We consider 1-d process $X$ $$ X(t) = b t + J_{t} + M_{t}$$ where $b$ is constant, $M$ is a continuous martingale process with $M(0) = 0$, and $J$ is a symmestric $\alpha$-stable process with its Levy symbol $\eta(u) = - |u|^{\alpha}$ for some constant $\alpha\in [1,2)$. Its Levy measure is $\nu(y) = \frac{1}{|y|^{1+\alpha}}.$ Consider $$\tau = \inf\{t>0: X(t) <0\}.$$ I want to know if the following claim is true.
[Claim] $\tau = 0$ almost surely.
[Discussion1] It is true if $M \equiv 0$. By Theorem VIII.5 of [Bertoin 1996], it yields that $$\lim\inf_{t\to 0+} \frac{J_{t}}{(t \ln 1/t)^{1/\alpha}} = - \infty.$$ Therefore, we also have $$\lim\inf_{t\to 0+} \frac{X(t)}{(t \ln 1/t)^{1/\alpha}} = -\infty,$$ which implies $\tau = 0$.
[Discussion1'] If $M$ is independent to $J$, then the result is trivially extended.
[Discussion2] It is true if $M = \int_{0}^{t} \sigma_{s} dB_{s}$ for some Brownian motion with $\sigma_{s} \in [1, 2]$ for all $s\ge 0$. In this case, with $$ \theta(t) = \int_{0}^{t} |\sigma(s)|^{2} ds,$$ there exists $\mathbb P$-Brownian motion $\hat B$ with respect time changed filtration such that $$M(t) = \hat B(\theta (t)), \ \mathbb P-a.s.$$ By Law of Iterated Logarithm , we have $$\lim\inf_{t \to 0+} \frac{\hat B(t)}{\sqrt{t}} = -\infty.$$ Together with the fact $1 \le \frac{\theta(t)}{t} \le 4,$ we obtain $$\lim\inf_{t\to 0^{+}} \frac{M(t)}{\sqrt{t}} = -\infty.$$
[Discussion 2'] It is true if $M = \int_{0}^{t} \sigma_{s} dB_{s}$ with $\sigma_{0} \neq 0$ and $\sigma$ is Cadlag process by localizing argument.
[Q] My question is if $\tau = 0$ holds for general continuous martingale? I am particularly interested in the case with $M_t = \int_{0}^{t} \sigma_{s} dB_{s}$ with $\sigma$ starting from $\sigma(0) = 0$ and possibly correlated to $J$. I need a proof, but also appreciate any clue or related reference you provided.
