Sharp bounds for transition probability densities of a class of diffusions

Abstract

Let p b ( x , t , y ) be the transition probability density of the one dimensional diffusion process d X t = d W t + b ( X t ) d t , 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.

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