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On Absolute Continuity of Feller's One‐Dimensional Diffusion Processes
Author(s) -
Groh Jürgen
Publication year - 1984
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19841160122
Subject(s) - mathematics , absolute continuity , stochastic differential equation , semimartingale , mathematical analysis , real line , generator (circuit theory) , operator (biology) , state space , diffusion , class (philosophy) , power (physics) , biochemistry , physics , chemistry , statistics , repressor , quantum mechanics , artificial intelligence , computer science , transcription factor , gene , thermodynamics
A class of F ELLER 's one‐dimemsional continuous strong M ARKOV processes generated by the generalized second order differential operator D m D 8 + is considered. In the case of natural boundaries of the state space ℜ and an identical road map s ( x ) = x , these diffusion processes are martingales. In a first part of this note some earlier results concerning the representation of such processes as weak solutions of stochastic differential equations are improved. The second part concerns with diffusions absolutely continuous with respect to a given one, determined by the generator D m D x + . Such absolutely continuous diffusions on the line were first described analytically by S. O REY in terms of the corresponding speed measures and road maps. By the aid of the derived stochastic equations an explicit expression for the corresponding R ADON ‐N IKODYM derivatives is possible. This allows a characterization of diffusions with non‐identical scale functions by stochastic differential equations.