论文标题
在Itô-Taylor扩展上，用于随机微分方程，具有Markovian Switching及其应用于$γ\ in \ {n/2：n \ in \ Mathbb {n} \} $ - 订购方案
On Itô-Taylor expansion for stochastic differential equations with Markovian switching and its application in $γ\in\{n/2:n \in\mathbb{N}\}$-order scheme
论文作者
论文摘要
马尔可夫开关（SDEWM）的随机微分方程的系数还取决于马尔可夫链，并且没有区分此类功能相对于马尔可夫链的概念。特别是，这意味着SDEWMS的ITô-Taylor扩展不是随机微分方程（SDES）的Itô-Taylor扩展的直接扩展。此外，文献中无法使用SDEWM的高阶数值方案，也许是因为没有Itô-Taylor扩展。在本文中，首先，我们通过开发新技术来克服这些挑战，并在某些适当的规律性假设下得出SDEWMS的ITô-Taylor扩展。其次，我们证明了我们的第一个结果在iTô-taylor扩展中的应用在SDEWM的数值近似中。我们使用Itô-Taylor扩展为SDEWM提供了明确的方案，并表明我们方案的收敛性强度等于\ {n/2：N \ in \ in \ Mathbb {n} \} $ in \ {n/2：n \ in \ in \ n/2：n \ in \ in \ mathbb {n} \} $，在某些合适的Lipschitz-type条件下，在系数和衍生物上的条件下。值得一提的是，对Itô-Taylor扩展的设计和分析以及\ {n/2：n \ in \ Mathbb {n} \} $ - SDEWMS的订购方案变得更加复杂，并且由于连续的动态和离散事件的纠缠而涉及。最后，当马尔可夫链的状态是单胎集时，我们的结果与SDE上的相应结果一致。
The coefficients of the stochastic differential equations with Markovian switching (SDEwMS) additionally depend on a Markov chain and there is no notion of differentiating such functions with respect to the Markov chain. In particular, this implies that the Itô-Taylor expansion for SDEwMS is not a straightforward extension of the Itô-Taylor expansion for stochastic differential equations (SDEs). Further, higher-order numerical schemes for SDEwMS are not available in the literature, perhaps because of the absence of the Itô-Taylor expansion. In this article, first, we overcome these challenges and derive the Itô-Taylor expansion for SDEwMS, under some suitable regularity assumptions on the coefficients, by developing new techniques. Secondly, we demonstrate an application of our first result on the Itô-Taylor expansion in the numerical approximations of SDEwMS. We derive an explicit scheme for SDEwMS using the Itô-Taylor expansion and show that the strong rate of convergence of our scheme is equal to $γ\in\{n/2:n\in\mathbb{N}\}$ under some suitable Lipschitz-type conditions on the coefficients and their derivatives. It is worth mentioning that designing and analysis of the Itô-Taylor expansion and the $γ\in\{n/2:n\in\mathbb{N}\}$-order scheme for SDEwMS become much more complex and involved due to the entangling of continuous dynamics and discrete events. Finally, our results coincide with the corresponding results on SDEs when the state of the Markov chain is a singleton set.
