I am reading Affine Diffusions and Related Processes: Simulation, Theory and Applications of Alfonsi, at some point (Chapter 1.2.2), the author writes [simplified]:
Given: $$ \mathbb{E}\left[\exp(uX_t^x)\right]=\exp\big(\phi_u(t)+\psi_u(t)x\big) $$ Using Markov property, we have: $$ \mathbb{E}\left[\exp(uX^x_T)\mid F_t\right]=\exp\big(\phi_u(T-t)+\psi_u(T-t)X^x_t\big) $$ Where, $t\in[0,T]$ and $X_0=x$
I want to verify this equality. I know that Markov property says that: $$ \operatorname{E}[f(X_t)\mid\mathcal{F}_s]=\operatorname{E}[f(X_t)\mid\sigma(X_s)] $$ But I can't figure out how I can use this to show the previous result.