英文标题：
《Optimization problem for a portfolio with an illiquid asset: Lie group
  analysis》
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作者：
Ljudmila A. Bordag, Ivan P. Yamshchikov
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最新提交年份：
2015
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英文摘要：
  Management of a portfolio that includes an illiquid asset is an important problem of modern mathematical finance. One of the ways to model illiquidity among others is to build an optimization problem and assume that one of the assets in a portfolio can not be sold until a certain finite, infinite or random moment of time. This approach arises a certain amount of models that are actively studied at the moment.   Working in the Merton\'s optimal consumption framework with continuous time we consider an optimization problem for a portfolio with an illiquid, a risky and a risk-free asset. Our goal in this paper is to carry out a complete Lie group analysis of PDEs describing value function and investment and consumption strategies for a portfolio with an illiquid asset that is sold in an exogenous random moment of time with a prescribed liquidation time distribution. The problem of such type leads to three dimensional nonlinear Hamilton-Jacobi-Bellman (HJB) equations. Such equations are not only tedious for analytical methods but are also quite challenging form a numeric point of view. To reduce the three-dimensional problem to a two-dimensional one or even to an ODE one usually uses some substitutions, yet the methods used to find such substitutions are rarely discussed by the authors.   We find the admitted Lie algebra for a broad class of liquidation time distributions in cases of HARA and log utility functions and formulate corresponding theorems for all these cases. We use found Lie algebras to obtain reductions of the studied equations. Several of similar substitutions were used in other papers before whereas others are new to our knowledge. This method gives us the possibility to provide a complete set of non-equivalent substitutions and reduced equations.
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中文摘要：
管理包含非流动资产的投资组合是现代数学金融学的一个重要问题。建立流动性不足模型的一种方法是建立一个优化问题，并假设投资组合中的一项资产在某个有限、无限或随机时刻之前无法出售。这种方法产生了一些目前正在积极研究的模型。在默顿的连续时间最优消费框架下，我们考虑了一个非流动资产、风险资产和无风险资产组合的优化问题。本文的目标是对描述价值函数和投资与消费策略的偏微分方程进行完整的李群分析，投资组合中的非流动资产在规定的清算时间分布的外生随机时刻出售。这类问题导致了三维非线性Hamilton-Jacobi-Bellman（HJB）方程。这样的方程不仅对分析方法来说是乏味的，而且从数值的角度来看也是相当具有挑战性的。为了将三维问题简化为二维问题，甚至简化为常微分方程，通常会使用一些代换，然而，作者很少讨论寻找此类代换的方法。在HARA和log效用函数的情况下，我们找到了一类广泛的清算时间分布的公认李代数，并对所有这些情况给出了相应的定理。我们使用发现的李代数来获得所研究方程的约化。之前有几篇论文使用过类似的替代方法，而其他的是我们所知的新方法。这种方法为我们提供了一套完整的非等价代换和约化方程的可能性。
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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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一级分类：Mathematics        数学
二级分类：Analysis of PDEs        偏微分方程分析
分类描述：Existence and uniqueness, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDE\'s, conservation laws, qualitative dynamics
存在唯一性，边界条件，线性和非线性算子，稳定性，孤子理论，可积偏微分方程，守恒律，定性动力学
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