VII: 11, 95-117, LNM 321 (1973)
EL KAROUI, Nicole;
REINHARD, Hervé
Processus de diffusion dans ${\bf R}^n$ (
Diffusion theory)
This paper concerns diffusions (without boundaries) whose generators have Borel bounded coefficients. It consists of two parts. The first one is devoted to the equivalence between the existence and uniqueness of the diffusion semigroup and the uniqueness in law of the solution of the corresponding Ito stochastic differential equation. This allows the authors to use in the elliptic case the deep results of Krylov on s.d.e.'s. The second part concerns mostly the Lipschitz case, and discusses several properties of the diffusion process in itself: the representation of additive functional martingales; the relations between the number of martingales necessary for the representation and the rank of the generator (locally); the existence of a dual diffusion; the support and absolute continuity properties of the semi-group
Comment: This paper is in part an improved version of a paper on degenerate diffusions by Bonami, El-Karoui, Reinhard and Roynette (
Ann. Inst. H. Poincaré, 7, 1971)
Keywords: Construction of diffusions,
Diffusions with measurable coefficients,
Degenerate diffusionsNature: Original
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XIX: 07, 91-112, LNM 1123 (1985)
SCHWARTZ, Laurent
Construction directe d'une diffusion sur une variété (
Stochastic differential geometry)
This seems to be the first use of Witney's embedding theorem to construct a process (a Brownian motion, a diffusion, a solution to some s.d.e.) in a manifold $M$ by embedding $M$ into some $
R^d$. Very general existence and uniqueness results are obtained
Comment: This method has since become standard in stochastic differential geometry; see for instance Émery's book
Stochastic Calculus in Manifolds (Springer, 1989)
Keywords: Diffusions in manifolds,
Stochastic differential equationsNature: Original Retrieve article from Numdam
XLIV: 02, 41-59, LNM 2046 (2012)
MIJATOVIĆ, Aleksandar;
NOVAK, Nika;
URUSOV, Mikhail
Martingale property of generalized stochastic exponentials (
Theory of martingales)
Keywords: Generalized stochastic exponentials,
Local martingales vs. true martingales,
One-dimensional diffusions
