## Stochastic volatility: approximation and goodness-of-fit test.(English)Zbl 1133.62065

Summary: Let $$X$$ be the unique solution started from $$x_0$$ of the stochastic differential equation $$dX_t= \theta(t,X_t)dB_t+ b(t,X_t)dt$$ with $$B$$ a standard Brownian motion. We consider an approximation of the volatility $$\theta(t,X_t)$$, the drift being considered as a nuisance parameter. The approximation is based on a discrete time observation of $$X$$ and we study its rate of convergence as a process. A goodness-of-fit test is also constructed.

### MSC:

 62M02 Markov processes: hypothesis testing 62M05 Markov processes: estimation; hidden Markov models 60J60 Diffusion processes 62G10 Nonparametric hypothesis testing 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 62G05 Nonparametric estimation 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 62P05 Applications of statistics to actuarial sciences and financial mathematics

### Keywords:

discrete time observations