问题 1.
The cost of the new portfolio $h(t)$, which has to be bought at today’s prices, is given by
$$
\sum_{i=1}^{N} h_{i}(t) S_{i}(t)=h(t) S(t)
$$
whereas the cost for the consumption rate $c(t)$ is given by $c(t) \Delta t$. The budget equation for period $t$ thus reads
$$
h(t-\Delta t) S(t)=h(t) S(t)+c(t) \Delta t .
$$
证明 .
If we introduce the notation
$$
\Delta X(t)=X(t)-X(t-A t),
$$
for an arbitrary process $\mathrm{X}$, we see that the budget equation reads
$$
S(t) \Delta h(t)+c(t) \Delta t=0
$$
MATH0085 COURSE NOTES ：
- The value of our old portfolio, as usual given by
$$
\mathrm{h}(\mathrm{t}-\Delta t) S(t)
$$ - The dividends eamed during the interval $(t-A t, t]$. These are given by
$$
\sum_{i=1}^{N} h_{i}(t-\mathrm{At})\left[D_{i}(t)-D_{i}(\mathrm{t}-\mathrm{At})\right]=\mathrm{h}(\mathrm{t}-\mathrm{At}) \Delta D(t)
$$
The relevant budget equation is thus given by
$$
\mathrm{h}(\mathrm{t}-\Delta t) S(t)+\mathrm{h}(\mathrm{t}-\Delta t) \Delta D(t)=h(t) S(t)+c(t) \Delta t .
$$