Estimation of integrated volatility in continuous-time financial models with applications to goodness-of-fit testing

Properties of a specification test for the parametric form of the variance function in diffusion processes are discussed. The test is based on the estimation of certain integrals of the volatility function. If the volatility function does not depend on the variable x it is known that the corresponding statistics have an asymptotic normal distribution. However, most models of mathematical finance use a volatility function which depends on the state x. In this paper we prove that in the general case, where sigma depends also on x the estimates of integrals of the volatility converge stably in law to random variables with a non-standard limit distribution. The limit distribution depends on the diffusion process X-t itself and we use this result to develop a bootstrap test for the parametric form of the volatility function, which is consistent in the general diffusion model.