By Mark Freidlin, Alexander Wentzell (auth.), Pao-Liu Chow, George Yin, Boris Mordukhovich (eds.)

To verify the previous fulfillment and to supply a highway map for destiny examine, an IMA engaging establishment convention entitled "Conference on Asymptotic research in Stochastic techniques, Nonparametric Estimation, and comparable difficulties" used to be held at Wayne country college, September 15-17, 2006. This convention was once additionally held to honor Professor Rafail Z. Khasminskii for his primary contributions to many features of stochastic procedures and nonparametric estimation thought at the get together of his seventy-fifth birthday. It assembled a powerful checklist of invited audio system, who're popular leaders within the fields of likelihood conception, stochastic techniques, stochastic differential equations, in addition to within the nonparametric estimation thought, and comparable fields. a few invited audio system have been early builders of the fields of chance and stochastic methods, developing the root of the fashionable likelihood thought. After the convention, to commemorate this distinct occasion, an IMA quantity devoted to Professor Rafail Z. Khasminskii was once prepare. It comprises 9 papers on a number of subject matters in chance and statistics. They comprise authoritative expositions in addition to major study papers of present curiosity. it's feasible that the amount can have an enduring influence at the extra improvement of stochastic research and nonparametric estimation.

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**Example text**

3), we obtain f (x ) = Ex[f (XTr )] = E[J(eSNrx + te rms independent of x )). 5) First , suppose that v > O. Differen t iation of Eq. 2, we get 111'11~ e- r 111'11< 00 . Letting here r -+ + 00 , we conclude that 111'11 = 0 and so If v = 0 then , differenti ating Eq . 6) we get 1" (x ) = E[e2SNr 1" (eSNr x f = const. + .. )), so t hat 111"11~ e- 2 r ll1"11< 00 . 9There is a slight pr oblem if X ,,; probability , so m ay be igno red . 7) but t his only ha pp ens wit h zero 46 L. BOGACHEV, G. DERFEL, S.

P. , An absorption probabi lity problem, J . Math . An al. Appl. 9 (1964 ) , 384-393 . K . GRINTSEVICHYUS, On the continuity of the distribution of a sum of dep endent variables con n ected with independent walks on lines, (Russian) Teor. Vero yatn. i Primenen. 1 9 (1974) , 163-168; (English translation) Theory P roba b. Appl. 19 (1974) , 163-168. [14J A . ISERLES, On th e generalized pantograph fun ctional-differential equation, European J . Appl. Math. 4 (1993) , 1- 38. K. LIU, On pantograph integro-differential equations, J.

2To be more precise, a certain vect or analog of Eq. 1). 29 30 L. BOGACHEV, G. DERFEL, S. MOLCHANOV, AND J. OCKENDON in a mathematical model for the dynamics of an overhead current collection system on an electric locomotive (with the physically relevant value q < 1) . At about the same time, a systematic analysis of solutions to the pantograph equation was started by Fox et at. [11], where various analytical, perturbation, and numerical techniques were discussed at length (for both q < 1 and q > 1).