Lie Symmetry Analysis of Differential Equations in Finance

Lie group theory is applied to differential equations occurring as mathematical models in financial problems. We begin with the complete symmetry analysis of the one-dimensional Black–Scholes model and show that this equation is included in Sophus Lie's classification of linear second-order partial differential equations with two independent variables. Consequently, the Black–Scholes transformation of this model into the heat transfer equation follows directly from Lie's equivalence transformation formulas. Then we carry out the classification of the two-dimensional Jacobs–Jones model equations according to their symmetry groups. The classification provides a theoretical background for constructing exact (invariant) solutions, examples of which are presented.     

Algebraic linearization criteria for systems of ordinary differential equations

Algebraic linearization criteria by means of general point transformations for systems of two second-order nonlinear ordinary differential equations (ODEs) are revisited. In previous work due to Wafo Soh and Mahomed (Int. J. Non-Linear Mech. 36:671, 2001) two four-dimensional Lie algebras that result in linearizability in terms of arbitrary point transformation for such systems were studied. Here we consider three more algebras of dimension four that result in linearization. Therefore our results supplement those of Wafo Soh and Mahomed (Int. J. Non-Linear Mech. 36:671, 2001). Moreover, it is shown that these are the only other possibilities for dimension four. Hence we provide the complete algebraic linearization criteria for dimension four algebras. Necessary and sufficient conditions for linearization via invertible maps of a nonlinear to a linear system are given. These are shown to be built up from the Lie algebraic criteria for linearization of scalar second-order ODEs. These results together with very recent work (Bagderina in J. Phys. A, Math. Theor. 43:465201, 2010) give a complete picture on linearizability properties via general point transformations for systems of two second-order ODEs. Furthermore, we provide natural extensions of these algebraic criteria for linearizing arbitrary systems of nonlinear second-order ODEs by means of point transformations. We also obtain algebraic criteria for the reduction of a linear system to the simplest system. Examples from Newtonian mechanics and geodesic equations are presented to illustrate our results.     

Invariant linearization criteria for a three-dimensional dynamical system of second-order ordinary differential equations and applications

Second-order dynamical systems are of paramount importance as they arise in mechanics and many applications. It is essential to have workable explicit criteria in terms of the coefficients of the equations to effect reduction and solutions for such types of equations. One important aspect is linearization by invertible point transformations which enables one to reduce a non-linear system to a linear system. The solution of the linear system allows one to solve the non-linear system by use of the inverse of the point transformation. It was proved that the n-dimensional system of second-order ordinary differential equations obtained by projecting down the system of geodesics of a flat (n+1)-dimensional space can be converted to linear form by a point transformation. This is a generalization of the Lie linearization criteria for a scalar second-order equation. In this case it is of the maximally symmetric class for a system and the linearizing transformation as well as the solution can be directly written down. This was explicitly used for two-dimensional dynamical systems. The criteria were written down in terms of the coefficients and the linearizing transformation allowed for the general solution of the original system. Here the work is extended to a three-dimensional dynamical system and we find explicit criteria, including the linearization test given in terms of the coefficients of the cubic in the first derivatives of the system and the construction of the transformations, that result in linearization. Applications to equations of classical mechanics and relativity are given to illustrate our results.     

Invariants of a Remarkable Family of Nonlinear Equations

In classical literature, invariants of families of differentialequations were considered for linear equations only, e.g. the renownedLaplace invariants for linear hyperbolic partial differential equationsand invariants of linear ordinary differential equations with variablecoefficients. The restriction to linear equations was essential inpioneering works of Cockle, Laguerre, Halphen, andForsyth for tackling the problem of invariants of differentialequations. Lie regretted that these authors did not use advantagesprovided by his theory of infinite continuous groups, but he himself didnot undertake further developments in this direction.Recently, the present author considered the possibility hinted byLie's remark and introduced the infinitesimal technique in thetheory of invariants of families of differential equations thatwas lacking in old methods. In consequence, a simple unifiedapproach was developed for calculation of invariants of algebraicand differential equations independent on the assumption oflinearity of equations. It was employed recently for calculationof Laplace type invariants for parabolic equations. Here, themethod is applied to calculation of invariants for the family ofnonlinear equations appearing in the problem on linearization ofnonlinear ordinary differential equations.     

Equivalent linearization for systems subjected to non-stationary random excitation

The method of eauivalent linearization is applied to the general problem of the response of non-linear discrete systems to non-stationary random excitation. Conditions for minimum equation difference are determined which do not depend explicitly on lime but only on the instantaneous statistics of the response process. Using the equivalent linear parameters, a deterministic non-linear ordinary differential equation for the covariance matrix is derived. An example is given of a damped Duffing oscillator subjected to modulated white noise.     

