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arXiv · 2205.02519

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

Alexander M. G. Cox·Benjamin A. Robinson
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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.

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Alexander M. G. Cox, Benjamin A. Robinson. 2023-07-19. SDEs with no strong solution arising from a problem of stochastic control. https://doi.org/10.1214/23-ejp995

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