AN OPTIMAL MARKOVIAN QUANTIZATION ALGORITHM FOR MULTI-DIMENSIONAL STOCHASTIC CONTROL PROBLEMS
AN OPTIMAL MARKOVIAN QUANTIZATION ALGORITHM FOR MULTI-DIMENSIONAL STOCHASTIC CONTROL PROBLEMS
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DOI：
10.1142/s0219493704001231
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发表时间：
2004-12
影响因子：
1.1
通讯作者：
G. Pagès;H. Pham;J. Printems
中科院分区：
文献类型：
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作者：
G. Pagès;H. Pham;J. Printems
我们提出了一种基于最优量化的概率数值方法来解决一些多维随机控制问题，例如，在数学金融投资组合优化。然后，我们考虑一些控制扩散与大多数组件控制自由。非受控扩散部分的欧拉格式近似为离散时间过程，该离散时间过程通过在最佳拟合其动力学的网格上的最近邻投影获得。由此产生的过程也被设计为保留马尔可夫财产相对于过滤的欧拉计划。这种马尔可夫量化方法导致一个近似的控制问题，可以解决数值的动态规划公式。这种方法在更高维度上似乎是有希望的。一个priorLp误差界的陈述，我们表明，空间离散误差项是最小的在某些特定的网格。设计了一个简单的递归算法来计算这些最佳网格的基础上的Monte Carlo模拟感应。通过一些数值例子来解决均值-方差对冲问题。
We propose a probabilistic numerical method based on optimal quantization to solve some multi-dimensional stochastic control problems that arise, for example, in mathematical finance for portfolio optimization. We then consider some controlled diffusions with most components control free. The Euler scheme of the uncontrolled diffusion part is approximated by a discrete time process obtained by a nearest neighbor projection on some grids optimally fitted to its dynamics. The resulting process is also designed to preserve the Markov property with respect to the filtration of the Euler scheme. This Markovian quantization approach leads to an approximate control problem that can be solved numerically by the dynamic programming formula. This approach seems promising in higher dimension. A prioriLp-error bounds are stated and we show that the spatial discretization error term is minimal at some specific grids. A simple recursive algorithm is devised to compute these optimal grids by induction based on a Monte Carlo simulation. Some numerical illustrations are processed for solving a mean-variance hedging problem.