Does a Levy process have the property $\mathbb{E}_x(X_t|\mathcal{G})=\mathbb{E}(X_t+x|\mathcal{G})$?

Question

Let $(X_t)_t$ be a Levy process on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, $\mathcal{G}\subseteq\mathcal{F}$ be a $\sigma$-algebra and $\mathbb{P}_x$ be the probability measure on $(\Omega,\mathcal{F})$ defined by $\mathbb{P}_x(X_{t_1}\in A_1,...,X_{t_n}\in A_n)=\mathbb{P}(X_{t_1}+x\in A_1,...,X_{t_n}+x\in A_n)$. I was wondering if then the equation $$ \mathbb{E}_x(X_t|\mathcal{G})=\mathbb{E}(X_t+x|\mathcal{G}) $$ holds. And if yes, in which sense? $\mathbb{P}_x$-a.s. or $\mathbb{P}$-a.s. ?

Answer 1

It is worth to note that $\mathbb P_x$ is defined on $\sigma(X)$, not on $\mathcal{F}$.

Speaking of your question, the answer is obviously negative. Take e.g. $\mathcal{G} = \sigma(X)$. Then $\mathbb{E}_x(X_t\mid\mathcal{G})=X_t\neq X_t +x = \mathbb{E}(X_t+x\mid \mathcal{G})$.