Convergence in Hölder norms for Markovian approximations of stochastic Volterra equations

RESUMO

We bound the difference between two stochastic Volterra processes with identical Lipschitz coefficients but different kernels. For non-convolution kernels, we establish estimates in $C^0([0,T];L^p(\Omega))$, $p\geq 2$, 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.