Pathwise vs. path-by-path uniqueness

TL;DR

This paper constructs stochastic differential equations that demonstrate nonuniqueness when considering individual paths, yet maintain a unique solution in the adapted process sense, highlighting differences in solution concepts.

Contribution

It provides explicit examples of SDEs with nonuniqueness in the path-by-path sense despite having pathwise uniqueness, clarifying solution concept distinctions.

Findings

Examples of SDEs with nonuniqueness path-by-path

Demonstrates pathwise uniqueness despite nonuniqueness in individual paths

Highlights differences between solution concepts in stochastic analysis

Abstract

We construct a series of stochastic differential equations of the form $dX_t = b(t, X_t) dt + dB_t$ which exhibit nonuniqueness in the path-by-path sense while having a unique adapted solution in the sense of stochastic processes, i.e. pathwise uniqueness holds.