Let $\{X_t,\mathcal{F}_t,0\le t\le T\}$ be a Feller process (or a strong Markov process), define the set of stopping times $$\mathcal{S}_{s,t}=\{\tau:s\le \tau\le t,\;\tau\;\text{is a stopping time}\}$$
I am trying to prove an important result in mathematical finance, that $$\underset{\tau\in\mathcal{S}_{t,T}}{\text{sup}}\mathbf{E}[e^{-r\tau}\varphi(X_\tau)|\mathcal{F}_t]=e^{-rt}f(T-t,X_t)\qquad (*)$$where $f(s,x)=\underset{\tau\in\mathcal{S}_{0,s}}{\text{sup}}\mathbf{E}^{x}[e^{-r\tau}\varphi(X_\tau)]$ and $r$ is a given positive number.
I have reduced the problem to the following:
- Let $\tau$ be an $\{\mathcal{F}_t^X\}$ stopping time such that $t\le \tau\le T$. If we have samples $\omega_1,\omega_2$ such that $X_s(\omega_1)=X_s(\omega_2)\;(\forall s\in [t,T])$, then $\tau(\omega_1)=\tau(\omega_2)$.
This statement looks similar to the renowned Galmarino's test, and I wonder if it is true. Please help by proving either the statement or the equality marked with (*). Thanks.