可解扩散的对偶随机变换
Dual Stochastic Transformations of Solvable Diffusions
摘要：
我们提出了新的扩展的方法，用于构建几个家庭的可解一维时间均匀扩散的过渡密度是在解析封闭形式。我们的方法是基于所谓的扩散正则变换方法，结合光滑的单调映射和测量通过Doob-h变换的变化的双重应用。这就产生了新的多参数可解扩散，一般分为两个主要类别;第一类是指定的仿射（线性）漂移与各种所得的非线性扩散系数函数，而第二类允许几个规格的（一般是非线性）扩散系数与所得的非线性漂移函数。该理论适用于具有奇异和/或非奇异端点的扩散。作为本文结果的一部分，我们还提出了一个完整的边界分类和鞅特征的新发展的扩散族。
英文摘要：
We present new extensions to a method for constructing several families of solvable one-dimensional time-homogeneous diffusions whose transition densities are obtainable in analytically closed-form. Our approach is based on a dual application of the so-called diffusion canonical transformation method that combines smooth monotonic mappings and measure changes via Doob-h transforms. This gives rise to new multi-parameter solvable diffusions that are generally divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The theory is applicable to diffusions with either singular and/or non-singular endpoints. As part of the results in this paper, we also present a complete boundary classification and martingale characterization of the newly developed diffusion families.
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