First passage times over stochastic boundaries for subdiffusive processes

Abstract

Let X = ( X t ) t ≥ 0 \mathbb {X}=(\mathbb {X}_t)_{t\geq 0} be the subdiffusive process defined, for any t ≥ 0 t\geq 0 , by X t = X ℓ t \mathbb {X}_t = X_{\ell _t} where X = ( X t ) t ≥ 0 X=(X_t)_{t\geq 0} is a Lévy process and ℓ t = inf { s > 0 ; K s > t } \ell _t=\inf \{s>0; \mathcal {K}_s>t \} with K = ( K t ) t ≥ 0 \mathcal {K}=(\mathcal {K}_t)_{t\geq 0} a subordinator independent of X X . We start by developing a composite Wiener-Hopf factorization to characterize the law of the pair ( T a ( b ) , ( X − b ) T a ( b ) ) (\mathbb {T}_a^{({b})}, (\mathbb {X} - {b})_{\mathbb {T}_a^{({b})}}) where T a ( b ) = inf { t > 0 ; X t > a + b t } \begin{equation*}\mathbb {T}_a^{({b})} = \inf \{t>0; \mathbb {X}_t > a+ {b}_t \} \end{equation*} with a ∈ R a \in \mathbb {R} and b = ( b t ) t ≥ 0 {b}=({b}_t)_{t\geq 0} a (possibly degenerate) subordinator independent of X X and K \mathcal {K} . We proceed by providing a detailed analysis of the cases where either X \mathbb {X} is a self-similar or is spectrally negative. For the later, we show the fact that the process ( T a ( b ) ) a ≥ 0 (\mathbb {T}_a^{({b})})_{a\geq 0} is a subordinator. Our proofs hinge on a variety of techniques including excursion theory, change of measure, asymptotic analysis and on establishing a link between subdiffusive processes and a subclass of semi-regenerative processes. In particular, we show that the variable T a ( b ) \mathbb {T}_a^{({b})} has the same law as the first passage time of a semi-regenerative process of Lévy type, a terminology that we introduce to mean that this process satisfies the Markov property of Lévy processes for stopping times whose graph is included in the associated regeneration set.