Connecting two semicontinuous processes with independent increments

We consider a stopped process X_t⁰ in the phase space E⁰=(–, +)/{0} such that X_t⁰=X_t¹ if X_t⁰ > 0 and X_t⁰=X_t² if X_t⁰ < 0, where X_t^j, j=1,2, are nonstopped stochastically continuous Markov processes with independent increments and with only negative jumps. We prove that there exists an extension of X_t⁰ into a homogeneous, stochastically continuous, and strong Markov Feller process X_t in the phase space (–; +) and that the extension can be characterized by a measure N(dy) and three constants b, c₁ c₂.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 596–600, May, 1991.