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

2022/05/05 by Alexander M. G. Cox, Cox, Alexander M. G., Benjamin A. Robinson +1
Economics, Econometrics and Finance · #60G44 #60H10 (Primary) #93E20 (Secondary) #Economic theories and models #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2205.02519

openalex publication_date 2022/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Capacities, Measurable Selection and Dynamic Programming Part II: Application in Stochastic Control Problems
• A regularized Kellerer theorem in arbitrary dimension

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