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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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