市场冲击博弈中纳什均衡的高频极限

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英文标题：
《High-frequency limit of Nash equilibria in a market impact game with
  transient price impact》
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作者：
Alexander Schied, Elias Strehle, and Tao Zhang
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最新提交年份：
2017
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英文摘要：
  We study the high-frequency limits of strategies and costs in a Nash equilibrium for two agents that are competing to minimize liquidation costs in a discrete-time market impact model with exponentially decaying price impact and quadratic transaction costs of size $\\theta\\ge0$. We show that, for $\\theta=0$, equilibrium strategies and costs will oscillate indefinitely between two accumulation points. For $\\theta>0$, however, strategies, costs, and total transaction costs will converge towards limits that are independent of $\\theta$. We then show that the limiting strategies form a Nash equilibrium for a continuous-time version of the model with $\\theta$ equal to a certain critical value $\\theta^*>0$, and that the corresponding expected costs coincide with the high-frequency limits of the discrete-time equilibrium costs. For $\\theta\\neq\\theta^*$, however, continuous-time Nash equilibria will typically not exist. Our results permit us to give mathematically rigorous proofs of numerical observations made in Schied and Zhang (2013). In particular, we provide a range of model parameters for which the limiting expected costs of both agents are decreasing functions of $\\theta$. That is, for sufficiently high trading speed, raising additional transaction costs can reduce the expected costs of all agents.
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中文摘要：
我们研究了在一个离散时间市场影响模型中，两个代理在纳什均衡中的策略和成本的高频极限，该模型具有指数衰减的价格影响和大小为$\\θ\\ ge0$的二次交易成本，竞争以最小化清算成本。我们表明，当$\\θ=0$时，均衡策略和成本将在两个累积点之间无限振荡。然而，当$\\θ>0$时，策略、成本和总交易成本将趋于独立于$\\θ$的限制。然后，我们证明，对于$\\θ$等于某个临界值$\\θ^*>0$的连续时间模型，限制策略形成了一个纳什均衡，并且相应的预期成本与离散时间均衡成本的高频极限一致。然而，对于$\\θ\\ neq \\θ^*$，通常不存在连续时间纳什均衡。我们的结果允许我们对Schied和Zhang（2013）中的数值观测进行严格的数学证明。特别是，我们提供了一系列模型参数，对于这些参数，两个代理的极限预期成本都是$\\θ$的递减函数。也就是说，对于足够高的交易速度，提高额外的交易成本可以降低所有代理的预期成本。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Trading and Market Microstructure        交易与市场微观结构
分类描述：Market microstructure, liquidity, exchange and auction design, automated trading, agent-based modeling and market-making
市场微观结构，流动性，交易和拍卖设计，自动化交易，基于代理的建模和做市
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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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关键词：纳什均衡 Mathematical Quantitative observations accumulation