Testing the parametric form of the volatility in continuous time diffusion models - an empirical process approach

Abstract

In this paper we present two new tests for the parametric form of the variance function in diffusion processes dX_t = b(t,X_t)+\omega(t,X_t)dW_t. Our approach is based on two stochastic processes of the integrated volatility. We prove weak convergence of these processes to centered processes whose conditional distributions given the process (X_t)_{t\in[0,1]} are Gaussian. In the special case of testing for a constant volatility the limiting process is the standard Brownian bridge in both cases. As a consequence an asymptotic distribution free test (for the problem of testing for homoscedasticity) and bootstrap tests (for the problem of testing for a general parametric form) can easily be implemented. It is demonstrated that the new tests are more powerful with respect to Pitman alternatives than the currently available procedures for this problem. The asymptotic advantages of the new approach are also observed for realistic sample sizes in a simulation study, where the finite sample properties of a Kolmogorov-Smirnov test are investigated.