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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.

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    $\begingroup$ 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. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 24, 2022 at 9:22
  • $\begingroup$ Ah, I do not know too many people working in SDE in person. I would be grateful if you could ask indeed. $\endgroup$
    – Nate River
    Commented Oct 24, 2022 at 9:32
  • $\begingroup$ The multi-dimensional case with $b$ of bounded variation has been studied in arXiv:1306.4816 $\endgroup$
    – Carlo Beenakker
    Commented Oct 24, 2022 at 17:57
  • $\begingroup$ 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 $\endgroup$
    – Nawaf Bou-Rabee
    Commented Oct 25, 2022 at 12:50
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    $\begingroup$ 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. $\endgroup$
    – Nawaf Bou-Rabee
    Commented Oct 25, 2022 at 13:22

1 Answer 1

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In the context of multidimensional SDEs,

  • For bounded and measurable SDE coefficients, there exists a unique solution to the corresponding martingale problem. For a detailed statement/proof, see Chapter 6 of Stroock & Varadhan 1997 entitled "The Martingale Formulation".
  • Moreover, existence/uniqueness of a weak solution is equivalent to existence/uniqueness of the corresponding martingale problem. For a precise statement/proof, see Karatzas & Shreve 1991 Section 5.4B entitled "Weak solutions and martingale problems".

References

Stroock, Daniel W.; Varadhan, S. R. Srinivasa, Multidimensional diffusion processes., Classics in Mathematics. Berlin: Springer (ISBN 3-540-28998-4/hbk). xii, 338 p. (2006). ZBL1103.60005.

Karatzas, Ioannis; Shreve, Steven E., Brownian motion and stochastic calculus., Graduate Texts in Mathematics, 113. New York etc.: Springer-Verlag. xxiii, 470 p. DM 68.00/pbk (1991). ZBL0734.60060.

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