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Limit theorems for discretely observed stochastic volatility models. (English) Zbl 0916.60075
The authors consider the following two-dimensional diffusion model for stock prices with stochastic volatility: $$Y_0 = 0$$, $$dY_t = \varphi(V_t) dt + V_t^{1/2} dB_t$$ and $$V_0 = \eta$$, $$dV_t = b(V_t) dt + a(V_t) dW_t$$ where $$B$$ and $$W$$ are independent Wiener processes. $$Y$$ is interpreted as the process of logarithmic stock prices. $$V$$ is assumed to be a strictly positive, ergodic, stationary diffusion. As typical for the finance background, the volatility process $$V$$ is unobservable, while one has a discrete series $$Y_{t_1}, \ldots, Y_{t_n}$$ of observations for $$Y$$. The authors’ aim is to derive information about the stationary distribution of the hidden process $$V$$ using the given discrete data for $$Y$$. They assume that the observation times are regularly spaced, $$t_i = i \Delta$$, and show that the empirical distribution $$\hat{P}$$ of the renormalized increments $$X_i = (Y_{t_i}-Y_{t_{i-1}})/\Delta^{1/2}$$ converges to a variance mixture of Gaussian laws $$P_\pi$$ as the length of the sampling interval $$\Delta$$ approaches zero and the total observation time $$t_n = n \Delta$$ tends to infinity. Moreover, $$P_\pi$$ is identified as the distribution of a random variable $$X = \varepsilon \eta^{1/2}$$ where $$\varepsilon$$ is standard normal distributed and independent of $$\eta=V_0$$. If the asymptotic framework is such that $$n \Delta^2$$ tends to zero, the authors also show that a central limit theorem holds.

##### MSC:
 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 91B28 Finance etc. (MSC2000) 60F05 Central limit and other weak theorems
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