Let $(N_t)_t$ be a Poisson process of intensity $\lambda$ with arrival time $(T_k)_{k\geq 0}$ and inter-arrival time $(X_k)_{k\geq 1}$. For a fixed time $t$, the $\textbf{age}$ is defined as $A_t=t-T_{N_t}$. I am trying to find the distribution of this random variable.
If we condition on $X_1$, we can find that \begin{equation} \mathbb{P}(A_t>y\,|\,X_1=x)=\left\{ \begin{aligned} &1,\quad & x\geq t \\ &0,\quad & t-y\leq x<t \\ &\mathbb{P}(A_{t-x}>y), &x<t-y. \end{aligned} \right. \end{equation} So $$ \mathbb{P}(A_t>y) = \int_{0}^{t-y} \mathbb{P}(A_{t-x}>y) dF(x)+\int_{t}^\infty dF(x). $$ But I have no idea what to do next. Can anyone help me?