一种构造高维Nilpotent正规型的方法

正规型方法是一种有效的简化一类非线方程的方法。今提出了一种简便的代数方法去构造高维非线性系统的Nilpotent范式。通过引入一系列简单的变换和代数运算,而无需求解任何偏微分方程,即可得到高维非线性系统的Nilpotent范式。以四维非线性系统为例介绍这个方法。该方法完全适用于分析高于四维的非线性系统的Nilpotent范式。     

Isospectral Euler-Bernoulli beams via factorization and the Lie method

We obtain isospectral Euler-Bernoulli beams by using factorization and Lie symmetry techniques. The canonical Euler-Bernoulli beam operator is factorized as the product of a second-order linear differential operator and its adjoint. The factors are then reversed to obtain isospectral beams. The factorization is possible provided the coefficients of the factors satisfy a system of non-linear ordinary differential equations. The uncoupling of this system yields a single non-linear third-order ordinary differential equation. This ordinary differential equation, called the principal equation, is analyzed, reduced or solved using Lie group methods. We show that the principal equation may admit a one-dimensional or three-dimensional symmetry Lie algebra. When the principal system admits a unique symmetry, the best we can do is to depress its order by one. We obtain a one-parameter family of invariant solutions in this case. The maximally symmetric case is shown to be isomorphic to a Chazy equation which is solved in closed form to derive the general solution of the principal equation. The transformations connecting isospectral pairs are obtained by numerically solving systems of ordinary differential equations using the fourth-order Runge-Kutta method.     

Construction of approximate analytical solutions to strongly nonlinear damped oscillators

An analytical approximate method for strongly nonlinear damped oscillators is proposed. By introducing phase and amplitude of oscillation as well as a bookkeeping parameter, we rewrite the governing equation into a partial differential equation with solution being a periodic function of the phase. Based on combination of the Newton’s method with the harmonic balance method, the partial differential equation is transformed into a set of linear ordinary differential equations in terms of harmonic coefficients, which can further be converted into systems of linear algebraic equations by using the bookkeeping parameter expansion. Only a few iterations can provide very accurate approximate analytical solutions even if the nonlinearity and damping are significant. The method can be applied to general oscillators with odd nonlinearities as well as even ones even without linear restoring force. Three examples are presented to illustrate the usefulness and effectiveness of the proposed method.     

Characteristic solutions of scalar newton equations in one space dimension

Newton equations are dynamical systems on the space of fields. The solutions of a given equation which are curves of characteristic fields for its force are planar and have constant angular momentum. Separable solutions are characteristic with angular momentum equal to zero. A Newton equation is separable if and only if its characteristic equation is homogeneous. Separable equations correspond to invariants of homogeneous ordinary differential equations, and those associated with a given homogenous equation correspond to its generalized dilation symmetries. A Newton equation is compatible with the characteristic condition if and only if its characteristic equation is linear. Such equations correspond to invariants of linear ordinary differential equations. Those associated with a given linear equation correspond to the central force problems on its solution space. Regardless of compatibility, any Newton equation with a plane of characteristic fields has non-separable characteristic solutions.     

Axisymmetric spreading of a thin power-law fluid under gravity on a horizontal plane

Separable solutions admitted by a nonlinear partial differential equation modeling the axisymmetric spreading under gravity of a thin power-law fluid on a horizontal plane are investigated. The model equation is reduced to a highly nonlinear second-order ordinary differential equation for the spatial variable. Using the techniques of Lie group analysis, the nonlinear ordinary differential equation is linearized and solved. As a consequence of this linearization, new results are obtained.     

Higher order linearization in non-linear random vibration

In this paper a higher order linearization method for analyzing non-linear random vibration problems is presented. The non-linear terms of the given equation are replaced by unknown linear terms. These are in turn described by extra non-linear differential equations. The combined system of equations is then linearized to arrive at a higher degree-of-freedom equation for the original system. The method is illustrated by considering the Duffing oscillator under white noise input. The equivalent two d.o.f linear system is derived by the present method. Numerical results on steady state variance and PSD functions are obtained. These are found to be better than the simple linearization results.     

Symmetry and ordinary differential constraints

The representative generalized symmetries of any ordinary differential equation are described in terms of its invariants. This identifies the evolution equations compatible with a given constraint. The restriction of the flow of a compatible equation to the solution space of the constraint is generated by the corresponding internal symmetry. This reduces the evolution equation to a finite dimensional system of first-order ordinary differential equations. The Euler–Lagrange equation of any conserved density of a given evolution equation yields such a reduction. Other examples include the generalized method of separation of variables, the characterization of separable evolution equations, and the characterization of equations with complete families of wave solutions. A Newton equation is compatible with an ordinary differential constraint if and only if the constraint is affine, with force field symmetry, in which case the equation reduces to a finite-dimensional dynamical system. Newton equations with complete families of characteristic solutions reduce to central force problems on solution spaces of linear constraints.     

