For all $t\in \mathbb{R}$ let $h_t = \frac{1}{2} + \int_0^t v_s\cdot dB_s$ be an Itô process, where $B_s$ is a standard Brownian of $\mathbb{R}^d$ and $v_t$ an $\mathbb{R}^d$ valued adapted process, such that :
- For all $t$, $h_t \in [0,1]$ almost surely
- There exists $b_0>0$ such that $d[h]_t = \vert v_t \vert^2 \leq \frac{1}{b_0 +t}h_t$
I would like to have an upper-bound on the probability :
$$P(h_t\leq \lambda) $$ for $t>0$ and $0<\lambda < \frac{1}{2}$
