Convergence in Hölder norms for Markovian approximations of stochastic Volterra equations

Abstract

We bound the difference between two stochastic Volterra processes with identical Lipschitz coefficients but different kernels. For non-convolution kernels, we establish estimates in $C^0([0,T];L^p(\Omega ))$, $p\ge 2$, and for convolution kernels in $L^p(\Omega ;L^q(0,T))$, $q\in [1,p]$, and $C^{\beta }([0,T];L^p(\Omega ))$, $L^p(\Omega ;C^{\beta }([0,T]))$, where the range of the Hölder exponent $\beta \in (0,1]$ is the maximal permitted by the regularity of the processes. For the fractional kernel, we then construct Markovian approximations whose error we show to decay as $e^{-a\sqrt{N}}$ in the aforementioned norms, using an $N$-node quadrature based on sinc methods. Numerical experiments for the fractional Brownian motion verify our findings.

arXiv comment42 pages, 5 figures