广义相关扩散和泛函Copula算子的张量逼近
Tensor Approximation of Generalized Correlated Diffusions and Functional Copula Operators
摘要：
本文研究了多维相关马氏过程的半鞅分解、逼近和鞅表示。利用鞅方法给出了过程间相关性的一种新的解释。我们表明，它是可以表示，在连续和离散空间中，多维相关广义扩散是一个线性组合的过程，起源于分解的起始多维半鞅。这一结果不仅与现有的扩散近似和分解理论相一致，而且定义了多维广义扩散的无穷小生成元的一般表示，我们也将证明Copula密度依赖结构的规范。这个新结果提供了相关随机微分方程近似解的直接表示。我们证明了所提出的多维半鞅分解近似的收敛结果。
英文摘要：
We investigate aspects of semimartingale decompositions, approximation and the martingale representation for multidimensional correlated Markov processes. A new interpretation of the dependence among processes is given using the martingale approach. We show that it is possible to represent, in both continuous and discrete space, that a multidimensional correlated generalized diffusion is a linear combination of processes that originate from the decomposition of the starting multidimensional semimartingale. This result not only reconciles with the existing theory of diffusion approximations and decompositions, but defines the general representation of infinitesimal generators for both multidimensional generalized diffusions and as we will demonstrate also for the specification of copula density dependence structures. This new result provides immediate representation of the approximate solution for correlated stochastic differential equations. We demonstrate desirable convergence results for the proposed multidimensional semimartingales decomposition approximations.
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