非流动性市场中的自融资套期保值策略模型：对称性约化与精确解
Models of self-financing hedging strategies in illiquid markets: symmetry reductions and exact solutions
摘要：
我们研究了Schoenbucher和Wilmott在2000年提出的非流动性市场中的自融资交易策略的一般模型。该模型框架下的套期保值策略满足一个包含函数g（alpha）的非线性偏微分方程（PDE）。这个函数与效用函数有很深的联系。我们描述了李对称代数的PDE，并提供了一套完整的减少PDE常微分方程（ODE）。此外，我们能够描述所有类型的函数g（alpha）的PDE承认一个扩展李群。三个特殊类型函数中的两个导致了不同作者以前介绍的模型，一个是新的。我们澄清了这三个特殊模型和一般模型之间的联系，交易策略在非流动性市场。我们用李群分析研究了描述自融资策略的偏微分方程的新特例。在一般模型和新的特殊模型中，我们给出了最优子代数系统，并研究了偏微分方程到不同常微分方程的完备约简。在所有的情况下，我们能够提供明确的解决方案，新的特殊模型。在其中一种情况下，解描述了幂导数乘积。
英文摘要：
We study the general model of self-financing trading strategies in illiquid markets introduced by Schoenbucher and Wilmott, 2000. A hedging strategy in the framework of this model satisfies a nonlinear partial differential equation (PDE) which contains some function g(alpha). This function is deep connected to an utility function. We describe the Lie symmetry algebra of this PDE and provide a complete set of reductions of the PDE to ordinary differential equations (ODEs). In addition we are able to describe all types of functions g(alpha) for which the PDE admits an extended Lie group. Two of three special type functions lead to models introduced before by different authors, one is new. We clarify the connection between these three special models and the general model for trading strategies in illiquid markets. We study with the Lie group analysis the new special case of the PDE describing the self-financing strategies. In both, the general model and the new special model, we provide the optimal systems of subalgebras and study the complete set of reductions of the PDEs to different ODEs. In all cases we are able to provide explicit solutions to the new special model. In one of the cases the solutions describe power derivative products.
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