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$<$d$<$1). In this work, we have studied the chaoticity properties of FIGARCH (p,d,q) processes by computing mutual information, correlation dimensions, FNNs (False Nearest Neighbour), the Lyapunov exponents, and for both the stochastic difference equation given above and for the financial time series. We have observed that maximal Lyapunov exponents are negative, therefore, it can be suggested that FIGARCH (p,d,q) is not deterministic chaotic process.