Out of Rogers - Williams: Diffusions, Markov Processes and Martingales: Page 279:
Let $\{\mathbb{P}^x:x\in I\}$ be a regular canonical diffusion. For $y\in J:=[a,b]\subset I$ let $X_t$ denote the respective canonical regular diffusion process with law $\mathbb{P}^y$ and let $H\equiv\inf\{t:X_t\in J\} $. Now I don't understand the following equality:
$$\mathbb{E}^y(H|\mathscr{F}_{t+}^°) \equiv\mathbb{E}^{X_{t\wedge H}}[H]+(t\wedge H)$$
I think markov property is used:
$$\mathbb{E}^y(H\circ \theta_t|\mathscr{F}_{t+}^°) \equiv\mathbb{E}^{X_{t\wedge H}}[H]$$
but we don't have time-shift in the above equality, I moreover have no clue where $(t\wedge H)$ does come from.
Thank you for every help!