Optimal systems of subalgebras for a nonlinear Black-Scholes equation

Asian Option Pricing under Uncertain Volatility Model

In this paper, we study the asymptotic behavior of Asian option prices in the worst case scenario under an uncertain volatility model. We give a procedure to approximate the Asian option prices with a small volatility interval. By imposing additional conditions on the boundary condition and cutting the obtained Black-Scholes-Barenblatt equation into two Black-Scholes-like equations, we obtain an approximation method to solve the fully nonlinear PDE.

The finite dimensional subalgebra classification of infinite dimensional symmetry algebra of two dimensional coupled nonlinear Schrödinger equations

The symmetry group structures of two dimensional coupled nonlinear Shrödinger equations are considered. We first show that the equations admit infinite dimensional symmetry algebra as well as the corresponding symmetry group depending on four arbitrary functions of one variable. Then we show some physical symmetries and an affine loop algebra contained in the symmetry algebra of the equations. Third, we give the complete classifications of finite dimension (less than four) subalgebras of the symmetry algebra under the adjoint group of the symmetry group. These results provide the theoretical a…

Lie symmetry analysis of nonlinear evolution equation for description nonlinear waves in a viscoelastic tube

In this paper, the Lie symmetry method is performed for the nonlinear evolution equation for description nonlinear waves in a viscoelastic tube. we will find one and two-dimensional optimal system of Lie subalgebras. Furthermore, preliminary classification of its group-invariant solutions are investigated.

风险模型中非线性偏微分方程的单调迭代格式

Monotone iteration scheme for nonlinear PDEs in risk models

本文提出将带有上下解的单调迭代方法应用于交易对手风险模型中的非线性 Black-Scholes 型方程，为每步迭代引入半显式解公式，并验证了其在看涨期权等合约上的可行性。

Symmetry reductions of a nonlinear option pricing model

The studied model was suggested to design a perfect hedging strategy for a large trader. In this case the implementation of a hedging strategy affects the price of the underlying security. The feedback-effect leads to a nonlinear version of the Black-Scholes partial differential equation. Using the Lie group theory we reduce the partial differential equation in special cases to ordinary differential equations. The found Lie group of the model equation gives rise to invariant solutions. Families of exact invariant solutions for special values of parameters are described.

Symmetry group classification and optimal reduction of a class of damped Timoshenko beam system with a nonlinear rotational moment

We consider a nonlinear Timoshenko system of partial differential equations (PDEs) with a frictional damping term in rotation angle. The nonlinearity is due to the arbitrary dependence on the rotation moment. A Lie symmetry group classification of the arbitrary function of rotation moment is presented. An optimal system of one-dimensional subalgebras of the nonlinear damped Timoshenko system is derived for all the non-linear cases. All possible invariant variables of the optimal systems for the three non-linear cases are presented. The corresponding reduced systems of ordinary differential equ…

非均匀Fisher-Kolmogorov方程中非线性波的Lie对称与行波解

Lie symmetries and travelling wave solutions of the nonlinear waves in the inhomogeneous Fisher-Kolmogorov equation

本文从偏微分方程对称性约化的角度，研究依赖空间变量的非均匀Fisher-Kolmogorov方程，通过二维阿贝尔子代数将其约化为常微分方程，推导并解释了新的行波解。

Complete group classification of a class of nonlinear wave equations

Preliminary group classification became prominent as an approach to symmetry analysis of differential equations due to the paper by Ibragimov, Torrisi and Valenti [J. Math. Phys. 32, 2988-2995] in which partial preliminary group classification of a class of nonlinear wave equations was carried out via the classification of one-dimensional Lie symmetry extensions related to a fixed finite-dimensional subalgebra of the infinite-dimensional equivalence algebra of the class under consideration. In the present paper we implement, up to both usual and general point equivalence, the complete group cl…

Group-invariant solutions of a nonlinear acoustics model

Based on a recent classification of subalgebras of the symmetry algebra of the Zabolotskaya-Khokhlov equation, all similarity reductions of this equation into ordinary differential equations are obtained. Large classes of group-invariant solutions of the equation are also determined, and some properties of the reduced equations and exact solutions are discussed.

Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations

Enhancing and essentially generalizing previous results on a class of (1+1)-dimensional nonlinear wave and elliptic equations, we apply several new techniques to classify admissible point transformations within this class up to the equivalence generated by its equivalence group. This gives an exhaustive description of its equivalence groupoid. After extending the algebraic method of group classification to non-normalized classes of differential equations, we solve the complete group classification problem for the class under study up to both usual and general point equivalences. The solution i…

Multidimensional linear and nonlinear partial integro-differential equation in Bessel potential spaces with applications in option pricing

The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional cas…

诺特对称性与一类时间依赖的多维非线性波方程的守恒律

Noether symmetries and conservation laws of a class of time-dependent multidimensional nonlinear wave equations

研究一类时间依赖的阻尼非线性多维波方程的守恒律，利用诺特定理发现其无穷小变分对称性构成欧几里得代数，产生动量守恒，特定情况下扩展到共形代数并发现更多守恒律。

求解高维非线性抛物型偏微分方程的深度遗传算法（Deep-GA）方法

A Deep-Genetic Algorithm (Deep-GA) Approach for High-Dimensional Nonlinear Parabolic Partial Differential Equations

提出深度遗传算法（deep-GA）将遗传算法嵌入 deep-BSDE 求解器以优化初始猜测选择，加速了高维非线性抛物型偏微分方程的求解，在保持相当精度的同时显著提升了计算效率。

Symmetry analysis of a system of modified shallow-water equations

We revise the symmetry analysis of a modified system of one-dimensional shallow-water equations (MSWE) recently considered by Raja Sekhar and Sharma [Commun. Nonlinear Sci. Numer. Simulat. 20 (2012) 630-636]. Only a finite dimensional subalgebra of the maximal Lie invariance algebra of the MSWE, which in fact is infinite dimensional, was found in the aforementioned paper. The MSWE can be linearized using a hodograph transformation. An optimal list of inequivalent one-dimensional subalgebras of the maximal Lie invariance algebra is constructed and used for Lie reductions. Non-Lie solutions are …

On nonlinear partial differential equations with an infinite-dimensional conditional symmetry

The invariance of nonlinear partial differential equations under a certain infinite-dimensional Lie algebra A_N(z) in N spatial dimensions is studied. The special case A_1(2) was introduced in J. Stat. Phys. {\bf 75}, 1023 (1994) and contains the Schrödinger Lie algebra sch_1 as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of A_N(z). However, a large class of strongly non-linear partial differential equations is found which are conditionally invariant with respect to the massless realization of A_N(z) such that the well…

Enhanced preliminary group classification of a class of generalized diffusion equations

The method of preliminary group classification is rigorously defined, enhanced and related to the theory of group classification of differential equations. Typical weaknesses in papers on this method are discussed and strategies to overcome them are presented. The preliminary group classification of the class of generalized diffusion equations of the form u_t=f(x,u)u_x^2+g(x,u)u_{xx} is carried out. This includes a justification for applying this method to the given class, the simultaneous computation of the equivalence algebra and equivalence group, as well as the classification of inequivale…

具有温度依赖材料特性的非线性耦合热弹性波系统的精确自相似解

Exact self similar solutions for nonlinear coupled heat elastic wave systems with temperature dependent material properties

研究热容量、热导率和弹性模量都依赖温度的耦合非线性热弹性系统，通过李对称分类等方法，确定三类本构函数，求出精确闭式解并构建行波解，探讨几何结构影响及热扩散等因素的相互作用。

KP-BBM方程的Lie对称性分析及若干新精确解

Lie Symmetry Analysis and Some New Exact Solutions to the KP-BBM Equation

针对Wazwaz提出的KP-BBM方程，本文结合Painleve检验、Lie对称性分析、Hu-Li-Chen算法及齐次平衡法与Tanh法，构造最优系统、相似约化和新精确解，并分析参数对孤立波的影响。

多层准地转问题的对称分析与精确解

Symmetry analysis and exact solutions of multi-layer quasi-geostrophic problem

本文基于多层准地转问题的对称分析，首次正确描述了守恒律和哈密顿结构，并通过原创方法计算了最大李不变代数和完整点对称伪群，从而获得包含多种已知线性方程的耦合系统及广泛精确解。

无弥散 Nizhnik 方程的点对称与接触对称伪群

Point- and contact-symmetry pseudogroups of dispersionless Nizhnik equation

本文首次用基于 megaideal 的代数方法完整计算无弥散 Nizhnik 方程的点对称伪群，并证明其接触对称伪群等于点对称伪群的一阶延拓，同时刻画了具有相同 Lie 不变代数的三阶偏微分方程及其定义性几何性质。