Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included.

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The Kuramoto model: A simple paradigm for synchronization phenomena

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

This paper develops analytical methods to forecast the distribution of future returns for a new continuous-time process, the Poisson multifractal. Out model captures the thick tails and volatility persistence exhibited by many financial time series. We assume that the forecaster knows the true generating process with certainty, but only observes past returns. The challenge in this environment is long memory and the corresponding infinite dimension of the state space. We show that a discretized version of the model has a finite state space, which allows an analytical solution to the conditioning problem. Further, the discrete model converges to the continuous-time model as time scale goes to zero, so that forecasts are consistent. The methodology is implemented on simulated data calibrated to the Deutschemark/US Dollar exchange rate. Applying these results to option pricing, we find that the model captures both volatility smiles and long-memory in the term structure of implied volatilities.

Forecasting Multifractal Volatility

Mean-Field Backward Stochastic Differential Equations and Related Partial Differential Equations ∗

Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution (Y, Z) of a mean-field backward stochastic differential equation driven by a forward stochastic differential of McKean-Vlasov type with solution X we study a special approximation by the solution (X-N, Y-N, Z(N)) of some decoupled forward-backward equation which coefficients are governed by N independent copies of (X-N, Y-N, Z(N)). We show that the convergence speed of this approximation is of order 1/root N. Moreover, our special choice of the approximation allows to characterize the limit behavior of root N(X-N - X, Y-N - Y, Z(N) - Z). We prove that this triplet converges in law to the solution of some forward-backward. stochastic differential equation of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field.

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Mean-field backward stochastic differential equations : a limit approach

We apply large-deviation theory to particle systems with a random mean-field interaction in the McKean-Vlasov limit. In particular, we describe large deviations and normal fluctuations around the McKean-Vlasov equation. Due to the randomness in the interaction, the McKean-Vlasov equation is a collection of coupled PDEs indexed by the state space of the single components in the medium. As a result, the study of its solution and of the finite-size fluctuation around this solution requires some new ingredient as compared to existing techniques for nonrandom interaction.

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McKean-Vlasov limit for interacting random processes in random media

https://hal.archives-ouvertes.fr/hal-01407443/document

Multi-scaling of moments in stochastic volatility models

Explicit solutions for multivariate, discrete-time control problems under uncertainty

https://www.math.unipd.it/~daipra/CDPMS.pdf

Proliferation and Death in a Binary Environment: A Stochastic Model of Cellular Ecosystems

We investigate solutions to the equation ∂tℰ−\(\mathcal{D}\)Δℰ=λS2ℰ, where S(x, t) is a Gaussian stochastic field with covariance C(x−x′, t, t′), and x∈\(\mathbb{R}\)d. It is shown that the coupling λcN(t) at which the N-th moment diverges at time t, is always less or equal for \(\mathcal{D}\)>0 than for \(\mathcal{D}\)=0. Equality holds under some reasonable assumptions on C and, in this case, λcN(t)=Nλc(t) where λc(t) is the value of λ at which diverges. The \(\mathcal{D}\)=0 case is solved for a class of S. The dependence of λcN(t) on d is analyzed. Similar behavior is conjectured when diffusion is replaced by diffraction, \(\mathcal{D}\)→i\(\mathcal{D}\), the case of interest for backscattering instabilities in laser-plasma interaction.

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P. Dai Pra

Papers

McKean-Vlasov limit for interacting random processes in random media

Multi-scaling of moments in stochastic volatility models

Explicit solutions for multivariate, discrete-time control problems under uncertainty

Proliferation and Death in a Binary Environment: A Stochastic Model of Cellular Ecosystems

Diffusion Effects on the Breakdown of a Linear Amplifier Model Driven by the Square of a Gaussian Field