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Pathwise uniqueness for a SDE with non-Lipschitz coefficients

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  • Swart, J. M.

Abstract

We consider the ordinary stochastic differential equation on the closed unit ball E in . While it is easy to prove existence and distribution uniqueness for solutions of this SDE for each c[greater-or-equal, slanted]0, pathwise uniqueness can be proved by standard methods only in dimension n=1 and in dimensions n[greater-or-equal, slanted]2 if c=0 or if c[greater-or-equal, slanted]2 and the initial condition is in the interior of E. We sharpen these results by proving pathwise uniqueness for c[greater-or-equal, slanted]1. More precisely, we show that for X1,X2 solutions relative to the same Brownian motion, the function is almost surely nonincreasing. Whether or not pathwise uniqueness holds in dimensions n[greater-or-equal, slanted]2 for 0

Suggested Citation

  • Swart, J. M., 2002. "Pathwise uniqueness for a SDE with non-Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 131-149, March.
    • Handle: RePEc:eee:spapps:v:98:y:2002:i:1:p:131-149
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    Cited by:

    1. B. Rajeev & K. Suresh Kumar, 2016. "A class of stochastic differential equations with pathwise unique solutions," Indian Journal of Pure and Applied Mathematics, Springer, vol. 47(2), pages 343-355, June.
    2. Alexandre Pannier & Antoine Jacquier, 2019. "On the uniqueness of solutions of stochastic Volterra equations," Papers 1912.05917, arXiv.org, revised Apr 2020.
    3. Larsson, Martin & Pulido, Sergio, 2017. "Polynomial diffusions on compact quadric sets," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 901-926.
    4. He, Hui, 2009. "Strong uniqueness for a class of singular SDEs for catalytic branching diffusions," Statistics & Probability Letters, Elsevier, vol. 79(2), pages 182-187, January.

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