One-Dimensional SDEs and Diffusions
Weak existence and uniqueness; pathwise uniqueness and comparison; scale function and speed measure; time-change representation; boundary classification; entrance boundaries and Feller properties; ratio ergodic theorem; recurrence and ergodicity
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- The study of continuous Markov processes and the associated parabolic differential equations, initiated by Kolmogorov (1931a) and Feller (1936), took a new direction with the seminal papers of Feller (1952, 1954), who studied the generators of one-dimensional diffusions within the framework of the newly developed semigroup theory. In particular, Feller gave a complete description in terms of scale function and speed measure, classified the boundary behavior, and showed how the latter is determined by the domain of the generator. Finally, he identified the cases when explosion occurs, corresponding to the absorption cases in Theorem 23.15.Google Scholar
- A more probabilistic approach to these results was developed by Dynkin (1955b, 1959), who along with Ray (1956) continued Feller’s study of the relationship between analytic properties of the generator and sample path properties of the process. The idea of constructing diffusions on a natural scale through a time change of Brownian motion is due to Hunt (1958) and Volkonsky (1958), and the full description in Theorem 23.9 was completed by Volkonsky (1960) and Itô and McKean (1965). The present stochastic calculus approach is based on ideas in Méléard (1986).Google Scholar
- The ratio ergodic Theorem 23.14 was first obtained for Brownian motion by Derman (1954), by a method originally devised for discrete-time chains by Doeblin (1938). It was later extended to more general diffusions by Motoo and Watanabe (1958). The ergodic behavior of recurrent one-dimensional diffusions was analyzed by Maruyama and Tanaka (1957).Google Scholar
- For one-dimensional SDEs, Skorohod (1965) noticed that Itô’s original Lipschitz condition for pathwise uniqueness can be replaced by a weaker Hölder condition. He also obtained a corresponding comparison theorem. The improved conditions in Theorems 23.3 and 23.5 are due to Yamada and Watanabe (1971) and Yamada (1973), respectively. Perkins (1982) and Le Gall (1983) noted how the use of semimartingale local time simplifies and unifies the proofs of those and related results. The fundamental weak existence and uniqueness criteria in Theorem 23.1 were discovered by Engelbert and Schmidt (1984, 1985), whose (1981) zero-one law is implicit in Lemma 23.2.Google Scholar
- Elementary introductions to one-dimensional diffusions appear in Breiman (1968), Freedman (1971b), and Rogers and Williams (2000b). More detailed and advanced accounts are given by Dynkin (1965) and Itô and McKean (1965). Further information on one-dimensional SDEs may be obtained from the excellent books by Karatzas and Shreve (1991) and Revuz and Yor (1999).Google Scholar