Extremes of stochastic volatility models
The authors study the extreme of the law of a discrete time stationary stochastic volatility process. Let \(X_t= \ln Y^2_t= \alpha_t+ \ln\xi^2_t\), where \(\alpha_t= \sum^\infty_{j=0} \theta_jZ_{t-j}\), \((\xi_t)_t\) are i.i.d. \(N(0;1)\), \((Z_t)_t\) are i.i.d. \(N(0;\sigma^2_Z)\); the main theorem states that there exist \(a_n\) and \(b_n\) such that \(P(a_n(M_n- b_n)\leq x)\to \exp(- e^{-x})\) as \(n\to\infty\), where \(M_n= \max(X_1,\dots, X_n)\). Explicit expressions of \(a_n\), \(b_n\) are given.
scientific article; zbMATH DE number 1319828
scientific article