  In this paper, by extending the classic stochastic integrals, we investigate
three kinds of more general stochastic integrals: Lebesgue-Stieltjes integrals
on predictable sets of interval type (in short: PSITs), stochastic integrals on
PSITs of predictable processes with respect to local martingales, and
stochastic integrals on PSITs of predictable processes with respect to
semimartingales. Such stochastic integrals on PSITs are defined only on
restricted stochastic subsets, and their values outside the subsets do not
matter. Our study reveals that a stochastic integral on a PSIT can be
characterized by a coupled sequence of classic stochastic integrals.
Furthermore, the It\^{o}'s formula for semimartingales on PSITs is developed
for stochastic calculus, and stochastic integrals on PSITs can be applied to
more general problems in mathematical finance.
