Differentiability of SDEs with drifts of super-linear growth
Differentiability of SDEs with drifts of super-linear growth
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DOI：
10.1214/18-ejp261
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发表时间：
2018-03
影响因子：
1.4
通讯作者：
P. Imkeller;Gonccalo dos Reis;William Salkeld
中科院分区：
文献类型：
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作者：
P. Imkeller;Gonccalo dos Reis;William Salkeld
我们用超线性增长（和随机系数）的漂移弥补了随机微分方程（SDE）文献中意想不到的空白，即我们证明了此类 SDE 的 Malliavin 和参数可微性。前者通过证明射线绝对连续性和随机G\^ateaux可微性来证明。这种方法使人们能够对概率而不是均方进行限制，从而绕过无界漂移中潜在的不可积误差项。这个问题与 Nualart 2006 年工作中的标准方法（针对此设置的引理 1.2.3）的困难密切相关。提供了几个例子来说明我们的结果的范围和范围。我们以参数可微性结束，并恢复连接导数以及 Bismut-Elworthy-Li 公式的表示。
We close an unexpected gap in the literature of stochastic differential equations (SDEs) with drifts of super linear growth (and random coefficients), namely, we prove Malliavin and Parametric Differentiability of such SDEs. The former is shown by proving Ray Absolute Continuity and Stochastic G\^ateaux Differentiability. This method enables one to take limits in probability rather than mean square which bypasses the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology from Nualart's 2006 work, Lemma 1.2.3 for this setting. Several examples illustrating the range and scope of our results are presented. We close with parametric differentiability and recover representations linking both derivatives as well as a Bismut-Elworthy-Li formula.