The Euler–Maruyama scheme for SDEs with irregular drift: convergence rates via reduction to a quadrature problem.

We study the strong convergence order of the Euler–Maruyama (EM) scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev–Slobodeckij-type regularity of order $\kappa \in (0,1)$ for the nonsmooth part of the drift, our analysis of the quadrature problem yields the convergence order $\min \{3/4,(1+\kappa )/2\}-\epsilon$ for the equidistant EM scheme (for arbitrarily small $\epsilon>0$). The cut-off of the convergence order at $3/4$ can be overcome by using a suitable nonequidistant discretization, which yields the strong convergence order of $(1+\kappa )/2-\epsilon$ for the corresponding EM scheme.

Seeking Primary Functions of Complete Differential with Six Variables

On the basis of quadrature problem of function with two, three, four and five variables, this paper discuss quadrature problem of function with six variables. Firstly, we have expounded the method of differential quadrature problem. Finally, we solve the primary functions of a specific complete differential with six variables.