Let stochastic process $(X_t)_{t\geq 0}$ be the unique strong solution of the SDE, and let $c\in\mathbb{R}$: $$ dX_t=\mathbb{1}_{\{X_t>c\}}dt+\sigma dB_t \quad \text{with} \quad X_0=x\in\mathbb{R}. $$ Define process $A_t=\int_0^t\mathbb{1}_{\{X_u>c\}}du$ for all $t\geq 0$ for some bounded deterministic function $g$.
Suppose for any $t\geq 0$, $\mathbb{E}\left[e^{A_t}\right]=f(t,x)$ for some known function $f$.
Then is it true that: for any $0\leq t\leq s<\infty$, $$ \mathbb{E}\left[e^{A_s-A_t}|\mathcal{F}_t\right]=f(s-t,X_t)? $$