论文标题
梯形规则与粗糙积分的收敛
Convergence of trapezoid rule to rough integrals
论文作者
论文摘要
粗糙的路径技术使能够定义由信号$ x $驱动的随机微分方程的解决方案，这些方程$ x $不是半明星，其$ p $变化仅适用于$ p $的大值。在这种情况下，粗糙的积分通常是带有校正术语的Riemann-Stieltjes积分，有时被视为不自然。与那些有点人工纠正的术语相反，我们在本注中的努力是为由一般$ d $维的高斯流程驱动的粗糙积分制定梯形规则。也就是说，我们将近似Riemann的通用粗糙积分$ \ int y \，dx $，避免使用通常的高阶校正术语，使表达式更易于使用和更自然。我们的近似值适用于所有受控流程$ y $，以及广泛的高斯流程$ x $，包括带有hurst参数$ h> 1/4 $的分数布朗运动。作为梯形规则的推论，我们还考虑了$ \ int f（x）dx $积分的中点规则的收敛性。
Rough paths techniques give the ability to define solutions of stochastic differential equations driven by signals $X$ which are not semimartingales and whose $p$-variation is finite only for large values of $p$. In this context, rough integrals are usually Riemann-Stieltjes integrals with correction terms that are sometimes seen as unnatural. As opposed to those somewhat artificial correction terms, our endeavor in this note is to produce a trapezoid rule for rough integrals driven by general $d$-dimensional Gaussian processes. Namely we shall approximate a generic rough integral $\int y \, dX$ by Riemann sums avoiding the usual higher order correction terms, making the expression easier to work with and more natural. Our approximations apply to all controlled processes $y$ and to a wide range of Gaussian processes $X$ including fractional Brownian motion with a Hurst parameter $H>1/4$. As a corollary of the trapezoid rule, we also consider the convergence of a midpoint rule for integrals of the form $\int f(X) dX$.
