Comparison of classical and path-by-path solutions to SDEs

TL;DR

This paper explores the differences between classical and path-by-path solutions to stochastic differential equations, providing counterexamples that highlight cases where solutions exist or are unique in one sense but not the other.

Contribution

It introduces explicit counterexamples demonstrating the divergence between classical and path-by-path solution concepts for SDEs, extending previous results to one-dimensional cases.

Findings

Existence of solutions without weak solutions for certain drifts

Weak solutions with pathwise uniqueness but not path-by-path uniqueness

Extension of previous counterexamples to one-dimensional SDEs

Abstract

We consider the Stochastic Differential Equation $X_t = X_0 + \int_0^t b(s,X_s) ds + B_t$, in $\mathbb{R}^d$. We give an example of a drift $b$ such that there does not exist a weak solution, but there exists a solution for almost every realization of the Brownian motion $B$. We also give an explicit example of a drift such that the SDE has a pathwise unique weak solution, but path-by-path uniqueness (i.e. uniqueness of solutions to the ODE for almost every realization of the Brownian motion) is lost. These counterexamples extend the results obtained in arXiv:2001.02869 to dimension $d=1$.