We study the stochastic effect on the threedimensional inviscid primitive equations (PEs, also called the hydrostatic Euler equations). Specifically, we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the illposedness in Sobolev spaces of PEs without horizontal viscosity. Under proper conditions, we prove the local existence of martingale solutions and pathwise uniqueness. By adding vertical viscosity,
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Abstract i.e. , considering the hydrostatic NavierStokes equations, we can relax the restriction on initial conditions to be only analytic in the horizontal variables with Sobolev regularity in the vertical variable, and allow the transport noise in the vertical direction. We establish the local existence of martingale solutions and pathwise uniqueness, and show that the solutions become analytic in the vertical variable instantaneously as and the vertical analytic radius increases as long as the solutions exist.$$t>0$$ $t>0$ 
Stochastic differential games have been used extensively to model agents' competitions in finance, for instance, in P2P lending platforms from the Fintech industry, the banking system for systemic risk, and insurance markets. The recently proposed machine learning algorithm, deep fictitious play, provides a novel and efficient tool for finding Markovian Nash equilibrium of large
player asymmetric stochastic differential games [J. Han and R. Hu, Mathematical and Scientific Machine Learning Conference, pages 221245, PMLR, 2020]. By incorporating the idea of fictitious play, the algorithm decouples the game into\begin{document}$ N $\end{document} suboptimization problems, and identifies each player's optimal strategy with the deep backward stochastic differential equation (BSDE) method parallelly and repeatedly. In this paper, we prove the convergence of deep fictitious play (DFP) to the true Nash equilibrium. We can also show that the strategy based on DFP forms an\begin{document}$ N $\end{document} Nash equilibrium. We generalize the algorithm by proposing a new approach to decouple the games, and present numerical results of large population games showing the empirical convergence of the algorithm beyond the technical assumptions in the theorems.\begin{document}$ \epsilon $\end{document} 
Jin, Shi (Ed.)In this paper, we apply the idea of fictitious play to design deep neural networks (DNNs), and develop deep learning theory and algorithms for computing the Nash equilibrium of asymmetric Nplayer nonzerosum stochastic differential games, for which we refer as deep fictitious play, a multistage learning process. Specifically at each stage, we propose the strategy of letting individual player optimize her own payoff subject to the other players’ previous actions, equivalent to solving N decoupled stochastic control optimization problems, which are approximated by DNNs. Therefore, the fictitious play strategy leads to a structure consisting of N DNNs, which only communicate at the end of each stage. The resulting deep learning algorithm based on fictitious play is scalable, parallel and modelfree, i.e., using GPU parallelization, it can be applied to any Nplayer stochastic differential game with different symmetries and heterogeneities (e.g., existence of major players). We illustrate the performance of the deep learning algorithm by comparing to the closedform solution of the linear quadratic game. Moreover, we prove the convergence of fictitious play under appropriate assumptions, and verify that the convergent limit forms an openloop Nash equilibrium. We also discuss the extensions to other strategies designed upon fictitious play and closedloopmore »

We propose a deep neural networkbased algorithm to identify the Markovian Nash equilibrium of general large 𝑁player stochastic differential games. Following the idea of fictitious play, we recast the 𝑁player game into 𝑁 decoupled decision problems (one for each player) and solve them iteratively. The individual decision problem is characterized by a semilinear HamiltonJacobiBellman equation, to solve which we employ the recently developed deep BSDE method. The resulted algorithm can solve large 𝑁player games for which conventional numerical methods would suffer from the curse of dimensionality. Multiple numerical examples involving identical or heterogeneous agents, with riskneutral or risksensitive objectives, are tested to validate the accuracy of the proposed algorithm in large group games. Even for a fiftyplayer game with the presence of common noise, the proposed algorithm still finds the approximate Nash equilibrium accurately, which, to our best knowledge, is difficult to achieve by other numerical algorithms.