Stochastic Differential Equations and Martingale Problems
Linear equations and Ornstein-Uhlenbeck processes; strong existence, uniqueness, and nonexplosion criteria; weak solutions and local martingale problems; well-posedness and measurability; pathwise uniqueness and functional solution; weak existence and continuity; transformation of SDEs; strong Markov and Feller properties
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- Long before the existence of any general theory for SDEs, Langevin (1908) proposed his equation to model the velocity of a Brownian particle. The solution process was later studied by Ornstein and Uhlenbeck (1930) and was thus named after them. A more rigorous discussion appears in Doob (1942a).Google Scholar
- The general idea of a stochastic differential equation goes back to Bernstein (1934, 1938), who proposed a pathwise construction of diffusion processes by a discrete approximation, leading in the limit to a formal differential equation driven by a Brownian motion. However, Itô(1942a, 1951b) was the first author to develop a rigorous and systematic theory, including a precise definition of the integral, conditions for existence and uniqueness of solutions, and basic properties of the solution process, such as the Markov property and the continuous dependence on initial state. Similar results were obtained, later but independently, by Gihman (1947, 1950–51).Google Scholar
- The notion of a weak solution was introduced by Girsanov (1960), and a version of the weak existence Theorem 21.9 appears in Skorohod (1965). The ideas behind the transformations in Propositions 21.12 and 21.13 date back to Glrsanov (1960) and Volkonsky (1958), respectively. The notion of a martingale problem can be traced back to Levy’s martingale characterization of Brownian motion and Dynkin’s theory of the characteristic operator. A comprehensive theory was developed by Stroock and Varadhan (1969), who established the equivalence with weak solutions to the associated SDEs, obtained general criteria for uniqueness in law, and deduced conditions for the strong Markov and Feller properties. The measurability part of Theorem 21.10 is a slight extension of an exercise in Stroock and Varadhan (1979).Google Scholar
- Yamada and Watanabe (1971) proved that weak existence and path-wise uniqueness imply strong existence and uniqueness in law. Under the same conditions, they further established the existence of a functional solution, possibly depending on the initial distribution of the process; that dependence was later removed by Kallenberg (1996a). Ikeda and Watanabe (1989) noted how the notions of pathwise uniqueness and uniqueness in law extend by conditioning from degenerate to arbitrary initial distributions.Google Scholar
- The basic theory of SDEs is covered by many excellent textbooks on different levels, including Ikeda and Watanabe (1989), Rogers and Williams (1987), and Karatzas and Shreve (1991). More information on the martingale problem is available in JACOD (1979), Stroock and Varadhan (1979), and Ethier and Kurtz (1986).Google Scholar