英文标题：
《Symmetry reduction and exact solutions of the non-linear Black--Scholes
  equation》
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作者：
Oleksii Patsiuk, Sergii Kovalenko
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最新提交年份：
2018
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英文摘要：
  In this paper, we investigate the non-linear Black--Scholes equation: $$u_t+ax^2u_{xx}+bx^3u_{xx}^2+c(xu_x-u)=0,\\quad a,b>0,\\ c\\geq0.$$ and show that the one can be reduced to the equation $$u_t+(u_{xx}+u_x)^2=0$$ by an appropriate point transformation of variables. For the resulting equation, we study the group-theoretic properties, namely, we find the maximal algebra of invariance of its in Lie sense, carry out the symmetry reduction and seek for a number of exact group-invariant solutions of the equation. Using the results obtained, we get a number of exact solutions of the Black--Scholes equation under study and apply the ones to resolving several boundary value problems with appropriate from the economic point of view terminal and boundary conditions.
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中文摘要：
本文研究了非线性Black-Scholes方程：$$u_t+ax^2u_{xx}+bx^3u_{xx}^2+c（xu_x-u）=0、\\quad a，b>0、\\c\\geq0.$$并证明了通过适当的变量点变换，这个方程可以简化为$$u_t+（u_{xx}+u_x）^2=0$$。对于得到的方程，我们研究了它的群论性质，即在李意义下找到其不变性的极大代数，进行对称约化，并寻求方程的若干精确群不变解。利用所得结果，我们得到了所研究的Black-Scholes方程的一些精确解，并从经济角度和边界条件出发，将这些精确解应用于解决几个具有适当边界条件的边值问题。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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