Complements sur les martingales conformes. (Complements on conformal martingales). (French) Zbl 0621.60055

Schwartz, Laurent

Osaka J. Math. 23, 77-116 (1986).

This is a study of continuous semimartingales \(M_ t(\omega)\) and in particular of conformal martingales which take values in an N-dimensional complex manifold V, and which are defined for (t,\(\omega)\) in some open subset \(A\subset {\mathbb{R}}_+\times \Omega\). It continues the authors work ”Semimartingales sur des variétés, et martingales conformes sur des variétés analytiques complexes.” Lecture Notes Math. 780 (1980; Zbl 0433.60047).
\(\{\) \(M_ t,t\geq 0\}\) is a conformal martingale if, roughly, \(f(M_ t)\) is an ordinary conformal martingale for each holomorphic f on V. This definition is localized to handle processes defined on random parameter sets, and various alternative characterizations are discussed.
In particular, the author gives a characterization of conformal martingales which used the 2 tangent space \(T^{2,0}(V)\). Since this avoids mention of holomorphic functions it applies to V which are not Stein manifolds and which may not have enough globally-defined holomorphic functions.
A quite different characterization uses connections: a V-semimartingale is conformal iff it is a martingale for all \(C^{\infty}\) connections compatible with the complex structure.
The final section addresses the question of semimartingale extensions: when can a semimartingale defined on the half open stochastic interval [S,T) be extended to the closed interval [S,T]?

Reviewer: J.Walsh

MSC:

60G48	Generalizations of martingales
60H99	Stochastic analysis
60G44	Martingales with continuous parameter
32C99	Analytic spaces

Keywords:

semimartingales; conformal martingales; Stein manifolds

Citations:

Zbl 0433.60047