| The article first introduces fundamentals of stochastic differential equations driven by Brownian motions and also presents the concepts and properties of Euler scheme as the simplest and most classic discretization method. On this basis, we give out an Euler scheme with variable taking logarithms for such stochastic differential equations with positive solutions in this article. We then calculate the conditions for strong convergence of the scheme and prove that the Logarithm-Euler scheme has a strong convergence rate of1/2under an enhanced linear growth condition and bounded exponential moment, which supports the Logarithm-Euler scheme in theory. Some implementation in the new scheme against the classic Euler scheme are listed in the second half of the article to verify reduction of discretization error and get better simulation results. |
