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Comparison theorem and estimates for transition probability densities of diffusion processes
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  • Published: 26 August 2003

Comparison theorem and estimates for transition probability densities of diffusion processes

  • Zhongmin Qian1,
  • Francesco Russo2 &
  • Weian Zheng3,4 

Probability Theory and Related Fields volume 127, pages 388–406 (2003)Cite this article

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Abstract

We establish several comparison theorems for the transition probability density p b (x,t,y) of Brownian motion with drift b, and deduce explicit, sharp lower and upper bounds for p b (x,t,y) in terms of the norms of the vector field b. The main results are obtained through carefully estimating the mixed moments of Bessel processes. All constants are explicit in our lower and upper bounds, which is different from most of the previous estimates, and is important in many applications for example in statistical inferences for diffusion processes.

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Author information

Authors and Affiliations

  1. Laboratoire de statistique et probabilités, Université Paul-Sabatier, 118, route de Narbonne, 31062, Toulouse and CNRS, France

    Zhongmin Qian

  2. Institut Galilée, Mathématiques, Université de Paris, 13, 99, av. J.B. Clément, 99430, Villetaneuse, France

    Francesco Russo

  3. Department of Statistics, East China Normal University, Shanghai, China

    Weian Zheng

  4. Department of Mathematics, University of California, Irvine, CA, 92697, USA

    Weian Zheng

Authors
  1. Zhongmin Qian
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  2. Francesco Russo
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  3. Weian Zheng
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Corresponding author

Correspondence to Zhongmin Qian.

Additional information

Research partially supported by N.S.F. Grants DMS-0203823, and by Doctoral Program Fundation of the Ministry of Education of China, Grant No. 20020269015.

Mathematics Subject Classification (2000): Primary: 60H10, 60H30; Secondary: 35K05

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Cite this article

Qian, Z., Russo, F. & Zheng, W. Comparison theorem and estimates for transition probability densities of diffusion processes. Probab. Theory Relat. Fields 127, 388–406 (2003). https://doi.org/10.1007/s00440-003-0291-1

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  • Received: 22 September 2002

  • Revised: 17 June 2003

  • Published: 26 August 2003

  • Issue Date: November 2003

  • DOI: https://doi.org/10.1007/s00440-003-0291-1

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Keywords

  •  Transition probabilities
  • Diffusion processes
  • Comparison theorems
  • Bessel processes
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