# What work has been done on SDE with diffusion coefficients of bounded variation in $\mathbb R^d$?

Consider the $$d$$-dimensional SDE, $$d > 1$$,

$$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t$$

where $$W$$ is a standard $$d$$-dimensional Brownian motion.

I am interested in the case where $$\sigma: \mathbb R^d \to \mathbb R^{d \times d}$$ is a bounded variation function, and $$b: \mathbb R^d \to \mathbb R^d$$ is assumed as nice as possible.

Question:

Has there been any work done on this case? In one dimension, there are existence results for $$\sigma$$ of bounded variation, and $$b$$ moderately irregular (Sobolev/Holder regularity). However I have not been able to find much in the multidimensional case.

The reason I ask is I believe I have a feasible plan to prove existence in the multidimensional case under some additional conditions on the diffusion coefficient, modulo several (hard) lemmas. However I would like to ensure that the result is new, and also would be of interest.

One potential application I have in mind is bounded variation stochastic control in multiple dimensions.

• One way to figure this out might be to email people working on related stuff. Have you tried this? If you like, I can ask Alexei M. Kulik from my department when I see him, he is doing parametrix method for SDEs. Oct 24, 2022 at 9:22
• Ah, I do not know too many people working in SDE in person. I would be grateful if you could ask indeed. Oct 24, 2022 at 9:32
• The multi-dimensional case with $b$ of bounded variation has been studied in arXiv:1306.4816 Oct 24, 2022 at 17:57
• For bounded and measurable SDE coefficients, it is a classical result that there exists a unique solution to the corresponding martingale problem. See: Chapter 6 of Stroock & Varadhan 1997 entitled "The Martingale Formulation" link.springer.com/content/pdf/10.1007/3-540-28999-2.pdf Oct 25, 2022 at 12:50
• Yes, this is also a classical result: existence/uniqueness of a weak solution is equivalent to existence/uniqueness of the corresponding martingale problem. For a precise statement and proof, see Karatzas & Shreve 1991 "Brownian motion and stochastic calculus" Chapter 5 Section 4.B. Oct 25, 2022 at 13:22