2025/10/07 by Frederik Krabbe, Krabbe, Frederik · 1 citation
Mathematics · Economics, Econometrics and Finance · #Statistical Methods and Inference #Financial Risk and Volatility Modeling #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2510.05716
Bougerol (1993) and Straumann and Mikosch (2006) gave conditions under which there exists a unique stationary and ergodic solution to the stochastic difference equation Yt \overseta.s.= Φt (Yt-1), t ∈ ℤ where (Φt)t ∈ ℤ is a sequence of stationary and ergodic random Lipschitz continuous functions from (Y,|| ⋅ ||) to (Y,|| ⋅ ||) where (Y,|| ⋅ ||) is a complete subspace of a real or complex separable Banach space. In the case where (Y,|| ⋅ ||) is a real or complex separable Banach space, Straumann and Mikosch (2006) also gave conditions under which any solution to the stochastic difference equation Yt \overseta.s.= Φt (Yt-1), t ∈ ℕ with Y0 given where (Φt)t ∈ ℕ is only a sequence of random Lipschitz continuous functions from (Y,|| ⋅ ||) to (Y,|| ⋅ ||) satisfies γt || Yt - Yt || \overseta.s.→ 0 as t → ∞ for some γ> 1. In this note, we give slightly different conditions under which this continues to hold in the case where (Y,|| ⋅ ||) is only a complete subspace of a real or complex separable Banach space by using close to identical arguments as Straumann and Mikosch (2006).