论文标题
粗糙微分方程的runge-kutta方法
Runge-Kutta methods for rough differential equations
论文作者
论文摘要
我们研究了用于粗糙微分方程的runge-kutta方法，这些方法可用于计算由比布朗运动更粗糙的过程驱动的随机微分方程的解决方案。我们使用泰勒级数表示（B系列）来用于数值方案和粗微分方程的解决方案，以确定为基础runge-kutta方法的局部误差所需顺序的条件。随后，我们证明了鉴于本地利率的全局错误的顺序。此外，我们通过引入基于基于粗微分方程驱动程序的增量来简化数值近似。这种简化的方法可以轻松实现，并且计算便宜，因为它是无导数的。我们提供了这种可实施的runge-kutta方法的全面表征，这意味着我们提供了必要和足够的代数条件，以便以最佳的收敛顺序，以防驱动程序，例如驱动程序，例如，hurst索引$ \ frac {1} {4} {4} {4} <H \ leq \ frac \ frac \ frac \ frac \ frac \ frac {1} $。我们通过进行数值实验来验证收敛的理论速率来结束本文。
We study Runge-Kutta methods for rough differential equations which can be used to calculate solutions to stochastic differential equations driven by processes that are rougher than a Brownian motion. We use a Taylor series representation (B-series) for both the numerical scheme and the solution of the rough differential equation in order to determine conditions that guarantee the desired order of the local error for the underlying Runge-Kutta method. Subsequently, we prove the order of the global error given the local rate. In addition, we simplify the numerical approximation by introducing a Runge-Kutta scheme that is based on the increments of the driver of the rough differential equation. This simplified method can be easily implemented and is computational cheap since it is derivative-free. We provide a full characterization of this implementable Runge-Kutta method meaning that we provide necessary and sufficient algebraic conditions for an optimal order of convergence in case that the driver, e.g., is a fractional Brownian motion with Hurst index $\frac{1}{4} < H \leq \frac{1}{2}$. We conclude this paper by conducting numerical experiments verifying the theoretical rate of convergence.
