Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients
Provides a theoretical convergence guarantee for explicit numerical methods applied to a class of SDEs with superlinear coefficients, addressing a known bottleneck in stochastic numerics.
The paper proposes explicit Euler schemes for SDEs with superlinearly growing drift and diffusion coefficients, proving convergence in probability and L^p with strong order 1/2 for uniform L^p-convergence.
A new class of explicit Euler schemes, which approximate stochastic differential equations (SDEs) with superlinearly growing drift and diffusion coefficients, is proposed in this article. It is shown, under very mild conditions, that these explicit schemes converge in probability and in $\mathcal{L}^p$ to the solution of the corresponding SDEs. Moreover, rate of convergence estimates are provided for $\mathcal{L}^p$ and almost sure convergence. In particular, the strong order $1/2$ is recovered in the case of uniform $\mathcal{L}^p$-convergence.