Well-posedness of Multidimensional Diffusion Processes with Weakly Differentiable Coefficients
By Dario Trevisan. 2015
131 citations recorded by the source corpus.
We investigate well-posedness for martingale solutions of stochastic differential equations, under low regularity assumptions on their coefficients, widely extending some results first obtained by A. Figalli. Our main results are a very general equivalence between different descriptions for multidimensional diffusion processes, such as Fokker-Planck equations and martingale problems, under minimal regularity and integrability assumptions, and new existence and uniqueness results for diffusions having weakly differentiable coefficients, by means of energy estimates and commutator inequalities. Our approach relies upon techniques recently developed, jointly with L. Ambrosio, to address well-posedness for ordinary differential equations in metric measure spaces: in particular, we employ in a systematic way new representations and inequalities for commutators between smoothing operators and diffusion generators.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Finance
View Paper: https://www.semanticscholar.org/paper/7e106db78e1050755a1b1d99aa094bee21cb4a00
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In Applied Mathematics
- Xicheng Zhang
- Michael Röckner
- Viorel Barbu
- Daniel Lacker
- Jiacheng Zhang
- Mykhaylo Shkolnikov
- Feng-Yu Wang
- Wei Liu
In Computational Theory and Mathematics
- Xicheng Zhang
- Michael Röckner
- Viorel Barbu
- Jiacheng Zhang
- Daniel Lacker
- Mykhaylo Shkolnikov
- Feng-Yu Wang
- Wei Liu
In Finance
- Michael Röckner
- Xicheng Zhang
- Mykhaylo Shkolnikov
- Jiacheng Zhang
- Viorel Barbu
- Feng-Yu Wang
- Daniel Lacker
- Wei Liu
Citations in the corpus
- Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, methodology): “The results of [33] have been extended to a wide range of SDEs with unbounded coefficients in [66] (see also [12, 61] and the comparison with the previous work [58] in [66, Remark 3.”
- From nonlinear Fokker–Planck equations to solutions of distribution dependent SDE (2018, methodology): “We would like to mention at this point that, by the very same result from [30], one can also easily prove that, if (1.”
- Well-posedness of distribution dependent SDEs with singular drifts (2018, background): “[23] Trevisan D.: Well-posedness of multidimensional diffusion processes with weakly differentiable coeffi- cients.”