Efficient simulation of a new class of Volterra-type SDEs

We propose a new theoretical framework that exploits convolution kernels to transform a Volterra-type path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. Remarkably, it is also possible to go back, i.e., the transformation is reversible. We discuss existence and path-wise regularity of solutions for our class of stochastic differential equations. In the fractional kernel case, when $H\in (0,\frac{1}{2})$, where $H$ is the Hurst coefficient, we propose a numerical simulation scheme which exhibits a remarkable strong convergence rate of order $1\mathrm{/}2$, which constitutes a bold improvement when compared with the performance of available Euler schemes, whose strong rate of convergence is $H$.