Projective stochastic equations and nonlinear long memory

A projective moving average $\{X_t, t \in \mathbb{Z}\}$ is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel $Q$ and a linear combination of projections of $X_t$ on "intermediate" lagged innovation subspaces with given coefficients $\alpha_i, \beta_{i,j}$. The class of such equations include usual moving-average processes and the Volterra series of the LARCH model. Solvability of projective equations is studied, including a nested Volterra series representation of the solution $X_t$. We show that under natural conditions on $Q, \alpha_i, \beta_{i,j}$, this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process

Netiesiniai ilgos atminties modeliai

The thesis introduces new nonlinear models with long memory which can be used for modelling of financial returns and statistical inference. Apart from long memory, these models are capable to exhibit other stylized facts such as asymmetry and leverage. The processes studied in the thesis are defined as stationary solutions of certain nonlinear stochastic difference equations involving a given i.i.d. “noise”. Apart from solvability issues of these equations which are not trivial by itself, it is proved that their solutions exhibit long memory properties. Finally, for a particularly tractable nonlinear parametric model with long memory (GQARCH) we prove consistency and asymptotic normality of quasi-ML estimators

Modelling of nonlinear long memory

The thesis introduces new nonlinear models with long memory which can be used for modelling of financial returns and statistical inference. Apart from long memory, these models are capable to exhibit other stylized facts such as asymmetry and leverage. The processes studied in the thesis are defined as stationary solutions of certain nonlinear stochastic difference equations involving a given i.i.d. “noise”. Apart from solvability issues of these equations which are not trivial by itself, it is proved that their solutions exhibit long memory properties. Finally, for a particularly tractable nonlinear parametric model with long memory (GQARCH) we prove consistency and asymptotic normality of quasi-ML estimators