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首页> 外文期刊>Journal of Theoretical Probability >SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations
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SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations

机译：时变Lévy过程驱动的SDE及其相关的时间分形阶伪微分方程

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摘要

It is known that the transition probabilities of a solution to a classical Itô stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coefficients determined by the corresponding SDE. Time-fractional Kolmogorov-type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distributed order differential equations, the associated class of SDEs can be described within the framework of SDEs driven by semimartingales. These semimartingales are time-changed Lévy processes where the independent time-change is given respectively by the inverse of a single or mixture of independent stable subordinators. Examples are provided, including a fractional analogue of the Feynman–Kac formula.
机译：众所周知，经典Itô随机微分方程（SDE）的解的转移概率在弱意义上满足相关的Kolmogorov方程。 Kolmogorov方程是偏微分方程，其系数由相应的SDE确定。时间分数Kolmogorov型方程用于在许多领域中对复杂过程进行建模。但是，除少数特殊情况外，与这些方程式相关的SDE的类别是未知的。本文表明，在时间分数阶或更一般的时间分布阶微分方程的情况下，可以在由半mart驱动的SDE框架内描述SDE的相关类。这些半市场是时变的Lévy过程，其中独立的时间变化分别由独立的稳定从属者的一个或多个的倒数给出。提供了示例，包括Feynman–Kac公式的分数类似物。

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