A note on 𝐿₂-estimates for stable integrals with drift

Abstract

Let X X be of the form X t = ∫ 0 t b s d Z s + ∫ 0 t a s d s , t ≥ 0 , X_t=\int _0^tb_sdZ_s+\int _0^ta_sds, t\ge 0, where Z Z is a symmetric stable process of index α ∈ ( 1 , 2 ) \alpha \in (1,2) with Z 0 = 0 Z_0=0 . We obtain various L 2 L_2 -estimates for the process X X . In particular, for m ∈ N , t ≥ 0 , m\in \mathbb N, t\ge 0, and any measurable, nonnegative function f f we derive the inequality \[ E ∫ 0 t ∧ τ m ( X ) | b s | α f ( X s ) d s ≤ N ‖ f ‖ 2 , m . {\mathbf E}\int _0^{t\land \tau _m(X)}|b_s|^{\alpha }f(X_s)ds\le N\|f\|_{2,m}. \] As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation d X t = b ( X t − ) d Z t + a ( X t ) d t dX_t=b(X_{t-})dZ_t+a(X_t)dt for any initial value x 0 ∈ R x_0\in \mathbb R .