On maximal inequalities for stable stochastic integrals (Q867115)

Let \(Z= (Z_t)_{t\geq 0}\) be a càdlàg real stable process of index \(\alpha\in(0,2)\) without component and \(H=(H_t)_{t\geq 0}\) be a sufficiently integrable predictable càdlàg process. In this paper the maximal inequalities in large range \[ P\Biggl(\sup_{0\leq s\leq t}\,\Biggl| \int^s_0 H_s dZ_s\Biggr|\geq x\Biggr)\leq {K\| H\|^\alpha_{(\alpha+ p,t)}\over x^\alpha},\quad p> 2-\alpha \] and small range (if \(Z\) is symmetric) \[ \begin{multlined} P\Biggl(\sup_{0\leq s\leq t}\,\Biggl|\int^s_0 H_s dZ_s\Biggr|\geq x\Biggr)\leq {2c\over\alpha} \Biggl({x\over \lambda\| H\|_{L^\infty(\Omega, L^\alpha([0, t]))}^0}\Biggr)^{{\alpha\over\alpha- 1}}+\\ \exp\Biggl({-\lambda\log(1+{(2-\alpha) \lambda\over 2c})\over 2}\Biggl({x\over \lambda\| H\|_{L^\infty(\Omega, L^\alpha([0, t]))^0}}\Biggr)^{{\alpha\over\alpha- 1}}\Biggr)\end{multlined} \] for stable stochastic integrals \(H\cdot Z= (\int^t_0 H_s dZ_s)_{t\geq 0}\) and \(\alpha\in(1,2)\) are proved. The results are used to estimate first passage times of symmetric stable processes above several positive continuous curves.