Delayed switching identities and multi-marginal solutions to the Skorokhod embedding problem

In this article, we consider a generalisation of the Skorokhod embedding problem (SEP) with a delayed starting time. In the delayed SEP, we look for stopping times which embed a given measure in a stochastic process, which occur after a given delay time. Our first contribution is to show that the switching identities introduced in a recent paper of Backhoff, Cox, Grass and Huesmann extend to the case with a delay. We then show that the delayed switching identities can be used to establish an optimal stopping representation of Root and Rost solutions to the multi-marginal Skorokhod embedding problem. We achieve this by rephrasing the multi-period problem into a one-period framework with delay. This not only recovers the known multi-marginal representation of Root, but also establishes a previously unknown optimal stopping representation associated to the multi-marginal Rost solution. The Rost case is more complex than the Root case since it naturally requires randomisation for general initial measures, and we develop the necessary tools to develop these solutions. Our work also provides a comprehensive and complete treatment of discrete Root and Rost solutions, embedding discrete measures into simple symmetric random walks.