Abstract
CARMA($p,q$) processes are compactly defined through a stochastic differential equation (SDE) involving $q+1$ derivatives of the Levy process driving the noise, despite this latter one has in general no differentiable paths. We replace the Levy 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.
Highlights
Continuous-time autoregressive moving-average processes, CARMA in short, represent the continuous-time version of the well-known ARMA models
In Benth et al [6] and Benth and Benth [7], the authors exploit the link between discrete-time ARMA and continuous-time ARMA to model the observed processes in a continuous manner, starting from discrete-time observations-based estimates, in the context of energy markets and weather derivatives
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 the latter having in general no differentiable paths
Summary
Continuous-time autoregressive moving-average processes, CARMA in short, represent the continuous-time version of the well-known ARMA models. 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 the latter having in general no differentiable paths Numerical schemes, such as the Euler scheme, are applied both for simulations and estimation of CARMA processes. Since CARMA processes have significant application within finance, from the modeling of the electricity spot price to the modeling of wind speed and temperature, with the present study, we justify its mathematical formulation and application in a rigorous way This may open for further applications in the context of sensitivity analysis with respect to paths of the driving noise process. Appendix A in Supplementary Material contains the proofs of the main results
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
