Stability problems for Cantor stochastic differential equations

TL;DR

This paper investigates stability and uniqueness in Cantor stochastic differential equations, establishing necessary and sufficient conditions for pathwise uniqueness, analyzing stability via Hölder and Nakao-Le Gall conditions, and constructing smooth solutions using Malliavin calculus.

Contribution

It provides new criteria for pathwise uniqueness, compares stability conditions, and constructs smooth solutions under weak assumptions for Cantor SDEs.

Findings

Feller test characterizes pathwise uniqueness for certain SDEs.

Nakao-Le Gall condition's sharpness and stability are confirmed.

Smooth solutions to degenerate Fokker-Planck equations are constructed.

Abstract

We consider driftless stochastic differential equations and the diffusions starting from the positive half line. It is shown that the Feller test for explosions gives a necessary and sufficient condition to hold pathwise uniqueness for diffusion coefficients that are positive and monotonically increasing or decreasing on the positive half line and the value at the origin is zero. Then, stability problems are studied from the aspect of H\"older-continuity and a generalized Nakao-Le Gall condition. Comparing the convergence rate of H\"older-continuous case, the sharpness and stability of the Nakao-Le Gall condition on Cantor stochastic differential equations is confirmed.Furthermore, using the Malliavin calculus, we construct a smooth solution to degenerate second order Fokker-Planck equations under weak conditions on the coefficients.