how can one show, that the present value of an annuity certain of 1 payable annually in advance starting at T can be written as
$$\ddot{a}(T) = \sum_{n=0}^{\infty} p(T, T+n) {}_n p_x$$
where $p(T,U)$ is the value of a Zero coupon-bond with maturity $U$ at the time $T \leq U$.
${}_np_x$ is the probabilty of an x year old Person to survive n years, i.e. ${}_n p_x = P(T_x > n)$, where $T_x$ is the remaining life time of an x year old Person. And $p(T,U)$ is the value of an Zero coupon bond. A ZCB with maturity $T$ is defined as a contract, what pays out 1 at timte $T$. The value/Price of this ZCB at time $t \leq T$ is $p(t,T)$, in particular $p(T,T)=1$. Furthermore we assume that for fixed $T$, $\{p(t,T) \mid t \in [0,T]\}$ is positive and for fixed $t$ the function $p(t,T)$ is continuously differentiable for the $T$ variable.
Does someone have an idea how to start?
I believe this is understood as "The probability to pay the amount $1$ at time $T+n$ is $_np_x$", that is we expect to pay $1\cdot \,_np_x$ for the person $x$.
The present value (at time $T$) of this amount is $p(T,T+n)\cdot 1\cdot _xp_n$.
Totally, we expect to pay the sum given to the right.