Risk sharing between agents is important in the fields of risk management, finance, and insurance; with applications to problems such as the sharing of climate-related losses across insurers, the optimal allocation of credit risk in loan portfolios, and the redistribution of shocks in interconnected financial systems. The characterization of optimal allocations, which has roots in the seminal work of Borch (1962) and Wilson (1968), has been well studied in the literature for decades. Our work builds on recent developments in the field, which focus on obtaining crisp characterizations that can be numerically implemented (Liu, 2020; Ghossoub and Zhu, 2026). In Liu 2020, the author studies a weighted risk sharing problem and gives an explicit characterization when agent preferences are given by distortion risk measures. This characterization is further generalized to positively homogeneous monetary utilities in Ghossoub and Zhu 2026.

While the aforementioned literature considers a single-period framework (which we henceforth refer to as the static setting), the nature of investment and consumption is dynamic in reality. As a result, the literature has also devoted some effort to the study of the multi-period optimal allocation problem. However, in the existing literature on optimal consumption or allocation in a dynamic framework, Pareto efficiency is defined from the perspective of time-$0$ preferences. Future consumption streams, future endowment streams, or future risk profiles are first discounted to time $0$, and then evaluated. Our contention is that this leads to an inherently myopic view that fails to capture the dynamic structure of endowment profiles. We propose a novel approach that allows us to incorporate the dynamic structure of risk profiles into the decision-making process, through dynamic risk measures; and we introduce the associated notions of dynamic individual rationality and dynamic Pareto optimality.

Specifically, we study a dynamic optimal allocation problem where a collection of agents, each with an initial endowment, pool together their endowments and wish to reallocate the aggregate endowment based on individual risk preferences. We assume that each agent's initial endowment is given by a discrete time stochastic process, and agent preferences over discrete-time stochastic processes are represented by strongly time-consistent dynamic risk measures. We define the notion of dynamic Pareto optimality (or just PO for simplicity), which not only considers the risk of the allocation process at time $0$, but also incorporates the dynamic structure of risk profiles into the decision making process. In our first main result, we show that an allocation is PO if and only if it is the solution to a series of recursive (backward in time) optimization problems. This result generalizes the classical result that Pareto optimal allocations optimize the social welfare function in a single period setting.

Next, we extend the classical notion of comonotonicity to our dynamic setting and derive a comonotone improvement theorem for dynamic risk measures that preserve the convex order. We further study Pareto optima in the comonotone market, and show that each PO allocation in the comonotone market is also PO in the unconstrained market, therefore providing justification for studying the comonotone market. Our second main result provides a crisp characterization of comonotone Pareto optima when agent preferences are coherent and satisfy a property that we call equidistribution preserving. This allows for a straightforward algorithmic implementation, which we demonstrate in a two period example. Our numerical example highlights the differences between dynamic and static models, and it demonstrates how future risks affect allocations in earlier periods.