SDEs with no strong solution arising from a problem of stochastic control

TL;DR

This paper investigates a two-dimensional stochastic differential equation that admits a unique weak solution but lacks a strong solution, highlighting its similarities to Tsirelson's example and exploring implications for stochastic control.

Contribution

The paper constructs a two-dimensional SDE with a unique weak solution but no strong solution, extending Tsirelson's example to a Markovian setting and analyzing its control applications.

Findings

No strong solution exists for the SDE.

Weak solution's filtration is generated by a Brownian motion.

Application to stochastic control with fixed quadratic variation.

Abstract

We study a two-dimensional stochastic differential equation that has a unique weak solution but no strong solution. We show that this SDE shares notable properties with Tsirelson's example of a one-dimensional SDE with no strong solution. In contrast to Tsirelson's equation, which has a non-Markovian drift, we consider a strong Markov martingale with Markovian diffusion coefficient. We show that there is no strong solution of the SDE and that the natural filtration of the weak solution is generated by a Brownian motion. We also discuss an application of our results to a stochastic control problem for martingales with fixed quadratic variation in a radially symmetric environment.