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

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