Abstract.
This paper proves that-valued solutions to the SDE are unique in distribution, when D⊂ d is convex and open, θ∈D, c>0, is positive and locally Lipschitz on D and zero on ∂D, and {x∈D:g(x)≥r} is convex for r sufficiently small. The proof (for θ=0) is based on the transformation X t ↦e ct X t , which removes the drift, and a random time change. Although the set-up is rather specialized the result gives uniqueness for some SDE’s that cannot be treated by any of the conventional techniques.
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Mathematics Subject Classification (2000): 60J60, 60H10
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Swart, J. Uniqueness for isotropic diffusions with a linear drift. Probab. Theory Relat. Fields 128, 517–524 (2004). https://doi.org/10.1007/s00440-003-0315-x
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DOI: https://doi.org/10.1007/s00440-003-0315-x