Stochastics and Applications

In this talk I will explain how a certain stochastic pressure equation appears in modelling enhanced geothermal heating (EGS) and how we approach the existence problem. EGS consists of pushing water through crystalline crustal rock at depths of 6-8km, the heat from the rock can then be extracted. Based on empirical observations it seems that the porosity satisfies a log like correlation from at least mm to km scale, the permeability (diffusion coefficient) is assumed to be the exponential of permeability. As a “simple” model, we model the porosity using a Gaussian log-correlated field, the properly normalized exponential thus becomes the so called multiplicative chaos measure. Using Gaussian analysis, we transform the Wick renormalized stochastic problem into a family of weighted elliptic equations, and I will show how regularity of these equations imply existence for the stochastic solution. Joint work with: Benny Avelin (Uppsala), Tuomo Kuusi (Helsinki), Patrik Nummi (Aalto), Eero Saksman (Helsinki), and Lauri Viitasaari (Aalto).

We study a mean field game in continuous time over a finite horizon, T, where the state of each agent is binary and where players base their strategic decisions on two, possibly competing, factors: the willingness to align with the majority (conformism) and the aspiration of sticking with the own type (stubbornness). We also consider a quadratic cost related to the rate at which a change in the state happens: changing opinion may be a costly operation. Depending on the parameters of the model, the game may have more than one Nash equilibrium, even though the corresponding N-player game does not. Moreover, it exhibits a very rich phase diagram, where polarized/unpolarized, coherent/incoherent equilibria may coexist, except for T small, where the equilibrium is always unique. We fully describe such phase diagram in closed form and provide a detailed numerical analysis of the N-player counterpart of the mean field game. Joint work with Paolo Dai Pra (Verona) and Elena Sartori (Padova).

We will discuss some recent results concerning weak and strong well-posedness of nonlinear stable driven SDEs with convolution interaction kernel, where the kernel belongs to a suitable Besov space. We will in particular characterize how singular the kernel can be in function of the stability index of the driving noise. In connection with some concrete models, some convergence rates for an approximating particle system will be discussed.

In this paper, we introduce a novel observation-driven model for high-dimensional correlation matrices, wherein the largest conditional eigenvalues are modelled dynamically. We impose equal correlations for any pair of assets from the same sector(s), which facilitates the use of a highly efficient alternative expression of the likelihood of a tν-distributed random vector. This alternative expression utilises the canonical form for block correlation matrices by Archakov and Hansen (2020). The dynamics of the eigenvalues is obtained from the Generalised Autoregressive Score (GAS) framework by Creal et al. (2011). We provide an empirical application by constructing Global Minimum Variance (GMV) portfolios using daily returns of 200 assets. In its simplest form, where just a single eigenvalue is updated, our model is extremely fast to estimate. It surpasses the Dynamic Equicorrelation (DECO) model model by Engle and Kelly (2012) and rivals their Block DECO (BDECO) model’s performance in achieving low variance in GMV portfolio returns. Joint work with: Stan Thijssen and Andre Lucas.

In the context of finite horizon mean field games with continuous time dynamics driven by additive Wiener noise, we introduce a notion of coarse correlated equilibrium in open-loop strategies. For non-cooperative many-player games, a coarse correlated equilibrium can be seen as a lottery on strategy profiles run according to a publicly known mechanism by a moderator who uses the (non-public) lottery outcomes to tell players in private which strategy to play. Players have to decide in advance whether to pre-commit to the mediator's recommendations or to play without seeing them. We justify our definition by showing that any coarse correlated solution of the mean field game induces approximate coarse correlated equilibria for the underlying N-player games. An existence result for coarse correlated mean field game solutions, not relying on the existence of classical solutions, will be given; an explicitly solvable example will be discussed as well. Joint work with Luciano Campi and Federico Cannerozzi (University of Milan "La Statale").

Modeling traffic dynamics has highlighted some universal properties of emergent phenomena, like the stop and go congestion when the vehicle density overcomes a certain threshold. The congestion formation on a urban road network is one of the main issues for the development of a sustainable mobility in the future smart cities and different models have been proposed. The quantification of the congestion degree for a city has been considered by various authors and data driven models have been develpoed using the large data sets on individual mobility provided by the Information Communication Technologies. However the simulation results suggest the existence of universal features for the transition to global congested states on a road network. We cope with the question if simple transport models on graph can reproduce universal features of congestion formation and the existence of control parameters is still an open problem. We propose a reductionist approach to this problem studying a simple transport model on a homogeneous road network by means of a random process on a graph. Each node represents a location and the links connect the different locations. We assume that each node has a finite transport capacity and it can contain a finite number of particles (vehicles). The dynamics is realized by a random walk on graphs where each node has a finite flow and move particles toward the connected nodes according to given transition rates (link weights). Each displacement is possible if the number of particles in the destination nodes is smaller than their maximal capacity. The graph structure can be very simple, like a uniform grid, but we have also considered random graphs with maximum in and out degree, to simulate more realistic transport networks. We study the properties the stationary distributions of the particles on the graph and the possibility of the applying the entropy concept of Statistical Mechanics to characterize the stationary distributions and to understand the congestion formation.

TBA

This lecture starts from two famous discrete-time dynamic models in economics, namely the Cobweb model to describe price dynamics and the Cournot duopoly model to describe competition between two firms producing homogeneous goods, and shows how their study has stimulated new fruitful streams of literature rooted in the field of qualitative analysis of nonlinear discrete dynamical systems. In the case of the Cobweb model, starting from the standard one-dimensional dynamic model the introduction of new kinds of expectations and learning mechanisms open new mathematical research about two-dimensional maps with a vanishing denominator, leading to the study of new kinds of singularities called focal points and prefocal curves. Analogously, in the case of the two-dimensional Cournot duopoly model, some recent developments are described concerning the introduction of nonlinearities leading to multistability, i.e. the coexistence of several stable equilibria, with the related problem of the delimitation of basins of attraction, which requires a global dynamical analysis based on the method of critical curves. Moreover, in the particular case of identical players, some recent results about chaos synchronization and related bifurcations (such as riddling or blowout bifurcation) are illustrated, with extensive reference to the rich and flourishing recent stream of literature.

We prove a second-order smooth-fit principle for a class of infinite-dimensional singular stochastic control problems. These can be thought of as spatial monotone-follower problems and find applications in spatial models of production and climate transition. Let (D, M, μ) be a finite measure space and consider the Hilbert space H := L^2(D, M, μ; R). Let then X be a H-valued stochastic process on a suitable complete probability space, whose evolution is determined through an SPDE driven by a linear operator A and affected by a cylindrical Brownian motion. The evolution of X is controlled linearly via a vector-valued control consisting of the direction and the intensity of action, a real-valued nondecreasing right-continuous stochastic process, adapted to the underlying filtration. The goal is to minimize an infinite time-horizon, discounted convex cost-functional. By combining properties of semiconcave functions and techniques from viscosity theory, we first show that the value function of the problem V is a C^{1,Lip}(H)-viscosity solution to the corresponding dynamic programming equation, which here takes the form of a variational inequality with gradient constraint. Then, allowing the decision maker to choose only the intensity of the control, and requiring that the given direction of control n is an eigenvector of the linear operator A, we establish that the directional derivative V_n is of class C^1(H), hence a second-order smooth-fit principle in the controlled direction holds for V . This result is obtained by exploiting a connection to optimal stopping and combining results and techniques from convex analysis and viscosity theory.