Sharp bounds for transition probability densities of a class of diffusions

• Published in: Comptes rendus. Mathématique Vol. 335; no. 11; pp. 953 - 957
• Main Authors: Qian, Zhongmin, Zheng, Weian
• Format: Journal Article
• Language: English
• Published: Paris Elsevier SAS 01.12.2002; Elsevier
• Subjects: Exact sciences and technology; Markov processes; Mathematical analysis; Mathematics; Ordinary differential equations; Probability and statistics; Probability theory and stochastic processes; Probability theory on algebraic and topological structures; Sciences and techniques of general use; Brownian motion; Probability density; Lebesgue measure; Differential equation; Probability theory; Transition probability; Diffusion process; Joint distribution; Probability measure; Stochastic equation
• ISSN: 1631-073X; 1778-3569; 1778-3569
• DOI: 10.1016/S1631-073X(02)02579-7

Let p b ( 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 p b ( 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. To cite this article: Z. Qian, W. Zheng, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 953–957. Soit p b ( 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 p b ( 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. Pour citer cet article : Z. Qian, W. Zheng, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 953–957.