A nonclassical solution to a classical SDE and a converse to Kolmogorov's zero-one law

TL;DR

This paper constructs a weak solution for a discrete-time, discrete-space SDE lacking a classical strong solution, revealing the importance of measurability and establishing a converse to Kolmogorov's zero-one law.

Contribution

It introduces a non-measurable, non-anticipative weak solution for a specific SDE and presents a novel converse to Kolmogorov's zero-one law.

Findings

Weak solution constructed for a non-classical SDE

Measurability is crucial in non-discrete probability

A new converse to Kolmogorov's zero-one law established

Abstract

For a discrete-negative-time discrete-space SDE, which admits no strong solution in the classical sense, a weak solution is constructed that is a (necessarily nonmeasurable) non-anticipative function of the driving i.i.d. noise. The result highlights the strong r\^ole measurability plays in (non-discrete) probability. En route one -- quite literally -- stumbles upon a converse to the celebrated Kolmogorov's zero-one law for sequences with independent values.