Let $(\Omega, \mathcal{F}, \mathbb{P} )$ be a probability space and let $X$ and $Y$ be two (not necessarily independent) processes taking values on $\mathbb{R}$. Let $\phi: \mathbb{R} \to \mathbb{R}$ be a continuous function. Let $\mathcal{G}$ be a sigma-algebra defined by $$ \mathcal{G} := \sigma(Y_t, 0 \leq t \leq T).$$ We define the process $(X^{0,x})$ to denote process $X$ starting at $x$ at time $0$. Suppose that $X_0$ has a distribution denoted by $\mu$. Then, we know that by the tower property of conditional expectation, for any $0 \leq t \leq T$, $$ \mathbb{E} \phi(X_t) = \int_{ \mathbb{R}} \mathbb{E} \Big[ \phi \big( (X_t)^{0,x} \big) \Big] \mu (dx). $$ But is it true that $$ \mathbb{E} \Big[ \phi(X_t) | \mathcal{G} \Big] = \int_{ \mathbb{R}} \mathbb{E} \Big[ \phi \big( (X_t)^{0,x} \big) \Big| \mathcal{G} \Big] \mu (dx) \, \, ? $$

I can't be able to prove or disprove this statement by the properties of conditional expectation. If this is not true, do you know any similar expression for the conditional case?


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