A. In classical optimal transport, the contributions of Benamou-Brenier and Mc-Cann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas.Stretched Brownian motion provides an analogue for the martingale version of this problem. In this article we provide a characterization in terms of gradients of convex functions, similar to the characterization of optimizers in the classical transport problem for quadratic distance cost.

We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval [0, 1]. The problem is related to the study of extremal martingales associated with a peacock ("process increasing in convex order", by Hirsch, Profeta, Roynette and Yor [16]). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordère, Ob lój, Spoida and Touzi [13]. Under technical conditions, some explicit characteristics of the solutions to the optimal SEP as well as to its dual problem are obtained. We also discuss the associated martingale inequality.

We combine forward investment performance processes and ambiguityaverse portfolio selection. We introduce robust forward criteria which address ambiguity in the specification of the model, the risk preferences and the investment horizon. They encode the evolution of dynamically consistent ambiguity-averse preferences.We focus on establishing dual characterisations of the robust forward criteria, which is advantageous as the dual problem amounts to the search for an infimum whereas the primal problem features a saddle point. Our approach to duality builds on ideas developed in Schied (Finance Stoch. 11:107-129, 2007) and Žitković (Ann. Appl. Probab. 19:2176-2210). We also study in detail the so-called timemonotone criteria. We solve explicitly the example of an investor who starts with logarithmic utility and applies a quadratic penalty function. Such an investor builds a dynamic estimate of the market price of riskλ and updates her stochastic utility in accordance with the so-perceived elapsed market opportunities. We show that this leads to a time-consistent optimal investment policy given by a fractional Kelly strategy associated withλ and with the leverage being proportional to the investor's confidence in her estimate. Keywords

Abstract. We consider the problem of finding model-independent bounds on the price of an Asian option, when the call prices at the maturity date of the option are known. Our methods differ from most approaches to modelindependent pricing in that we consider the problem as a dynamic programming problem, where the controlled process is the conditional distribution of the asset at the maturity date. By formulating the problem in this manner, we are able to determine the model-independent price through a PDE formulation. Notably, this approach does not require specific constraints on the payoff function (e.g. convexity), and would appear to generalise to many related problems.

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