A note on $L_2$-estimates for stable integrals with drift

Let $X$ be of the form $X_t=\int_0^tb_sdZ_s+\int_0^ta_sds, t\ge 0,$ where $Z$ is a symmetric stable process of index $\alpha\in(1,2)$ with $Z_0=0$. We obtain various $L_2$-estimates for the process $X$. In particular, for $m\in\mathbb{N}, t\ge 0,$ and any measurable, nonnegative function $f$ we derive the inequality

${\mathbf E}\int_0^{t\land\tau_m(X)}\vert b_s\vert^{\alpha}f(X_s)ds\le N\Vert f\Vert _{2,m}.$

As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation $dX_t=b(X_{t-})dZ_t+a(X_t)dt$ for any initial value $x_0\in\mathbb{R}$.