论文标题
随机缝纫引理的扩展以及对分数随机演算的应用
An extension of the stochastic sewing lemma and applications to fractional stochastic calculus
论文作者
论文摘要
我们给出了Lê的随机缝纫引理的扩展[Electron。 J. Probab。 25：1-55，2020]。随机缝纫引理在$ l_m的Riemann类型中证明了融合$ \ sum _ {[s，t] \ inπ} a_ {s，t} $，用于在某些情况下，在某些情况下，在某些情况下，适用于$ a_ { $ \ MATHCAL {F} _s $。我们的扩展名代替了给定$ \ Mathcal f_s $的条件期望，而给定的$ \ Mathcal f_v $对于$ v <s $，它允许使用$ a_ {s，t} $和$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \} _v $的渐近脱字属性。我们提供了三种应用，其中Lê的随机缝纫引理似乎不足。首先是证明在低规律性假设下，随机积分沿分数布朗尼动作的Itô或Stratonovich近似值的收敛。第二个是通过离散化获得分数布朗动作的本地时代的新表示。第三个是改善对由布朗运动驱动的随机微分方程的扩散系数的规律性假设，这些方程式是布朗运动的路径唯一性和强烈存在的。
We give an extension of Lê's stochastic sewing lemma [Electron. J. Probab. 25: 1 - 55, 2020]. The stochastic sewing lemma proves convergence in $L_m$ of Riemann type sums $\sum _{[s,t] \in π} A_{s,t}$ for an adapted two-parameter stochastic process $A$, under certain conditions on the moments of $A_{s,t}$ and of conditional expectations of $A_{s,t}$ given $\mathcal {F}_s$. Our extension replaces the conditional expectation given $\mathcal F_s$ by that given $\mathcal F_v$ for $v < s$, and it allows to make use of asymptotic decorrelation properties between $A_{s,t}$ and $\mathcal {F}_v$ by including a singularity in $(s-v)$. We provide three applications for which Lê's stochastic sewing lemma seems to be insufficient.The first is to prove the convergence of Itô or Stratonovich approximations of stochastic integrals along fractional Brownian motions under low regularity assumptions. The second is to obtain new representations of local times of fractional Brownian motions via discretization. The third is to improve a regularity assumption on the diffusion coefficient of a stochastic differential equation driven by a fractional Brownian motion for pathwise uniqueness and strong existence.
