Let $N(t)$ be a Poisson process with rate $\lambda$ and let $$u(x, t) = \mathbb E [f(x + N(t))]$$ where $f:\mathbb R\to \mathbb R$ is a continuous and bounded function, $t\geq 0$, $x \in \mathbb R$. I am asked to prove that $u(t, x)$ is a solution of $$\frac{\partial }{\partial t} u(t, x)= \lambda [u(t, x+1)-u(t, x)]$$ with initial condition $u(0, x)= f(x)$.
My idea was to define $p_k(t)=\mathbb P(N(t)=k)$ and then use the fact that $p_n'(t) = -\lambda p_n(t)+ \lambda p_{n-1}(t)$. To this end, I tried to express the expectation as follows $$\mathbb E [f(x + N(t))] =\sum_{n \in E} f(x + n) p_n(t)$$ where $E$ is the state space. In this case, the proof would be straightforward. However, I am not sure if I can actually consider the term $f(x+n)$ as independent from $t$, or if I am missing some steps.
Any suggestion will be appreciated!