Convergence Implications via Dual Flow Method

#### T. Amaba, D. Taguchi, G. Yuki

2019, v.25, Issue 3

ABSTRACT

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.
In \cite{AW}, Akahori and Watanabe 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.

Keywords: dual stochastic flow, Siegmund's duality, absorbing diffusion, reflecting diffusion, Euler\tire Maruyama approximation