the commutators of their generators are nonlinear functions of the same generators. The linear functions correspond to Lie algebras, whereas the quadratics are associated with the so-called quadratic algebras having wide applications in quantum field theory. A nontrivial example of integration of the Klein-Gordon equation in a curved space not allowing separation of variables is considered. A classification of four- and five-dimensional quadratic algebras is performed.A method of dimensional reduction for noncommutatively integrable many-dimensional partial differential equations is suggested. Generally, the reduced equation has a complicated functional symmetry algebra. The method permits integration of the reduced equation without the use of the explicit form of its functional algebra.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 3–15, January, 1996.     

Extension of Euler’s method to parabolic equations

Euler generalized d’Alembert’s solution to a wide class of linear hyperbolic equations with two independent variables. He introduced in 1769 the quantities that were rediscovered by Laplace in 1773 and became known as the Laplace invariants. The present paper is devoted to an extension of Euler’s method to linear parabolic equations with two independent variables. The new method allows one to derive an explicit formula for the general solution of a wide class of parabolic equations. In particular, the general solution of the Black–Scholes equation is obtained.     

Lie group analysis and propagation of weak discontinuity in one-dimensional ideal isentropic magnetogasdynamics

The aim of this paper is to carry out symmetry group analysis to obtain important classes of exact solutions from the given system of nonlinear partial differential equations (PDEs). Lie group analysis is employed to derive some exact solutions of one dimensional unsteady flow of an ideal isentropic, inviscid and perfectly conducting compressible fluid, subject to a transverse magnetic field for the magnetogasdynamics system. By using Lie group theory, the full one-parameter infinitesimal transformations group leaving the equations of motion invariant is derived. The symmetry generators are used for constructing similarity variables which leads the system of PDEs to a reduced system of ordinary differential equations; in some cases, it is possible to solve these equations exactly. Further, using the exact solution, we discuss the evolutionary behavior of weak discontinuity.