Express the present value of the sums $c_{t_{1}},c_{t_2},...,c_{t_n}$ due at times $t_1,t_2,...,t_n$ in terms of the force of interest at time $t$, denoted by $\delta (t)$.
The answer given is $$\sum_{j=1}^n(c_{t_j}exp(-\int_{0}^{t_j}\delta(s)ds)$$
Can anyone explain why and how? Thanks in advance.
First of all we may assume $n=1$ since we are just adding up the value of the sums. So we want to justify that the present value of $c$ due at time $t$ is $$c\exp\left(-\int_0^t\delta(s)\,ds\right).$$ The present time is time 0 and the present value $x$ should evolve into $c$ at time $t$: $$x\exp\left(\int_0^t\delta(s)\,ds\right) = c.$$ This is just expressing that force-of-interest $\delta$ is related to interest $i$ by $$e^{\delta\cdot t}=(1+i)^t,$$ and that when $\delta$ is not constant, $\delta\cdot t$ is replaced by $\int_0^t\delta(s)\,ds$.
