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Register a new userA Mean Field Game of Optimal Portfolio Liquidation
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We consider a mean field game (MFG) of optimal portfolio liquidation under asymmetric information. We prove that the solution to the MFG can be characterized in terms of a FBSDE with possibly singular terminal condition on the backward component or, equivalently, in terms of a FBSDE with finite terminal value, yet singular driver. Extending the method of continuation to linear-quadratic FBSDE with singular driver we prove that the MFG has a unique solution. Our existence and uniqueness result allows to prove that the MFG with possibly singular terminal condition can be approximated by a sequence of MFGs with finite terminal values.
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We analyze novel portfolio liquidation games with self-exciting order flow. Both the N-player game and the mean-field game are considered. We assume that players trading activities have an impact on the dynamics of future market order arrivals thereby generating an additional transient price impact. Given the strategies of her competitors each player solves a mean-field control problem. We characterize open-loop Nash equilibria in both games in terms of a novel mean-field FBSDE system with unknown terminal condition. Under a weak interaction condition we prove that the FBSDE systems have unique solutions. Using a novel sufficient maximum principle that does not require convexity of the cost function we finally prove that the solution of the FBSDE systems do indeed provide existence and uniqueness of open-loop Nash equilibria.
A decentralized blockchain is a distributed ledger that is often used as a platform for exchanging goods and services. This ledger is maintained by a network of nodes that obeys a set of rules, called a consensus protocol, which helps to resolve inconsistencies among local copies of a blockchain. In this paper, we build a mathematical framework for the consensus protocol designer that specifies (a) the measurement of a resource which nodes strategically invest in and compete for in order to win the right to build new blocks in the blockchain; and (b) a payoff function for their efforts. Thus the equilibrium of an associated stochastic differential game can be implemented by selecting nodes in proportion to this specified resource and penalizing dishonest nodes by its loss. This associated, induced game can be further analyzed by using mean field games. The problem can be broken down into two coupled PDEs, where an individual nodes optimal control path is solved using a Hamilton-Jacobi-Bellman equation, where the evolution of states distribution is characterized by a Fokker-Planck equation. We develop numerical methods to compute the mean field equilibrium for both steady states at the infinite time horizon and evolutionary dynamics. As an example, we show how the mean field equilibrium can be applied to the Bitcoin blockchain mechanism design. We demonstrate that a blockchain can be viewed as a mechanism that operates in a decentralized setup and propagates properties of the mean field equilibrium over time, such as the underlying security of the blockchain.
This paper extends the project initiated in arXiv:2002.00201 and studies a lifecycle portfolio choice problem with borrowing constraints and finite retirement time in which an agent receives labor income that adjusts to financial market shocks in a path dependent way. The novelty here, with respect to arXiv:2002.00201, is the fact that we have a finite retirement time, which makes the model more realistic, but harder to solve. The presence of both path-dependency, as in arXiv:2002.00201, and finite retirement, leads to a two-stages infinite dimensional stochastic optimal control problem, a family of problems which, to our knowledge, has not yet been treated in the literature. We solve the problem completely, and find explicitly the optimal controls in feedback form. This is possible because we are able to find an explicit solution to the associated infinite dimensional Hamilton-Jacobi-Bellman (HJB) equation, even if state constraints are present. Note that, differently from arXiv:2002.00201 , here the HJB equation is of parabolic type, hence the work to identify the solutions and optimal feedbacks is more delicate, as it involves, in particular, time-dependent state constraints, which, as far as we know, have not yet been treated in the infinite dimensional literature. The explicit solution allows us to study the properties of optimal strategies and discuss their financial implications.
Mean field games are concerned with the limit of large-population stochastic differential games where the agents interact through their empirical distribution. In the classical setting, the number of players is large but fixed throughout the game. However, in various applications, such as population dynamics or economic growth, the number of players can vary across time which may lead to different Nash equilibria. For this reason, we introduce a branching mechanism in the population of agents and obtain a variation on the mean field game problem. As a first step, we study a simple model using a PDE approach to illustrate the main differences with the classical setting. We prove existence of a solution and show that it provides an approximate Nash-equilibrium for large population games. We also present a numerical example for a linear--quadratic model. Then we study the problem in a general setting by a probabilistic approach. It is based upon the relaxed formulation of stochastic control problems which allows us to obtain a general existence result.
This paper is devoted to the singular perturbation problem for mean field game systems with control on the acceleration. This correspond to a model in which the acceleration cost vanishes. So, we are interested in analyzing the behavior of solutions to the mean field game systems arising from such a problem as the acceleration cost goes to zero. In this case the Hamiltonian fails to be strictly convex and superlinear w.r.t. the momentum variable and this creates new issues in the analysis of the problem. We obtain that the limit problem is the classical mean field game system.
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