Stochastic modelling of nonlinear dynamical systems

We develop a general theory dealing with stochastic models for dynamical systems that are governed by various nonlinear, ordinary or partial differential, equations. In particular, we address the problem how flows in the random medium (related to driving velocity fields which are generically bound to obey suitable local conservation laws) can be reconciled with the notion of dispersion due to a Markovian diffusion process.Comment: in D. S. Broomhead, E. A. Luchinskaya, P. V. E. McClintock and T. Mullin, ed., "Stochaos: Stochastic and Chaotic Dynamics in the Lakes", American Institute of Physics, Woodbury, Ny, in pres

Mean field games based on the stable-like processes

In this paper, we investigate the mean field games with K classes of agents who are weakly coupled via the empirical measure. The underlying dynamics of the representative agents is assumed to be a controlled nonlinear Markov process associated with rather general integro-differential generators of L´evy-Khintchine type (with variable coefficients), with the major stress on applications to stable and stable- like processes, as well as their various modifications like tempered stable-like processes or their mixtures with diffusions. We show that nonlinear measure-valued kinetic equations describing the dynamic law of large numbers limit for system with large number N of agents are solvable and that their solutions represent 1/N-Nash equilibria for approximating systems of N agents

Interacting stochastic processes on sparse random graphs

Large ensembles of stochastically evolving interacting particles describe phenomena in diverse fields including statistical physics, neuroscience, biology, and engineering. In such systems, the infinitesimal evolution of each particle depends only on its own state (or history) and the states (or histories) of neighboring particles with respect to an underlying, possibly random, interaction graph. While these high-dimensional processes are typically too complex to be amenable to exact analysis, their dynamics are quite well understood when the interaction graph is the complete graph. In this case, classical theorems show that in the limit as the number of particles goes to infinity, the dynamics of the empirical measure and the law of a typical particle coincide and can be characterized in terms of a much more tractable dynamical system of reduced dimension called the mean-field limit. In contrast, until recently not much was known about corresponding convergence results in the complementary case when the interaction graph is sparse (i.e., with uniformly bounded average degree). This article provides a brief survey of classical work and then describes recent progress on the sparse regime that relies on a combination of techniques from random graph theory, Markov random fields, and stochastic analysis. The article concludes by discussing ramifications for applications and posing several open problems.Comment: 26 pages, 4 figures; a version of this article will appear in the 2022 ICM Proceeding

Mean field games with controlled jump-diffusion dynamics: Existence results and an illiquid interbank market model

We study a family of mean field games with a state variable evolving as a multivariate jump diffusion process. The jump component is driven by a Poisson process with a time-dependent intensity function. All coefficients, i.e. drift, volatility and jump size, are controlled. Under fairly general conditions, we establish existence of a solution in a relaxed version of the mean field game and give conditions under which the optimal strategies are in fact Markovian, hence extending to a jump-diffusion setting previous results established in [30]. The proofs rely upon the notions of relaxed controls and martingale problems. Finally, to complement the abstract existence results, we study a simple illiquid inter-bank market model, where the banks can change their reserves only at the jump times of some exogenous Poisson processes with a common constant intensity, and provide some numerical results.Comment: 37 pages, 6 figure

Novel Lagrange sense exponential stability criteria for time-delayed stochastic Cohen–Grossberg neural networks with Markovian jump parameters: A graph-theoretic approach

This paper concerns the issues of exponential stability in Lagrange sense for a class of stochastic Cohen–Grossberg neural networks (SCGNNs) with Markovian jump and mixed time delay effects. A systematic approach of constructing a global Lyapunov function for SCGNNs with mixed time delays and Markovian jumping is provided by applying the association of Lyapunov method and graph theory results. Moreover, by using some inequality techniques in Lyapunov-type and coefficient-type theorems we attain two kinds of sufficient conditions to ensure the global exponential stability (GES) through Lagrange sense for the addressed SCGNNs. Ultimately, some examples with numerical simulations are given to demonstrate the effectiveness of the acquired result

A finite-dimensional approximation for partial differential equations on Wasserstein space

This paper presents a finite-dimensional approximation for a class of partial differential equations on the space of probability measures. These equations are satisfied in the sense of viscosity solutions. The main result states the convergence of the viscosity solutions of the finite-dimensional PDE to the viscosity solutions of the PDE on Wasserstein space, provided that uniqueness holds for the latter, and heavily relies on an adaptation of the Barles & Souganidis monotone scheme to our context, as well as on a key precompactness result for semimartingale measures. We illustrate this result with the example of the Hamilton-Jacobi-Bellman and Bellman-Isaacs equations arising in stochastic control and differential games, and propose an extension to the case of path-dependent PDEs