PCT No. PCT/JP97/03987 Sec. 371 Date Jul. 2, 1998 Sec. 102(e) Date Jul. 2, 1998 PCT Filed Oct. 31, 1997 PCT Pub. No. WO98/20068 PCT Pub. Date May 14, 1998A thermoplastic elastomer composition containing a component (A) (thermoplastic polyolefin resin), component (B) (epoxy-group-containing ethylene acrylic ester copolymer rubber), and component (C) (acid-modified polyolefin resin) comprising a continuous phase of the component (A) in which the component (B) is dispersed as a dispersed phase and is cross-linked by a peroxide; a thermoplastic elastomer composition comprising a continuous phase of a thermoplastic polyolefin resin (A) and a dispersed phase of an epoxy-group-containing ethylene-acrylic ester copolymer rubber component (B) and acrylonitrile butadiene rubber component (C), which is dynamically cross-linked; and a hose having an inner tube, at least one reinforcing layer, and an outer cover, wherein the inner tube and the outer cover contain a thermoplastic elastomer composition comprised of a thermoplastic resin in which is blended an at least partially cross-linked elastomer component, the reinforcing layer is a brass-plated wire, and the reinforcing layer and inner tube and the reinforcing layer and outer cover have between them a maleic-acid-modified polyolefin thermoplastic resin.