论文标题
关于由分数布朗动作驱动的积分和与不连续集成的过程的积分离散率的急剧收敛速率
On sharp rate of convergence for discretisation of integrals driven by fractional Brownian motions and related processes with discontinuous integrands
论文作者
论文摘要
我们考虑由Hölder连续的高斯过程$ h> \ frac12 $驱动的随机积分的等距近似值，并具有涉及有界变异函数的不连续集成。我们在$ l^1 $ distance中给出了确切的收敛速度，并提供了不同驱动程序的示例。事实证明，与不连续的集成媒体相比，与$ n^{1-2H} $相比，收敛的确切速率比$ n^{1-2H} $成正比，这是不连续集体的最佳结果的两倍，并且与平滑整合体的情况相对应。我们方法的新颖性是，我们不使用涉及积分的乘法估计值，而是应用变量公式的更改以及在凸功能上的某些事实，使我们能够明确计算期望。
We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order $H>\frac12$ with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the $L^1$-distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to $n^{1-2H}$ that is twice better compared to the best known results in the case of discontinuous integrands, and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.
