Bellman equation and viscosity solutions for mean-field stochastic control problem

We consider the stochastic optimal control problem of McKean-Vlasov stochastic differential equation where the coefficients may depend upon the joint law of the state and control. By using feedback controls, we reformulate the problem into a deterministic control problem with only the marginal distribution of the process as controlled state variable, and prove that dynamic programming principle holds in its general form. Then, by relying on the notion of differentiability with respect to pro\-bability measures recently introduced by P.L. Lions in [32], and a special It{\^o} formula for flows of probability measures, we derive the (dynamic programming) Bellman equation for mean-field stochastic control problem, and prove a veri\-fication theorem in our McKean-Vlasov framework. We give explicit solutions to the Bellman equation for the linear quadratic mean-field control problem, with applications to the mean-variance portfolio selection and a systemic risk model. We also consider a notion of lifted visc-sity solutions for the Bellman equation, and show the viscosity property and uniqueness of the value function to the McKean-Vlasov control problem. Finally, we consider the case of McKean-Vlasov control problem with open-loop controls and discuss the associated dynamic programming equation that we compare with the case of closed-loop controls.Comment: to appear in ESAIM: COC

At the Mercy of the Common Noise: Blow-ups in a Conditional McKean--Vlasov Problem

We extend a model of positive feedback and contagion in large mean-field systems, by introducing a common source of noise driven by Brownian motion. Although the driving dynamics are continuous, the positive feedback effect can lead to `blow-up' phenomena whereby solutions develop jump-discontinuities. Our main results are twofold and concern the conditional McKean--Vlasov formulation of the model. First and foremost, we show that there are global solutions to this McKean--Vlasov problem, which can be realised as limit points of a motivating particle system with common noise. Furthermore, we derive results on the occurrence of blow-ups, thereby showing how these events can be triggered or prevented by the pathwise realisations of the common noise.Comment: 43 pages, 3 figures, substantial revision with generalisation of results in Section 2 and improved presentation overal

A Spatiotemporal Analysis of the McKean Complex on the Northern Plains

Characterizing hunter-gatherer mobility has been problematic in archaeological research (Anthony 1990). For pre-contact cultures on the Northern Plains there is no documentation of the human decisions involved in movement processes. This thesis examines the known information available regarding the McKean Complex on the Northern Plains. Using radiocarbon ages and known site locations, Kriging analysis was used to create a predictive model to examine spread of this archaeological complex, directions of movement, and origins. This thesis re-examines existing theories regarding origin and migration with regards to this model. The geographic distribution of projectile point styles, floral remains and faunal remains are also examined. This research provides a comprehensive database of stratified sites with McKean components as well as a comprehensive database of McKean radiocarbon ages associated with McKean projectile points. This study offers new information regarding subsistence and expansion of the complex, providing a preliminary model towards re-examining the McKean Complex. The model will benefit from future research with regards to the McKean Complex as more radiocarbon ages taken from McKean sites can only help improve the current model and help provide a greater understanding of this Complex on the Northern Plains

A continuum of weakly coupled oscillatory McKean neurons

The McKean model of a neuron possesses a one dimensional fast voltage-like variable and a slow recovery variable. A recent geometric analysis of the singularly perturbed system has allowed an explicit construction of its phase response curve [S Coombes 2001 Phase-locking in networks of synaptically coupled McKean relaxation oscillators, Physica D, Vol 160, 173-188]. Here we use tools from coupled oscillator theory to study weakly coupled networks of McKean neurons. Using numerical techniques we show that the McKean system has traveling wave phase-locked solutions consistent with that of a network of more biophysically detailed Hodgkin-Huxley neurons

Mean Field description of and propagation of chaos in recurrent multipopulation networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons

We derive the mean-field equations arising as the limit of a network of interacting spiking neurons, as the number of neurons goes to infinity. The neurons belong to a fixed number of populations and are represented either by the Hodgkin-Huxley model or by one of its simplified version, the Fitzhugh-Nagumo model. The synapses between neurons are either electrical or chemical. The network is assumed to be fully connected. The maximum conductances vary randomly. Under the condition that all neurons initial conditions are drawn independently from the same law that depends only on the population they belong to, we prove that a propagation of chaos phenomenon takes places, namely that in the mean-field limit, any finite number of neurons become independent and, within each population, have the same probability distribution. This probability distribution is solution of a set of implicit equations, either nonlinear stochastic differential equations resembling the McKean-Vlasov equations, or non-local partial differential equations resembling the McKean-Vlasov-Fokker- Planck equations. We prove the well-posedness of these equations, i.e. the existence and uniqueness of a solution. We also show the results of some preliminary numerical experiments that indicate that the mean-field equations are a good representation of the mean activity of a finite size network, even for modest sizes. These experiment also indicate that the McKean-Vlasov-Fokker- Planck equations may be a good way to understand the mean-field dynamics through, e.g., a bifurcation analysis.Comment: 55 pages, 9 figure

Empirical approximation to invariant measures of non-degenerate McKean--Vlasov dynamics

This paper studies the approximation of invariant measures of McKean--Vlasov dynamics with non-degenerate additive noise. While prior findings necessitated a strong monotonicity condition on the McKean--Vlasov process, we expand these results to encompass dissipative and weak interaction scenarios. Utilizing a reflection coupling technique, we prove that the empirical measures of the McKean--Vlasov process and its path-dependent counterpart can converge to the invariant measure in the Wasserstein metric. The Curie--Weiss mean-field lattice model serves as a numerical example to illustrate empirical approximation.Comment: 21 pages, 1 figur

A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion

In this thesis, we mainly study some properties for certain stochastic di↵er-ential equations.The types of stochastic di↵erential equations we are interested in are (i) stochastic di↵erential equations driven by Brownian motion, (ii) stochastic functional di↵erential equations driven by fractional Brownian motion, (iii) McKean-Vlasov stochastic di↵erential equations driven by Brownian motion,(iv) McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion.The properties we investigate include the weak approximation rate of Euler-Maruyama scheme, the central limit theorem and moderate deviation principle for McKean-Vlasov stochastic di↵erential equations. Additionally, we investigate the existence and uniqueness of solution to McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion, and then the Bismut formula of Lion’s derivatives for this model is also obtained.The crucial method we utilised to establish the weak approximation rate of Euler-Maruyama scheme for stochastic equations with irregular drift is the Girsanov transformation. More precisely, giving a reference stochastic equa-tions, we construct the equivalent expressions between the aim stochastic equations and associated numerical stochastic equations in another proba-bility spaces in view of the Girsanov theorem.For the Mckean-Vlasov stochastic di↵erential equation model, we first construct the moderate deviation principle for the law of the approxima-tion stochastic di↵erential equation in view of the weak convergence method. Subsequently, we show that the approximation stochastic equations and the McKean-Vlasov stochastic di↵erential equations are in the same exponen-tially equivalent family, and then we establish the moderate deviation prin-ciple for this model.Based on the result of Well-posedness for Mckean-Vlasov stochastic di↵er-ential equation driven by fractional Brownian motion, by using the Malliavin analysis, we first establish a general result of the Bismut type formula for Lions derivative, and then we apply this result to the non-degenerate case of this model