论文标题
分数布朗运动的大偏差的几何光学元件
Geometrical optics of large deviations of fractional Brownian motion
论文作者
论文摘要
最近已经显示，最佳波动方法 - 基本上是几何光学，为布朗运动的巨大偏差提供了宝贵的见解。在这里，我们将几何光学形式主义扩展到双面的$ - \ infty <t <t <\ infty $，该线上的分数布朗运动（FBM），该线将其“将”“推到”大型偏差方向上，通过强加的约束。我们在可用的确切解决方案的三个示例上测试形式主义：FBM的两点概率和三点概率分布以及在指定的时间间隔内在FBM下的区域分布。然后，我们通过评估以下分布的大型驱动尾巴来将形式主义应用于几个以前未解决的问题：（i）的第一阶段时间，（ii）的最大和（iii）区域的最大和（iii），布朗桥和分数布朗尼桥和（iv）fbm的第一站区域分布的（iv）。几何光学计算的固有部分是确定最佳路径，这是该过程中最有可能实现的，该过程主导了条件过程的概率分布。由于FBM的非马克维亚性质，FBM的最佳路径受到有限间隔的约束$ 0 <t \ leq t $，涉及过去的$ - \ infty <t <0 $和未来的$ t <t <t <\ t <\ infty $。
It has been shown recently that the optimal fluctuation method -- essentially geometrical optics -- provides a valuable insight into large deviations of Brownian motion. Here we extend the geometrical optics formalism to two-sided, $-\infty<t<\infty$, fractional Brownian motion (fBM) on the line, which is "pushed" to a large deviation regime by imposed constraints. We test the formalism on three examples where exact solutions are available: the two- and three-point probability distributions of the fBm and the distribution of the area under the fBm on a specified time interval. Then we apply the formalism to several previously unsolved problems by evaluating large-deviation tails of the following distributions: (i) of the first-passage time, (ii) of the maximum of, and (iii) of the area under, fractional Brownian bridge and fractional Brownian excursion, and (iv) of the first-passage area distribution of the fBm. An intrinsic part of a geometrical optics calculation is determination of the optimal path -- the most likely realization of the process which dominates the probability distribution of the conditioned process. Due to the non-Markovian nature of the fBm, the optimal paths of a fBm, subject to constraints on a finite interval $0<t\leq T$, involve both the past $-\infty<t<0$ and the future $T<t<\infty$.
