标题： Wiener过程的分数幂存在性的数值证明 显示英文标题

标题： Numerical proof of existence of fractional powers of Wiener processes

摘要： 使用欧拉-马鲁亚米方法，我们证明了一类维纳过程存在，这些过程是通过计算它们的任意正幂得到的。 这可以通过一组适当的定义来实现，这些定义使这些过程在离散时间点上的实现具有意义并可计算。 对于整数次幂，伊藤微积分的标准结果仍然成立，因为我们只是对它们进行了扩展。 我们提供了大量样本的蒙特卡洛模拟结果。 我们通过在幂提升后从这些过程中恢复出我们最初的标准布朗运动来证明这些过程的存在性。 我们获得的数值结果完全一致，这是这些过程存在的明确证据。 这可能为随机积分和相关过程的概念推广铺平道路。 显示英文摘要

摘要： Using the Euler--Maruyama technique, we show that a class of Wiener processes exist that are obtained by computing an arbitrary positive power of them. This can be accomplished with a proper set of definitions that makes meaningful the realization at discrete times of these processes and make them computable. Standard results from It\=o calculus for integer powers hold as we are just extending them. We provide the results from a Monte Carlo simulation with a large number of samples. We yield evidence for the existence of these processes by recovering from them the standard Brownian motion we started with after power elevation. The perfect coincidence of the numerical results we obtained is a clear evidence of existence of these processes. This could pave the way to a generalization of the concepts of stochastic integral and relative process.