Assume a borrower takes out an installment loan of size $1$ and makes continuous-time payments on it. The installment loan starts at time $0$, ends at time $T$, and has an interest rate of $r$, compounding continually. We'll let $P(t)$ be the principal owed by the borrower at time $t$, and $I(t)$ be the interest they've paid on the loan so far. Unlike in discrete time lending, we don't need to keep track of the amount of unpaid interest owed by the borrower; it's always zero since the borrower makes continuous payments. To check your comprehension, $P(0) = 1$, $P(T) = 0$, $I(0) = 0$, and $I(T)$ is the total amount of interest the borrower will pay on their loan (assuming no defaults, fees, or prepayment).
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