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A representation formula for transition probability densities of diffusions and applications

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  • Qian, Zhongmin
  • Zheng, Weian

Abstract

We establish a representation formula for the transition probability density of a diffusion perturbed by a vector field, which takes a form of Cameron-Martin's formula for pinned diffusions. As an application, by carefully estimating the mixed moments of a Gaussian process, we deduce explicit, strong lower and upper estimates for the transition probability function of Brownian motion with drift of linear growth.

Suggested Citation

  • Qian, Zhongmin & Zheng, Weian, 2004. "A representation formula for transition probability densities of diffusions and applications," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 57-76, May.
    • Handle: RePEc:eee:spapps:v:111:y:2004:i:1:p:57-76
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    Cited by:

    1. Qian, Zhongmin & Xu, Xingcheng, 2023. "Probability bounds for reflecting diffusion processes," Statistics & Probability Letters, Elsevier, vol. 199(C).
    2. Zhongmin Qian & Yuhan Yao, 2022. "McKean–Vlasov type stochastic differential equations arising from the random vortex method," Partial Differential Equations and Applications, Springer, vol. 3(1), pages 1-22, February.
    3. Song, Ruili & Ying, Jiangang, 2007. "A formula for transition density function under Girsanov transform," Statistics & Probability Letters, Elsevier, vol. 77(6), pages 658-666, March.
    4. Taguchi, Dai & Tanaka, Akihiro, 2020. "Probability density function of SDEs with unbounded and path-dependent drift coefficient," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5243-5289.
    5. Albeverio, S. & Marinelli, C., 2005. "Reconstructing the drift of a diffusion from partially observed transition probabilities," Stochastic Processes and their Applications, Elsevier, vol. 115(9), pages 1487-1502, September.
    6. Downes, A.N., 2009. "Bounds for the transition density of time-homogeneous diffusion processes," Statistics & Probability Letters, Elsevier, vol. 79(6), pages 835-841, March.

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    Keywords

    Heat kernel estimates Diffusion;

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