SDEs with no strong solution arising from a problem of stochastic control
Probability
2025-06-10 v2
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.
Cite
@article{arxiv.2205.02519,
title = {SDEs with no strong solution arising from a problem of stochastic control},
author = {Alexander M. G. Cox and Benjamin A. Robinson},
journal= {arXiv preprint arXiv:2205.02519},
year = {2025}
}
Comments
20 pages, 1 figure