Convergence in Hölder norms for Markovian approximations of stochastic Volterra equations

Source 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 $C^0([0,T];L^p(Ω))$, $p\geq 2$, and for convolution kernels in $L^p(Ω;L^q(0,T))$, $q \in [1,p]$, and $C^β([0,T];L^p(Ω))$, $L^p(Ω;C^β([0,T]))$, where the range of the Hölder exponent $β\in (0,1]$ 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.