论文标题
混合粗糙微分方程的精确拉普拉斯近似
Precise Laplace approximation for mixed rough differential equation
论文作者
论文摘要
这项工作着重于与AS的混合粗糙路径驱动的粗糙微分方程（RDE）的拉普拉斯近似。首先，基于从混合分数布朗运动（FBM）提起的几何粗糙路径，给出了对RDE解决方案的第一级路径定律的雪橇型大偏差原理（LDP）。由于混合粗糙路径的特殊性，执行拉普拉斯近似的主要困难是证明了Itômap的Hessian Matrix的Hilbert-Schmidt特性，该特性限制在混合FBM的Cameron-Martin空间上。为此，我们将Cameron-Martin空间嵌入到更大的希尔伯特空间中，然后Hessian可以计算。随后，显示了Hessian的概率表示。最后，构建了拉普拉斯近似值，这在指数级别上断言了更精确的渐近渐近物。
This work focuses on the Laplace approximation for the rough differential equation (RDE) driven by mixed rough path with as . Firstly, based on geometric rough path lifted from mixed fractional Brownian motion (fBm), the Schilder-type large deviation principle (LDP) for the law of the first level path of the solution to the RDE is given. Due to the particularity of mixed rough path, the main difficulty in carrying out the Laplace approximation is to prove the Hilbert-Schmidt property for the Hessian matrix of the Itô map restricted on the Cameron-Martin space of the mixed fBm. To this end, we imbed the Cameron-Martin space into a larger Hilbert space, then the Hessian is computable. Subsequently, the probability representation for the Hessian is shown. Finally, the Laplace approximation is constructed, which asserts the more precise asymptotics in the exponential scale.
