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.
| Original language | English |
|---|---|
| Article number | 104 |
| Pages (from-to) | 1-24 |
| Number of pages | 24 |
| Journal | Electronic Journal of Probability |
| Volume | 28 |
| DOIs | |
| Publication status | Published - 8 Aug 2023 |
Bibliographical note
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.
Funding
*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. †Department of Mathematical Sciences, University of Bath, Bath, UK. E-mail: [email protected] ‡Universität Wien, Vienna, Austria. E-mail: [email protected]