On rate of convergence of finite difference scheme for degenerate parabolic-hyperbolic PDE with Levy noise

Abstract: In this article, we consider a semi discrete finite difference scheme for a degenerate parabolic-hyperbolic PDE driven by L\'evy noise in one space dimension. Using bounded variation estimations and a variant of classical Kru\v{z}kov's doubling of variable approach, we prove that expected value of the $L^1$-difference between the unique entropy solution and approximate solution converges at a rate of $(\Delta x)^\frac{1}{7}$, where $\Delta x$ is the spatial mesh size.