CARMA Approximations and Estimation - Lavagnini, Silvia (Frontiers in Applied Mathematics and Statistics, 2020)
CARMA($p,q$) processes are compactly defined through a stochastic differential equation (SDE) involving $q+1$ derivatives of the Lévy process driving the noise, despite this latter one has in general no differentiable paths. We replace the Lévy noise with a continuously differentiable process obtained by stochastic convolution. The solution of the new SDE is then a stochastic process which converges to the original CARMA in an $L^2$ sense. CARMA processes are largely applied in finance, and this article mathematically justifies their representation via SDE. Consequently, the use of numerical schemes in simulation and estimation by approximating derivatives with discrete increments, is also justified for every time step. This provides a link between discrete-time ARMA models and CARMA models in continuous-time. We then analyse an Euler method and find a rate of convergence proportional to the square of the time step for discretization and to the inverse of the convolution parameter. These must then be adjusted to get accurate approximations.