Copula函数描述的相关Lévy噪声非线性滤波
Nonlinear filtering with correlated Lévy noise characterized by copulas
摘要：
随机滤波的目的是重建关于未观察到的（随机）过程的信息，称为信号过程，给定该过程的某个噪声变换的当前可用观测。通常X和Y由布朗运动或跳跃（或Levy）过程驱动的随机微分方程建模。我们感兴趣的情况下，状态过程X和观察过程Y的扰动耦合Levy过程。更确切地说，L=（L_1，L_2）是一个二维Levy过程，其中相关结构用Levy Copula描述.我们导出了密度过程的Zakai方程，并建立了依赖于Copula和$L$的Zakai方程相应解的可解性的充分条件.特别地，我们给出了密度过程的存在唯一性条件，如果人们有兴趣估计像P（X（t）>a）这样的量，其中a是阈值。
英文摘要：
The objective in stochastic filtering is to reconstruct information about an unobserved (random) process, called the signal process, given the current available observations of a certain noisy transformation of that process. Usually X and Y are modeled by stochastic differential equations driven by a Brownian motion or a jump (or Levy) process. We are interested in the situation where both the state process X and the observation process Y are perturbed by coupled Levy processes. More precisely, L=(L_1,L_2) is a 2--dimensional Levy process in which the structure of dependence is described by a Levy copula. We derive the associated Zakai equation for the density process and establish sufficient conditions depending on the copula and $L$ for the solvability of the corresponding solution to the Zakai equation. In particular, we give conditions of existence and uniqueness of the density process, if one is interested to estimate quantities like P( X(t)>a), where a is a threshold.
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