Abstract:
The parametrix method is a powerful analytical approach for constructing and analyzing fundamental solutions to parabolic equations and transition probability densities of solutions to stochastic differential equations. The “continuous” version of the method has a long history and dates back to the work of the Italian mathematician Eugenio Ella Levi (1907). However, the continuous version, which made it possible to develop a discrete analogue of the method, belongs to H. McKean, I. Singer (1967). A discrete version of the method was proposed in the article by K. and S. Molchanov (TV and MS, 1984), and a more detailed and general version in the work by K. and E. Mammen (PTRF, 2000). The method is effective for non-smooth (Hölder) coefficients of drift and diffusion. Modern research adapts the method to Kolmogorov degenerate diffusions and Markov chains. The work in question is motivated by the desire to find a concrete problem in which these methods work. The object of the study was the well-known stochastic approximation procedure proposed by Robbins and Monroe in 1951 and named after them. Markov chains related to this procedure were found and, apparently, for the first time, local limit theorems on convergence to the Gaussian diffusion process were obtained, Based on these results, strong invariance principles were obtained. The talk is based on joint works with Enno Mammen (Heidelberg university, Germany).
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