  In the present paper, we propose a two-component generalization of the
reduced Ostrovsky equation, whose differential form can be viewed as the
short-wave limit of a two-component Degasperis-Procesi (DP) equation. They are
integrable due to the existence of Lax pairs. Moreover, we have shown that
two-component reduced Ostrovsky equation can be reduced from an extended BKP
hierarchy with negative flow through a pseudo 3-reduction and a hodograph
(reciprocal) transform. As a by-product, its bilinear form and $N$-soliton
solution in terms of pfaffians are presented. One- and two-soliton solutions
are provided and analyzed. In the second part of the paper, we start with a
modified BKP hierarchy, which is a B\"acklund transformation of the above
extended BKP hierarchy, an integrable semi-discrete analogue of two-component
reduced Ostrovsky equation is constructed by defining an appropriate discrete
hodograph transform and dependent variable transformations. Especially, the
backward difference form of above semi-discrete two-component reduced Ostrovsky
equation gives rise to the integrable semi-discretization of the short wave
limit of a two-component DP equation. Their $N$-soliton solutions in terms of
pffafians are also provided.
