On stochastic continuity of generalized diffusion processes constructed as the strong solution to an SDE

TL;DR

This paper proves the continuous dependence on initial conditions for a class of generalized diffusion processes defined as strong solutions to SDEs, using comparison theorems for skew Brownian motions.

Contribution

It establishes stochastic continuity for generalized diffusion processes via comparison theorems, providing new insights into their dependence on initial conditions.

Findings

Comparison theorem for skew Brownian motions proved

Estimate on ${\ ext{Cal}}_1$-distance between skew Brownian motions derived

Continuous dependence on starting point established for certain diffusion processes

Abstract

The comparison theorem for skew Brownian motions is proved. As the corollary we get the estimate on ${\Cal L}_1-$distance between two skew Brownian motions started from different points. Using this result we prove the continuous dependence on starting point of one class of generalized diffusion processes constructed as the strong solution to an SDE.