Local martingales measures. (English) Zbl 0996.60054

Štěpán, J.; Ševčík, P.

Acta Univ. Carol., Math. Phys. 41, No. 1, 37-55 (2000).

Local martingale problems arising in stochastic analysis and financial mathematics can be formulated in a canonical framework considering the measurable space of the continuous functions on \([0,+\infty)\) endowed with the topology of uniform convergence on compact sets and its corresponding Borel \(\sigma\)-algebra. In this setting, solutions to local martingale problems are Borel probability measures on this measurable space.
The present paper studies some geometrical and topological properties of sets of solutions to local martingale problems. It is proved that, in many important cases, these sets of solutions are convex and weakly closed sets. As a consequence, some relationships between solutions with deterministic and arbitrary stochastic initial conditions are presented. Then, the authors apply their results to three interesting instances of local martingale problems: the set of local martingale distributions, the set of probability distributions of weak solutions of a stochastic differential equation and the set of probability distributions under which a given process becomes a local martingale with a given quadratic variation.
This article has a continuation in the paper reviewed below, where the study of measure convexity for the sets of solutions of local martingale problems is treated.

Reviewer: Miguel Ángel Mirás Calvo (Vigo)

MSC:

60G48	Generalizations of martingales
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
91B28	Finance etc. (MSC2000)

Keywords:

local martingale problems; weakly closed sets of probability distributions; stochastic differential equations