Evolving communities with individual preferences

Abstract

The goal of this paper is to provide mathematically rigorous tools for modelling the evolution of a community of interacting individuals. We model the population by a measure space (Ω, F, ν) where ν determines the abundance of individual preferences. The preferences of an individual ω ∈ Ω are described by a measurable choice X(ω) of a rough path.We aim to identify, for each individual, a choice for the forward evolution Y t(ω) for an individual in the community. These choices Y t(ω) must be consistent so that Yt(ω) correctly accounts for the individual's preference and correctly models their interaction with the aggregate behaviour of the community.In general, solutions are continuum of interacting threads analogous to the huge number of individual atomic trajectories that together make up the motion of a fluid. The evolution of the population need not be governed by any over-arching partial differential equation (PDE). Although one can match the standard non-linear parabolic PDEs of McKean-Vlasov type with specific examples of communities in this case. The bulk behaviour of the evolving population provides a solution to the PDE.We focus on the case of weakly interacting systems, where we are able to exhibit the existence and uniqueness of consistent solutions.An important technical result is continuity of the behaviour of the system with respect to changes in the measure ν assigning weight to individuals. Replacing the deterministic ν with the empirical distribution of an independent and identically distributed sample from ν leads to many standard models, and applying the continuity result allows easy proofs for propagation of chaos.The rigorous underpinning presented here leads to uncomplicated models which have wide applicability in both the physical and social sciences. We make no presumption that the macroscopic dynamics are modelled by a PDE.This work builds on the fine probability literature considering the limit behaviour for systems where a large number of particles are interacting with independent preferences; there is also work on continuum models with preferences described by a semi-martingale measure. We mention some of the key papers.

Integrability and tail estimates for Gaussian rough differential equations

“…This obstacle eventually boils down to the need for sharp estimates on the integrability of the Jacobian of the flow J X t←0 (y 0 ). Cass, Lyons [4] and Inahama [21] establish such integrability for the Brownian rough path, but only by using the independence of the increments; for more general Gaussian processes a more careful analysis is needed. To understand the difficulty of this problem, we note from [12] that the standard deterministic estimate on J X t←0 (y 0 ) gives |J x t←0 (y 0 )| ≤ C exp(C x p pvar;[0,T ] ).…”

“…Proof. We will only prove the lemma for the most difficult case p ∈ [3,4). By definition we have that…”

Tail estimates for Markovian rough paths

“…This example makes clear the importance of the Laplace transform when analysing the tail behaviour of N r 0 (X a,x , [0, T ]). What is important, as we will show, is not to have a closed-form expression as in (16), but instead to have an upper bound controlling its asymptotic behaviour as λ → ∞.…”

“…We therefore expect uses of our results to be widespread. In [16] it was observed that M(X, [0, T ]) appears in optimal Lipschitz-estimates on the rough path distance between two different RDE solutions. This has uses in fixed-point arguments, e.g.…”

The Inverse Problem for Rough Controlled Differential Equations

“…would follow from (14), giving, for a choice of M = 2c 1 2 , the conclusion Ψ y A ts , B ts − ϕ ts (y) > c 1 |t − s| a , contradicting identity (12), where X ts , X ts belongs to U for δ small enough, and the fact that A ts , B ts is a minimizer. This proves Theorem 6 in the special case where d ≥ m and where for some y ∈ R d the family V i (y), V j , V k (y) ; 1 ≤ i ≤ , 1 ≤ j < k ≤ is free.…”