We denote $v>w$ for two vectors $v, w \in \mathbb{R}^{m}$, if $$v^{i}>w^{i}, \quad \forall 1\leq i \leq m,$$ and write componentwise expectation as $$\mathbb{E}[v | \mathcal{F}]=(\mathbb{E}[v^{1}|\mathcal{F}],\ \mathbb{E}[v^{2}|\mathcal{F}], \ \cdots, \ \mathbb{E}[v^{m}|\mathcal{F}])$$ for random vector $v \in \mathbb{R}^{m}$.
Suppose that firstly, on event $\{w_{t} >\vec{0}\}$, $$\mathbb{E}[w_{t+1}-w_{t} \ | \mathcal{F}_t] \leq \left({K \over \|K\|_{2}} -I\right)w_t$$ where $K$ is positive definite matrix and $\| \cdot \|$ is spectral norm. With assumption of bounded drift $a.e.$, $$|w_{t+1}-w_{t}|<R$$ for some $R>0$, and suppose that $w_{0}<\vec{0}$.
Then how can I get a hitting time estimate function $F$ for $\tau_{\rho}$ of $m$ dimensional stochastic process $w_{t}$ as; $$\tau_{\rho}:=\inf{\{t \geq 0: w_{t}>(\rho, \rho, \cdots, \rho)\}}$$$i.e.$, $$\mathbb{P}[\tau_{\rho}<t] \leq F(t, K, R, w_{0})$$