Chaos in Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastic Processes

Fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) arises in modeling of financial time series. FIGARCH is essentially governed by a system of nonlinear stochastic difference equations ${u_t}$ = ${z_t}$ $(1-\sum\limits_{j=1}^q \beta_j L^j)\sigma_{t}^2 = \omega+(1-\sum\limits_{j=1}^q \beta_j L^j - (\sum\limits_{k=1}^p \varphi_k L^k) (1-L)^d) u_t^2$, where $\omega\in$ R, and $\beta_j\in$ R are constant parameters, $\{u_t\}_{{t\in}^+}$ and $\{\sigma_t\}_{{t\in}^+}$ are the discrete time real valued stochastic processes which represent FIGARCH (p,d,q) and stochastic volatility, respectively. Moreover, L is the backward shift operator, i.e. $L^d u_t \equiv u_{t-d}$ (d is the fractional differencing parameter 0$