卡尔曼粒子滤波器在线参数估计及其在仿射模型中的应用
A Kalman particle filter for online parameter estimation with applications to affine models
摘要：
在本文中，我们解决的问题，估计的后验分布的连续时间状态空间模型与离散时间观测的算法，结合卡尔曼滤波和粒子滤波器的静态参数。该算法是半递归的，具有两层结构，外层提供未知参数的后验分布估计，内层提供状态变量的后验分布估计。该算法与递归嵌套粒子滤波器具有相似的结构，但与递归嵌套粒子滤波器两层都使用一个粒子滤波器不同，该算法引入了动态核对外层参数粒子进行采样，以获得更快的收敛速度。此外，该算法还实现了卡尔曼滤波器在内层，以减少计算时间。该算法也可用于参数突变的估计。证明了对于具有一定结构的状态空间模型，未知参数和状态变量的估计后验分布以$\mathcal{O}(N^{-\frac{1}{2}}+δ^{\frac{1}{2}})$阶速率收敛于$L_p$中的实际分布，其中$N$是外层参数的粒子数，$δ$是两个连续观测值之间的最大时间步长.我们目前的数值结果，该算法的实施，特别是我们实现了仿射利率模型，可能与随机波动率的算法，虽然该算法可以应用到更广泛的一类模型。
英文摘要：
In this paper we address the problem of estimating the posterior distribution of the static parameters of a continuous time state space model with discrete time observations by an algorithm that combines the Kalman filter and a particle filter. The proposed algorithm is semi-recursive and has a two layer structure, in which the outer layer provides the estimation of the posterior distribution of the unknown parameters and the inner layer provides the estimation of the posterior distribution of the state variables. This algorithm has a similar structure as the so-called recursive nested particle filter, but unlike the latter filter, in which both layers use a particle filter, this proposed algorithm introduces a dynamic kernel to sample the parameter particles in the outer layer to obtain a higher convergence speed. Moreover, this algorithm also implements the Kalman filter in the inner layer to reduce the computational time. This algorithm can also be used to estimate the parameters that suddenly change value. We prove that, for a state space model with a certain structure, the estimated posterior distribution of the unknown parameters and the state variables converge to the actual distribution in $L_p$ with rate of order $\mathcal{O}(N^{-\frac{1}{2}}+δ^{\frac{1}{2}})$, where $N$ is the number of particles for the parameters in the outer layer and $δ$ is the maximum time step between two consecutive observations. We present numerical results of the implementation of this algorithm, in particularly we implement this algorithm for affine interest models, possibly with stochastic volatility, although the algorithm can be applied to a much broader class of models.
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