Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions
Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions
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具有局部平滑系数的 SDE 的平滑度和密度渐近估计及其在平方根型扩散中的应用
DOI：
10.1214/10-aap717
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发表时间：
2011
影响因子：
1.8
通讯作者：
S. Marco
中科院分区：
文献类型：
--
作者：
S. Marco
我们研究 SDE 解的密度平滑度，其系数仅在开放域 $D$ 上平滑且非简并。我们证明 $D$ 上存在平滑密度，并给出该密度的上限。在一些附加条件下（主要处理系数及其导数的增长），我们制定了适合于获得大状态变量值的密度渐近估计（“尾部”估计）的上限。这些结果具体说明并扩展了 Kusuoka 和 Stroock 的一些结果 [J.事实。科学。大学。东京教派。 IA 数学。 32 (1985) 1--76]，但我们的方法有很大不同，并且基于受 Fournier [Electron.32 (1985) 1--76] 启发的估计傅里叶变换的技术。 J. 普罗巴布. 13 (2008) 135--156] 和 Bally [局部平滑定律的分部积分公式及其在跳跃方程中的应用 I (2007) 瑞典皇家科学院]。这项研究的动机是依赖于具有非 Lipschitz 系数的 SDE 的现有金融证券模型。事实上，我们将我们的结果应用于系数取决于状态变量的平方根型扩散（CIR 或 CEV），即基于 Malliavin 演算的密度估计标准技术不适用的情况。我们建立了平滑密度的存在性，对此我们给出指数估计并研究原点（奇点）处的行为。
We study smoothness of densities for the solutions of SDEs whose coefficients are smooth and nondegenerate only on an open domain $D$. We prove that a smooth density exists on $D$ and give upper bounds for this density. Under some additional conditions (mainly dealing with the growth of the coefficients and their derivatives), we formulate upper bounds that are suitable to obtain asymptotic estimates of the density for large values of the state variable ("tail" estimates). These results specify and extend some results by Kusuoka and Stroock [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985) 1--76], but our approach is substantially different and based on a technique to estimate the Fourier transform inspired from Fournier [Electron. J. Probab. 13 (2008) 135--156] and Bally [Integration by parts formula for locally smooth laws and applications to equations with jumps I (2007) The Royal Swedish Academy of Sciences]. This study is motivated by existing models for financial securities which rely on SDEs with non-Lipschitz coefficients. Indeed, we apply our results to a square root-type diffusion (CIR or CEV) with coefficients depending on the state variable, that is, a situation where standard techniques for density estimation based on Malliavin calculus do not apply. We establish the existence of a smooth density, for which we give exponential estimates and study the behavior at the origin (the singular point).