McKean-Vlasov limit for interacting systems with simultaneous jumps

Motivated by several applications, including neuronal models, we consider the McKean-Vlasov limit for mean-field systems of interacting diffusions with simultaneous jumps. We prove propagation of chaos via a coupling technique that involves an intermediate process and that gives a rate of convergence for the $W_1$ Wasserstein distance between the empirical measures of the two systems on the space of trajectories $\mathbf{D}([0,T],\mathbb{R}^d)$

Gelation, hydrodynamic limits and uniqueness in cluster coagulation processes

We consider the problem of gelation in the cluster coagulation model introduced by Norris [$\textit{Comm. Math. Phys.}$, 209(2):407-435 (2000)], where clusters take values in a measure space $E$, and merge to form a new particle $z$ according to a transition kernel $K(x,y, \mathrm{d} z)$. This model is general enough to incorporate various inhomogenieties in the evolution of clusters, for example, their shape, or their location in space. We derive general, sufficient criteria for stochastic gelation in this model, and for trajectories associated with this process to concentrate among solutions of a generalisation of the Flory equation; thus providing sufficient criteria for the equation to have gelling solutions. As particular cases, we extend results related to the classical Marcus-Lushnikov coagulation process and Smoluchowski coagulation equation, showing that reasonable `homogenous' coagulation processes with exponent $\gamma>1$ yield gelation; and also, coagulation processes with kernel $\alpha(m,n)~\geq~(m \wedge n) \log{(m \wedge n)}^{3 +\varepsilon}$ for $\varepsilon>0$. In another special case, we prove a law of large numbers for the trajectory of the empirical measure of the stochastic cluster coagulation process, by means of a uniqueness result for the solution of the aforementioned generalised Flory equation. Finally, we use coupling arguments with inhomogeneous random graphs to deduce sufficient criterion for strong gelation (the emergence of a particle of size $O(N)$).Comment: 43 page

McKean-Vlasov limits, propagation of chaos and long-time behavior of some mean field interacting particle systems

In this thesis we study mean field interacting particle systems and their McKean-Vlasov limiting processes, in particular we focus on three different interaction mechanisms, mainly emerging from biological modelling. The first type of interaction is given by the so called simultaneous jumps. We consider a system of interacting jump-diffusion processes that interact by means of the discontinuous component: each particle performs a main jump and it simultaneously induces in all the other particles a simultaneous jump whose amplitude is rescaled with the size of the system. This peculiar interaction is motivated by recent neuroscience models and here we depict a general framework for this type of processes. We focus on the well-posedness of the McKean-Vlasov limits of these particle systems under different assumptions on the coefficients and we prove a pathwise propagation of chaos result. The second interaction we consider is an asymmetric one. We describe a system of biased random walks on the positive integers, reflected at zero, where each particle may perform a leftward jump with a rate proportional to the fraction of particles which are strictly at its left. We study the critical interaction strength able to ensure ergodicity to this system, that would be transient in absence of interaction. We compare this model with existing models of diffusions interacting through their CDF and we highlight their differences, mainly caused by the presence of clusters of particles in the discrete model. The third interaction we account for is based on a dynamical version of the generalized Curie-Weiss model. We modify a Langevin dynamics for this model with a dissipative evolution of the interaction component, breaking the reversibility of the system. We prove that, in the mean field limit, this gives rise to stable limit cycles, explaining self-sustained periodic behaviors. In particular, we build a flexible model in which a suitable change in the interaction function can result in a system which, in certain regimes of parameters, displays coexistence of stable periodic orbits

Pricing reliability options under different electricity price regimes

Reliability Options are capacity remuneration mechanisms aimed at enhancing security of supply in electricity systems. They can be framed as call options on electricity sold by power producers to System Operators. This paper provides a comprehensive mathematical treatment of Reliability Options. Their value is first derived by means of closed-form pricing formulae, which are obtained under several assumptions about the dynamics of electricity prices and strike prices. Then, the value of the Reliability Option is simulated under a real-market calibration, using data of the Italian power market. We perform sensitivity analyses to highlight the role of the level and volatility of both power and strike price, of the mean reversion speeds and of the correlation coefficient on the Reliability Options' value. Finally, we calculate the parameter model risk to quantify the impact that a model misspecification has on the equilibrium value of the RO

Noise-induced periodicity in a frustrated network of interacting diffusions

We investigate the emergence of a collective periodic behavior in a frustrated network of interacting diffusions. Particles are divided into two communities depending on their mutual couplings. On the one hand, both intra-population interactions are positive; each particle wants to conform to the average position of the particles in its own community. On the other hand, inter-population interactions have different signs: the particles of one population want to conform to the average position of the particles of the other community, while the particles in the latter want to do the opposite. We show that this system features the phenomenon of noise-induced periodicity: in the infinite volume limit, in a certain range of interaction strengths, although the system has no periodic behavior in the zero-noise limit, a moderate amount of noise may generate an attractive periodic law.Comment: 32 pages, 8 figure

A large-deviations approach to gelation

A large-deviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t=1. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erd\H{o}s-R\'enyi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller Erd\H{o}s-R\'enyi graphs are connected

A large-deviations principle for all the components in a sparse inhomogeneous random graph

We study an inhomogeneous sparse random graph, GN, on [N] = { 1,...,N } as introduced in a seminal paper [BJR07] by Bollobás, Janson and Riordan (2007): vertices have a type (here in a compact metric space S), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit N → ∞ , we consider the sparse regime, where the average degree is O(1). We prove a large-deviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size ≈ N). In doing so, we derive explicit logarithmic asymptotics for the probability that GN is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of GN. In particular, we recover the criterion for the existence of the phase transition given in [BJR07]