Suppose that $X_t$ and $\tilde{X}_t$ are processes defined on the stochastic basis $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ such that
- for any time $t>0$, $1>p_t>0$, where $p_t$ is defined by $$ p_t\triangleq \mathbb{P}\left( \int_0^t X_s ds> \int_0^t \tilde{X}_s ds \right) , $$ (assuming the integrals are well defined).
- $X_0<\tilde{X}_0$ $\mathbb{P}$-a.s.
Define the hitting time $\tau$ by $$ \tau \triangleq \inf\left\{ t>0 : X_t >\tilde{X}_t \right\}. $$
How can we compute $\tau$?