Optimal Control of Trading Algorithms: A General Impulse Control Approach
Optimal Control of Trading Algorithms: A General Impulse Control Approach
复制标题
交易算法的最优控制：通用脉冲控制方法
DOI：
10.1137/090777293
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
Charles
中科院分区：
文献类型：
--
作者：
B. Bouchard;Ngoc;Charles
我们提出了一个通用的框架，为日内交易的基础上控制的交易算法。给定一组通用参数化算法（必须由控制器事先指定），我们的目标是优化它们启动的日期$（\tau_i）_i$、交易周期的长度$（\delta_i）_i $以及在时间间隔$[\tau_i，\tau_i + \delta_i）$内保持的参数$（{\cal E}_i）_i$的值。这为金融代理人提供了一个决策工具，用于选择在交易期的不同阶段应使用哪种算法（以及哪组参数和时间长度）。从数学的角度来看，这就产生了一个非经典脉冲控制问题，其中不仅区域${\cal E}_i$，而且周期$[\tau_i，\tau_i+ \delta_i）$必须由控制器在脉冲时间$\tau_i$确定。我们采用Bouchard和Touzi [SIAM J. Control Optim.，49（2011），pp. [948-962]给出了具有适当边界条件的偏微分方程组的不连续粘性解的相关值函数的一个特征，并证明了一个比较原理。我们还提出了一个数值方案的决议，上述系统，并表明它是收敛的。最后，我们提供了一个简单的例子，应用到一个问题的最佳股票交易的非线性市场影响函数。这表明参数如何适应市场。
We propose a general framework for intraday trading based on the control of trading algorithms. Given a set of generic parameterized algorithms (which have to be specified by the controller ex-ante), our aim is to optimize the dates $(\tau_i)_i$ at which they are launched, the length $(\delta_i)_i$ of the trading period, and the value of the parameters $({\cal E}_i)_i$ kept during the time interval $[\tau_i,\tau_i + \delta_i)$. This provides the financial agent a decision tool for selecting which algorithm (and for which set of parameters and time length) should be used in the different phases of the trading period. From the mathematical point of view, this gives rise to a nonclassical impulse control problem where not only the regime ${\cal E}_i$ but also the period $[\tau_i,\tau_i+ \delta_i)$ have to be determined by the controller at the impulse time $\tau_i$. We adapt the weak dynamic programming principle of Bouchard and Touzi [SIAM J. Control Optim., 49 (2011), pp. 948-962] to our context to provide a characterization of the associated value function as a discontinuous viscosity solution of a system of partial differential equations with appropriate boundary conditions, for which we prove a comparison principle. We also propose a numerical scheme for the resolution of the above system and show that it is convergent. We finally provide a simple example of application to a problem of optimal stock trading with a nonlinear market impact function. This shows how parameters adapt to the market.