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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 28, Fasc. 1,
pages 1 - 19
 

STOCHASTIC VOLATILITY: APPROXIMATION AND GOODNESS-OF-FIT TEST

Mihai Gradinaru
Ivan Nourdin

Abstract: 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.

2000 AMS Mathematics Subject Classification: Primary: 62M05; Secondary: 62M02, 62L20, 60H10, 60F15, 60F05.

Key words and phrases: Non-parametric estimation, goodness-of-fit test, stochastic volatility, discrete time observation.

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