非Gauss分布随机波浪力对海洋平台的非线性作用

为避免求解决定Maikov过程转移概率密度的Fokker—Planck方程，基于尺度分离的假设导出了一组描述非线性海洋平台受非Gauss分布随机波浪载荷作用所产生响应的矩量的常微分方程组。矩量方程清楚地反映出分别对应随机载荷和结构响应的两种不同统计特性的相互关系。由于矩量方程不依赖载荷的概率分布的具体细节，以它来模拟随机激励作用下的非线性系统将免于Monte Carlo方法所面临的正确模拟载荷概率分布的困难任务。将摄动法用于矩量方程可使线性化不再需要，这样就不会因为线性化而产生不可预料的误差。     

Parabolic method of solving problems of the flow of a sonic gas stream around a thin symmetric profile

A parabolic method consisting of replacement of the stream acceleration ?_xx in the non-linear member of (1.1) by a specially chosen constant has been proposed [1] for the solution of the mixed-type transonic equation with boundary conditions on the profile, and the solution of the linear parabolic-type equation obtained can be considered as a certain approximation to the solution of the initial problem. An improvement of the parabolic method is the method of local linearization [2] (see [3] also), in which the acceleration ?_xx fixed from the beginning is replaced by a function of the coordinate x which satisfies some condition. An ordinary first-order differential equation is obtained for the velocity distribution along the profile in [2]. Another method of “defrosting” the acceleration ?_xx “frozen” from the beginning is proposed in this paper; a second-order ordinary differential equation is obtained for the velocity on the profile, which permits getting rid of some disadvantages of the local linearization method. Several solutions of (2.3) are presented in comparison to known exact solutions.     

Numerical method for unsteady compressible boundary layers driven by compression or expansion wave

A numerical analysis is presented for the unsteady compressible laminar boundary layer driven by a compression or expansion wave. Approximate or series expansion methods have been used for the problems because of the characteristics of the governing equations, such as non-linearity, coupling with the thermal boundary layer equation and initial conditions. Here a transformation of the governing equations and the numerical linearization technique are introduced to deal with the difficulties. First, the governing equations are transformed for the initial conditions by Howarth and semisimilarity variables. These transformations reduce the number of independent variables from three to two and the governing equations from partial to ordinary differential equations at the initial point. Next, the numerical linearization technique is introduced for the non-linearity and the coupling with the thermal boundary layer equation. Because the non-linear terms are linearized without sacrifice of numerical accuracy, the solutions can be obtained without numerical iterations. Therefore the exact numerical solution, not approximate or series expansion, can be obtained. Compared with the approximate or series expansion method, this method is much improved. Results are compared with the series expansion solutions.     

Stability and bifurcation analysis in the delay-coupled nonlinear oscillators

This paper investigates the dynamical behavior of two oscillators with nonlinearity terms, which are coupled with finite delay parameters. Each oscillator is a general class of second-order nonlinear delay-differential equations. The system of delay differential equations is analyzed by reducing the delay equations to a system of ordinary differential equations on a finite-dimensional center manifold, the corresponding to an infinite-dimensional phase space. In addition, the characteristic equation for the linear stability of the trivial equilibrium is completely analyzed and the stability region is illustrated in the parameters space. Our analysis reveals necessary coefficients of the reduced vector field on the center manifold for studying the bifurcations of the trivial equilibrium such as transcritical, pitchfork, and Hopf bifurcation. Finally, we consider the delay-coupled van der Pol equations.     

Nonlinear Effects on Coexistence Phenomenon in Parametric Excitation

We investigate the effect of nonlinearites on a parametrically excited ordinary differential equation whose linearization exhibits the phenomenon of coexistence. The differential equation studied governs the stability of a mode of vibration in an unforced conservative two degree of freedom system used to model thefree vibrations of a thin elastica. Using perturbation methods, we show thatat parameter values corresponding to coexistence, nonlinear terms can cause the origin to become nonlinearly unstable,even though linear stability analysis predicts the origin to be stable.We also investigate the bifurcations associated with this instability.     

A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations

In this paper, we study the structural stability of the Cahn-Hilliard equation and the phase-field equations. We show that the Cahn-Hilliard equation and the phase-field equations are topologically conjugate to a decoupled system of a linear equation of infinite dimension and an ordinary differential equation which is the reduced equation on the inertial manifold; particularly, the flow nearby hyperbolic stationary solutions is structurally stable.     

Geometric Optics and Instability for Semi-Classical Schrödinger Equations

We prove some instability phenomena for semi-classical (linear or) nonlinear Schrödinger equations. For some perturbations of the data, we show that for very small times, we can neglect the Laplacian, and the mechanism is the same as for the corresponding ordinary differential equation. Our approach allows smaller perturbations of the data, where the instability occurs for times such that the problem cannot be reduced to the study of an ordinary differential equation.     

Analytic solution for axisymmetric flow over a nonlinearly stretching sheet

An analysis is performed for the boundary-layer flow of a viscous fluid over a nonlinear axisymmetric stretching sheet. By introducing new nonlinear similarity transformations, the partial differential equations governing the flow are reduced to an ordinary differential equation. The resulting ordinary differential equation is solved using the homotopy analysis method (HAM). Analytic solution is given in the form of an infinite series. Convergence of the obtained series solution is explicitly established. The solution for an axisymmetric linear stretching sheet is obtained as a special case.