Properties of separable Banach-valued martingales

Abstract

This thesis is conerned with the extension of the classical theory of martingales of real random variables as contained in Doob [4] to the abstract theory of martingales of random variables whose values lie in a real Banach space. Extensions of almost all the convergence theorems in [4] for discrete parameter martingales can be found in Chatterji [1] and [2], Scalora [10], Tulcea and Tulcea [12], and Driml and Hans [5]. In addition to extending the existing theory this thesis also attempts to further the correlation between the abstract and classical theories. To pursue this aim I follow much of the development of [4] and show how frequently its proofs can be abstracted in a straight forward manner. To do this satisfactorily, it has been necessary to define and use a type of measurability for a Banach-valued function analogous to the type of measurability for a real-valued function used in [4]. In chapters 2 and 3 I demonstrate the properties of such a measurable function and those of its conditional expectations.