Maximal Inequalities for Fractional Lévy and Related Processes

Abstract

In this paper we study processes which are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding 'convoluted martingale' is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.in terms of the kernel and the L p (P)-norm of the driving martingale. Let us illustrate our results for the case of a fractional Lévy process

A generalised Itō formula for Lévy-driven Volterra processes

“…If we choose β = 0 and γ = 1−d we deduce that equation (4) in condition (v) is satisfied. Moreover, we infer by means of the mean value theorem that Hence, equation (5) in condition (v) holds with θ = 1 − d. That condition (vi) if fulfilled follows from Example 9 in [10].…”

“…Proof In the case τ > −∞ the assertion follows from Remark 5 in [10]. In the case τ = −∞ it follows from Theorem 8 in [10] with the choice ϕ q ′ (t) = |t| q ′ ∨ 1 for…”

“…(i) We start with the right continuity and therefore can assume t ∈ [0, T ). With ε > 0 as in assumption (i)b) we write where the finiteness follows from (10). Therefore, we can apply the DCT to obtain by means of (i)a) that where the finiteness follows from (12).…”

Nonparametric Estimation of Linear Multiplier for Stochastic Differential Equations Driven by Fractional Lévy Process With Small Noise

“…x 0 e −LT ) ≤ P ( inf for some positive constants C 4 and C 5 . It easy to see that 0 < 4ρ 2ρ−2d+1 < 4 for ρ > 2d−1 since 0 < d < 1 2 . Hence we obtain the following result implying the uniform consistency of the estimatorθ(t) as an estimator of θ(t) as ϵ → 0.…”

“…2 ) < ∞ and the process {L(t), t ∈ R} has no Brownian component. For d ∈ (0, 1 2 ), define the stochastic process…”