Proving that a process has the Markov property

Question

Let $X_t=xe^{ct+aB_t}$ where $B_t$ is one dimensional Brownian motion. How would I prove this is a Markov process using the expectation definition of a Markov process, i.e., $E[f(X_t)|\sigma(B_s)]=E[f(X_t)|X_s]$ for appropriate functions f.

Answer 1

Here is a more general statement:

Consider some Markov process $(Y_t)$ and define, for every $t$, $Z_t=F(t,Y_t)$, where $F$ is measurable and, for every $t$, $F(t,\ )$ has a measurable inverse. Then, $\sigma(Z_s;s\leqslant t)=\sigma(Y_s;s\leqslant t)$ for every $t$ and the process $(Z_t)$ is Markov.

Can you prove this and apply it to your setting?