Say $x$ is a random process with independent increments, i.e. for every sequence $t_0<t_1<...<t_n\in\mathbb{T}$, we know that: $x_{t_n}-x_{t_{n-1}}\perp ... \perp x_{t_1}-x_{t_0} $.
This obviously also means that $E[x_{t_n}-x_{t_{n-1}}|x_{t_{n-1}}-x_{t_{n-2}},...,x_{t_1}-x_{t_0}]=E[x_{t_n}-x_{t_{n-1}}]$.
Is it also true that $E[x_{t_n}-x_{t_{n-1}}|x_{t_{n-1}},x_{t_{n-2}},...,x_{t_1},x_{t_0}]=E[x_{t_n}-x_{t_{n-1}}]$? I feel like having the actual values of a process has additional information when compared to the increments of a process.
Will the answer change if $x_{t_0}$ is known (with probability 1)?
If $x_{t_0} = c$ w.p. 1, then the equation holds because: $x_{t_i} = \sum_{j=1}^{i}(x_{t_j}-x_{t_{j-1}})+c, i=1,...,n-1$, which is a function of $x_{t_j}-x_{t_{j-1}}$, hence $x_{t_n}-x_{t_{n-1}}$ is independent of $x_{t_i}$