Convergence theorem of Pettis integrable multivalued pramart

Abstract

Purpose – In this work, the authors are interested in the notion of vector valued and set valued Pettis integrable pramarts. The notion of pramart is more general than that of martingale. Every martingale is a pramart, but the converse is not generally true. Design/methodology/approach – In this work, the authors present several properties and convergence theorems for Pettis integrable pramarts with convex weakly compact values in a separable Banach space. Findings – The existence of the conditional expectation of Pettis integrable mutifunctions indexed by bounded stopping times is provided. The authors prove the almost sure convergence in Mosco and linear topologies of Pettis integrable pramarts with values in (cwk(E)) the family of convex weakly compact subsets of a separable Banach space. Originality/value – The purpose of the present paper is to present new properties and various new convergence results for convex weakly compact valued Pettis integrable pramarts in Banach space.

Keywords

• Conditional expectation
• Martingale
• Pettis integrable pramart
• Subpramart
• Pettis integral
• Hausdorff distance