If I deposit \$1 at $t=0$ into an account which credits interest at the end of each year at a force of interest $\delta_t$ (assume it's integrable.) Then, if I reinvest the interest at an annual effective rate interest $j,$ what'd be the total accumulated amount?
To generalize further what if the first interest is given by $f(t)$ and reinvested interest is $g(t)?$