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A new analytical approximation tool, derived from the classical PDE theory, is introduced in order to build approximate transition densities of diffusions. The tool is useful for approximate pricing and hedging of financial derivatives and for maximum likelihood and method of moments estimates of diffusion parameters. The approximation is uniform with respect to time and space variables. Moreover, easily computable error bounds are available in any dimension.doi:10.1137/080742336 fatcat:u2ep665abnfv3lolq5ftyirsne