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Stochastic representation of diffusions corresponding to divergence form operators

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  • Rozkosz, Andrzej

Abstract

We show that a diffusion process X corresponding to a uniformly elliptic second-order divergence form operator is a Dirichlet process for each starting point. We establish also the Stratonovich integral with respect to X and prove the Itô formula.

Suggested Citation

  • Rozkosz, Andrzej, 1996. "Stochastic representation of diffusions corresponding to divergence form operators," Stochastic Processes and their Applications, Elsevier, vol. 63(1), pages 11-33, October.
    • Handle: RePEc:eee:spapps:v:63:y:1996:i:1:p:11-33
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    References listed on IDEAS

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    1. Rozkosz, Andrzej & Slominski, Leszek, 1991. "On existence and stability of weak solutions of multidimensional stochastic differential equations with measurable coefficients," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 187-197, April.
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    Cited by:

    1. Xavier Bardina & Maria Jolis, 2002. "Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itô's Formula," Journal of Theoretical Probability, Springer, vol. 15(1), pages 223-247, January.
    2. Lejay, Antoine, 2002. "BSDE driven by Dirichlet process and semi-linear parabolic PDE. Application to homogenization," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 1-39, January.
    3. Bardina, Xavier & Jolis, Maria, 1997. "An extension of Ito's formula for elliptic diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 69(1), pages 83-109, July.
    4. Moret, S. & Nualart, D., 2001. "Generalization of Itô's formula for smooth nondegenerate martingales," Stochastic Processes and their Applications, Elsevier, vol. 91(1), pages 115-149, January.
    5. Cheng Cai & Tiziano De Angelis, 2021. "A change of variable formula with applications to multi-dimensional optimal stopping problems," Papers 2104.05835, arXiv.org, revised Jul 2023.
    6. Lejay, Antoine, 2004. "A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE," Stochastic Processes and their Applications, Elsevier, vol. 110(1), pages 145-176, March.
    7. Tomasz Klimsiak, 2013. "On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators," Journal of Theoretical Probability, Springer, vol. 26(2), pages 437-473, June.
    8. Cai, Cheng & De Angelis, Tiziano, 2023. "A change of variable formula with applications to multi-dimensional optimal stopping problems," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 33-61.
    9. Stoica, I. L., 2003. "A probabilistic interpretation of the divergence and BSDE's," Stochastic Processes and their Applications, Elsevier, vol. 103(1), pages 31-55, January.

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    1. Lejay, Antoine, 2002. "BSDE driven by Dirichlet process and semi-linear parabolic PDE. Application to homogenization," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 1-39, January.
    2. Lejay, Antoine, 2004. "A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE," Stochastic Processes and their Applications, Elsevier, vol. 110(1), pages 145-176, March.

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