Indirect Inference for L\'evy-driven continuous-time GARCH models

We advocate the use of an Indirect Inference method to estimate the parameter of a COGARCH(1,1) process for equally spaced observations. This requires that the true model can be simulated and a reasonable estimation method for an approximate auxiliary model. We follow previous approaches and use linear projections leading to an auxiliary autoregressive model for the squared COGARCH returns. The asymptotic theory of the Indirect Inference estimator relies {on a uniform SLLN and asymptotic normality of the parameter estimates of the auxiliary model, which require continuity and differentiability of the COGARCH process} with respect to its parameter and which we prove via Kolmogorov's continuity criterion. This leads to consistent and asymptotically normal Indirect Inference estimates under moment conditions on the driving L\'evy process. A simulation study shows that the method yields a substantial finite sample bias reduction compared to previous estimators.