最优滤波导数的粒子逼近的一致稳定性
Uniform Stability of a Particle Approximation of the Optimal Filter Derivative
摘要：
序贯蒙特卡罗方法，也称为粒子方法，是一组广泛使用的计算工具，用于非线性非高斯状态空间模型的推断。在许多应用中，可能需要计算最优滤波器相对于状态空间模型的静态参数的灵敏度或导数;例如，为了获得感兴趣的最大似然模型参数，或者为了计算最优控制问题中的最优控制器。在Poyiadjis等人[2011]中，提出了一种计算滤波器导数的原始粒子算法，并且使用数值示例表明，粒子估计在数值上是稳定的，因为它不会随着时间的推移而恶化。在本文中，我们证实了这一说法与详细的理论研究。Lp界和中心极限定理，这种粒子近似的过滤器的衍生物。进一步证明了在混合条件下，这些Lp界和由中心极限定理刻画的渐近方差关于时间指标是一致有界的.我们demonstrate性能的理论预测与几个数值例子。我们也使用粒子近似的过滤器衍生物进行在线最大似然参数估计的随机波动率模型。
英文摘要：
Sequential Monte Carlo methods, also known as particle methods, are a widely used set of computational tools for inference in non-linear non-Gaussian state-space models. In many applications it may be necessary to compute the sensitivity, or derivative, of the optimal filter with respect to the static parameters of the state-space model; for instance, in order to obtain maximum likelihood model parameters of interest, or to compute the optimal controller in an optimal control problem. In Poyiadjis et al. [2011] an original particle algorithm to compute the filter derivative was proposed and it was shown using numerical examples that the particle estimate was numerically stable in the sense that it did not deteriorate over time. In this paper we substantiate this claim with a detailed theoretical study. Lp bounds and a central limit theorem for this particle approximation of the filter derivative are presented. It is further shown that under mixing conditions these Lp bounds and the asymptotic variance characterized by the central limit theorem are uniformly bounded with respect to the time index. We demon- strate the performance predicted by theory with several numerical examples. We also use the particle approximation of the filter derivative to perform online maximum likelihood parameter estimation for a stochastic volatility model.
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