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.
Funding Statement
BR is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/L015684/1, and by the Austrian Science Fund (FWF) projects Y782-N25 and P35519.
Acknowledgments
The authors are grateful to the anonymous referee, whose comments led to strengthening the main results of this article.
Citation
Alexander M. G. Cox. Benjamin A. Robinson. "SDEs with no strong solution arising from a problem of stochastic control." Electron. J. Probab. 28 1 - 24, 2023. https://doi.org/10.1214/23-EJP995
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