In Chapter 2, we introduce the martingale quantisation scheme which, given probabilities $\mu\leqcx\nu$ in convex order and defined on a separable Banach space $B$, constructs finitely-supported approximations $\mu_n\to\mu, \nu_n\to\nu$ which are in convex order $\mu_n\leqcx\nu_n$. We provide upper-bounds for the speed of convergence, in terms of the Wasserstein distance. We discuss the (dis)advantages of our algorithm and its link with the discretisation of the Martingale Optimal Transport problem. We study the operation which given $\mu/\nu$ and some (finite) partition of $B$, outputs $\mu_n/\nu_n$, showing that applied to a probability $\gamma$ and to all partitions it outputs the set of all finitely-supported probabilities $\zeta\leqcx\gamma$.

In Chapter 3, we investigate cases where we can explicitly construct a martingale transport between $\mu,\nu$ such that $\mu\leqcx \nu$ and propose suitable variants of the martingale quantisation which apply in those cases. We show that this happens for a vast parametric classes of stable and finite-moment log-stable distributions which have important applications in mathematical finance. For such classes of distributions, we provide sufficient conditions (and also necessary for the log-stable case) for the convex ordering $\mu\leqcx\nu$. We additionally propose a generalisation of the martingale quantisation and carry out numerical examples in which we illustrate the implementation of the proposed constructions.

In Chapter 4, we extend the notion of conditional expectation to all real-valued random variables. We do so by providing several different definitions and show that they are all equivalent. As an application, we use the extended definition of conditional expectation to generalise the martingale quantisation applied to probability measures $\mu,\nu$ which do not necessarily have finite first moment and such that $\nu$ is a dilation of $\mu$.