Equilibrium in Functional Stochastic Games with Mean-Field Interaction

We consider a general class of finite-player stochastic games with mean-field interaction, in which the linear-quadratic cost functional includes linear operators acting on controls in $L^2$. We propose a novel approach for deriving the Nash equilibrium of the game explicitly in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations of the second kind and deriving their closed form solution. Furthermore, by proving stability results for the system of stochastic Fredholm equations we derive the convergence of the equilibrium of the $N$-player game to the corresponding mean-field equilibrium. As a by-product we also derive an $\varepsilon$-Nash equilibrium for the mean-field game, which is valuable in this setting as we show that the conditions for existence of an equilibrium in the mean-field limit are less restrictive than in the finite-player game. Finally we apply our general framework to solve various examples, such as stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact.Comment: 48 page

Financial asset bubbles in banking networks

We consider a banking network represented by a system of stochastic differential equations coupled by their drift. We assume a core-periphery structure, and that the banks in the core hold a bubbly asset. The banks in the periphery have not direct access to the bubble, but can take initially advantage from its increase by investing on the banks in the core. Investments are modeled by the weight of the links, which is a function of the robustness of the banks. In this way, a preferential attachment mechanism towards the core takes place during the growth of the bubble. We then investigate how the bubble distort the shape of the network, both for finite and infinitely large systems, assuming a non vanishing impact of the core on the periphery. Due to the influence of the bubble, the banks are no longer independent, and the law of large numbers cannot be directly applied at the limit. This results in a term in the drift of the diffusions which does not average out, and that increases systemic risk at the moment of the burst. We test this feature of the model by numerical simulations.Comment: 33 pages, 6 table

Modeling Financial System with Interbank Flows, Borrowing, and Investing

In our model, private actors with interbank cash flows similar to, but nore general than (Carmona, Fouque, Sun, 2013) borrow from the outside economy at a certain interest rate, controlled by the central bank, and invest in risky assets. Each private actor aims to maximize its expected terminal logarithmic utility. The central bank, in turn, aims to control the overall economy by means of an exponential utility function. We solve all stochastic optimal control problems explicitly. We are able to recreate occasions such as liquidity trap. We study distribution of the number of defaults (net worth of a private actor going below a certain threshold).Comment: 27 pages, 29 figures. Keywords: systemic risk, stochastic control, principal-agent problem, stationary distribution, stochastic stability, Lyapunov functio

Stochastic Delay Differential Games: Financial Modeling and Machine Learning Algorithms

In this paper, we propose a numerical methodology for finding the closed-loop Nash equilibrium of stochastic delay differential games through deep learning. These games are prevalent in finance and economics where multi-agent interaction and delayed effects are often desired features in a model, but are introduced at the expense of increased dimensionality of the problem. This increased dimensionality is especially significant as that arising from the number of players is coupled with the potential infinite dimensionality caused by the delay. Our approach involves parameterizing the controls of each player using distinct recurrent neural networks. These recurrent neural network-based controls are then trained using a modified version of Brown's fictitious play, incorporating deep learning techniques. To evaluate the effectiveness of our methodology, we test it on finance-related problems with known solutions. Furthermore, we also develop new problems and derive their analytical Nash equilibrium solutions, which serve as additional benchmarks for assessing the performance of our proposed deep learning approach.Comment: 29 pages, 8 figure

Preopening and equilibrium selection

In this paper, the authors introduce a form of pre-play communication that we call "preopening". During the preopening, players announce their tentative actions to be played in the underlying game. Announcements are made using a posting system which is subject to stochastic failures. Posted actions are publicly observable and players payo¤s only depend on the opening outcome, i.e. the action pro…le that is posted at the end of the preopening phase. We show that when the posting failures hit players idiosyncratically all equilibria of the preopening game lead to the same opening outcome that corresponds to the most "sensible" pure Nash equilibrium of the underlying game. By contrast preopening does not operate an equilibrium selection when posting failure hits players simultaneously.Preopening; equilibrium selection; bargaining; cheap talk