  We propose a new theoretical framework that exploits convolution kernels to
transform a Volterra path-dependent (non-Markovian) stochastic process into a
standard (Markovian) diffusion process. This transformation is achieved by
embedding a Markovian "memory process" within the dynamics of the non-Markovian
process. We discuss existence and path-wise regularity of solutions for the
stochastic Volterra equations introduced and we provide a financial application
to volatility modeling. We also propose a numerical scheme for simulating the
processes. The numerical scheme exhibits a strong convergence rate of 1/2,
which is independent of the roughness parameter of the volatility process. This
is a significant improvement compared to Euler schemes used in similar models.
  We propose a new theoretical framework that exploits convolution kernels to
transform a Volterra path-dependent (non-Markovian) stochastic process into a
standard (Markovian) diffusion process. This transformation is achieved by
embedding a Markovian "memory process" (the goldfish) within the dynamics of
the non-Markovian process (the elephant). Most notably, it is also possible to
go back, i.e., the transformation is reversible. We discuss existence and
path-wise regularity of solutions for the stochastic Volterra equations
introduced and we propose a numerical scheme for simulating the processes,
which exhibits a remarkable convergence rate of $1/2$. In particular, in the
fractional kernel case, the strong convergence rate is independent of the
roughness parameter, which is a positive novelty in contrast with what happens
in the available Euler schemes in the literature in rough volatility models.
