Let pb(x,t,y) be the transition probability density of the one dimensional diffusion process , 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.
Soit pb(x,t,y) la densité de la probabilité de transition du processus de diffusion , 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.
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Zhongmin Qian 1; Weian Zheng 2, 3
@article{CRMATH_2002__335_11_953_0, author = {Zhongmin Qian and Weian Zheng}, title = {Sharp bounds for transition probability densities of a class of diffusions}, journal = {Comptes Rendus. Math\'ematique}, pages = {953--957}, publisher = {Elsevier}, volume = {335}, number = {11}, year = {2002}, doi = {10.1016/S1631-073X(02)02579-7}, language = {en}, }
Zhongmin Qian; Weian Zheng. Sharp bounds for transition probability densities of a class of diffusions. Comptes Rendus. Mathématique, Volume 335 (2002) no. 11, pp. 953-957. doi : 10.1016/S1631-073X(02)02579-7. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02579-7/
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