A duality approach to a price formation MFG model

where Q, uT , and m0 are given functions, m0 is a probability measure on R, and the triplet (u,m,̟) is the unknown. Here, the state of a typical agent is the variable x ∈ R and represents the assets of that agent. The distribution of assets in the population of the agents at time t is encoded in the probability measure m(·, t). The agents change their assets by trading at a price ̟(t). The trading is subject to a balance condition encoded in the third equation in (1.1). This integral constraint that guarantees supply Q(t) meets demand is represented by the term on the left-hand side of that condition. As introduced in [36], u is the value function of an agent who trades a commodity with supply Q and price ̟. The function u is characterized by the first equation in (1.1) and the terminal condition in (1.2). Each agent selects their trading rate in order to minimize a given cost functional (see (1.7) below). The optimal control selection is −Hp(x,̟(t) + ux(t, x)). Under this optimal control, the density m describing the population of agents evolves according to the second equation in (1.1) and the initial condition in (1.2). The third equation in (1.1), which we refer to as the balance condition, is an integral constraint that guarantees supply meets demand.

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