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Smooth transition densities for one-dimensional diffusions. (English) Zbl 0604.60077

Consider the one-dimensional stochastic differential equation \[ dX_ t=\sigma (X_ t)dB+b(X_ t)dt,\quad X_ 0=x \] which is a Markov process with transition kernel \(P_ t(x,dy)\). If \(\sigma\) is \(C^{r+2}\), b is \(C^{r+1}\), and \(\sigma\) (x) is positive and bounded away from 0, the author proves that the transition kernel has a density p(t,x,y) which is jointly continuous, and possesses continuous partial derivatives of order \(C^ r\) in all three variables.
This result is obtained without the use of Malliavin calculus, but the method does not appear to generalize to higher dimensions. The result is new in that Malliavin calculus [see the expository article by M. Zakai, Acta Appl. Math. 3, 175-207 (1985; Zbl 0553.60053)] only proves smoothness of p(t,x,y) as a function of x and y, for fixed t.
Reviewer: R.W.R.Darling

MSC:

60J60 Diffusion processes
60H20 Stochastic integral equations
60J35 Transition functions, generators and resolvents

Citations:

Zbl 0553.60053
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