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In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions.

However, I have the following statement from a set of notes:

$$g(t,S_t) = E[(S_T-K)^{+}\mid F_t]$$

Where $g$ is some Borel function.

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How can this be justified? Our $f$ here is $(x-K)^+$ which is not bounded.

Thank you

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Actually the Markov property for Itô diffusions rather reads $E[f(X_{t+h})\mid F_t] = E[f(X_{t+h})\mid X_t]$. (In a second phase, the *stationarity* of the transitions of these processes then yields $E[f(X_{t+h})\mid X_t] = E^{X_t}[f(X_h)]$, that is, more explicitely, $E[f(X_{t+h})\mid X_t] = g(X_t)$ where, for every $x$, $g(x)=E^x[f(X_h)]$.) This holds for every Borel function $f$ such that $f(X_s)$ is integrable for every $s$.

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