Sharp bounds for transition probability densities of a class of diffusions
[Bornes explicites pour la densité de la probabilité de transition d'un processus de diffusion]
Qian, Zhongmin1 ; Zheng, Weian2, 3
1 Laboratoire de statistique et probabilités CNRS, Université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse cedex 4, France
2 Department of Statistics, East China Normal University, Shanghai, China
3 Department of Mathematics, University of California, Irvine, CA 92697, USA
Comptes Rendus. Mathématique, Tome 335 (2002) no. 11, pp. 953-957.

Soit pb(x,t,y) la densité de la probabilité de transition du processus de diffusion dX t =dW t +b(X t )dt, où |b(·)|∞⩽1. Nous montrons que la borne supérieure (resp. la borne inférieure) de pb(x,t,y) est atteinte pour (x,t,y) fixés quand b(z)=sgn(y−z) (resp. quand b(z)=sgn(z−y)). Les bornes explicites sont présentées.

Let pb(x,t,y) be the transition probability density of the one dimensional diffusion process dX t =dW t +b(X t )dt, where |b(·)|∞⩽1. We show that the upper and lower bounds of pb(x,t,y) are achieved for fixed (x,t,y) when b(z)=sgn(y−z) and b(z)=sgn(z−y) respectively. Moreover, the precise bounds are given.

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DOI : 10.1016/S1631-073X(02)02579-7
Qian, Zhongmin 1 ; Zheng, Weian 2, 3

1 Laboratoire de statistique et probabilités CNRS, Université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse cedex 4, France
2 Department of Statistics, East China Normal University, Shanghai, China
3 Department of Mathematics, University of California, Irvine, CA 92697, USA 
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     title = {Sharp bounds for transition probability densities of a class of diffusions},
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Qian, Zhongmin; Zheng, Weian. Sharp bounds for transition probability densities of a class of diffusions. Comptes Rendus. Mathématique, Tome 335 (2002) no. 11, pp. 953-957. doi : 10.1016/S1631-073X(02)02579-7. http://www.numdam.org/articles/10.1016/S1631-073X(02)02579-7/

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