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Some estimates of the transition density of a nondegenerate diffusion Markov process. (English) Zbl 0738.60060

It is considered the difficult problem of estimating the transition density \(P_ t(x,y)\) for a nondegenerate diffusion process using Fleming’s logarithmic transformation and the corresponding nonlinear dynamic programming equation of a stochastic control problem. Estimate of \(D^ m_ x \log P_ t(x,y)\) instead of \(D^ m_ xP_ t(x,y)\) brings difficulties which are avoided using a stochastic parallel translation. Two theorems and seven lemmas are proved.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J35 Transition functions, generators and resolvents
60J60 Diffusion processes
60H07 Stochastic calculus of variations and the Malliavin calculus
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