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Convergence implications via dual flow method. (English) Zbl 1423.60087
Summary: Given a one-dimensional stochastic differential equation, one can associate to this equation a stochastic flow on \([0,+\infty)\), which has an absorbing barrier at zero. Then one can define its dual stochastic flow. J. Akahori and S. Watanabe [“On the strong solutions of stochastic differential equations” (Japanese), Soc. Syst. Stud. 4, 1–12 (2002)] showed that its one-point motion solves a corresponding stochastic differential equation of Skorokhod-type. In this paper, we consider a discrete-time stochastic-flow which approximates the original stochastic flow. We show that under some assumptions, one-point motions of its dual flow also approximates the corresponding reflecting diffusion. We prove and use relations between a stochastic flow and its dual in order to obtain weak and strong approximation results related to stochastic differential equations of Skorokhod-type.
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60J25 Continuous-time Markov processes on general state spaces
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