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Meanfield backward stochastic differential equations and related patial differential equations. Submitted. Available at http://arxiv.org/abs/0711.2167
, 2007
"... Mathematical meanfield 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 meanfield problem in a purely stochastic approa ..."
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Cited by 66 (8 self)
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Mathematical meanfield 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 meanfield problem in a purely stochastic approach: for the solution (Y,Z) of a meanfield 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 / √ N. Moreover, our special choice of the approximation allows to characterize the limit behavior of √ 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 meanfield type, which is not only governed by a Brownian motion but also by an independent Gaussian field. 1. Introduction. Our
2001. Metastability in stochastic dynamics of disordered meanfield models, Probab. Theory Related Fields 119
"... Abstract: We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of t ..."
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Cited by 32 (10 self)
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Abstract: We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the WentzellFreidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of “admissible transitions”. For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a factor √ N, where N denotes the volume of the system. The distribution rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field CurieWeiss model.
Sharp asymptotics for metastability in the random field Curie–Weiss model,”arXiv:0806.4478v1 [math.PR
, 2008
"... ABSTRACT. In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field CurieWeiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distrib ..."
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Cited by 10 (3 self)
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ABSTRACT. In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field CurieWeiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important. 1.
Large Portfolio Losses: A Dynamic Contagion Model
 Annals of Applied Probability
"... Using particle system methodologies we study the propagation of financial distress in a network of firms facing credit risk. We investigate the phenomenon of a credit crisis and quantify the losses that a bank may suffer in a large credit portfolio. Applying a large deviation principle we compute th ..."
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Cited by 6 (1 self)
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Using particle system methodologies we study the propagation of financial distress in a network of firms facing credit risk. We investigate the phenomenon of a credit crisis and quantify the losses that a bank may suffer in a large credit portfolio. Applying a large deviation principle we compute the limiting distributions of the system and determine the time evolution of the credit quality indicators of the firms, deriving moreover the dynamics of a global financial health indicator. We finally describe a suitable version of the “Central Limit Theorem ” useful to study large portfolio losses. Simulation results are provided as well as applications to portfolio loss distribution analysis. 1. Introduction. 1.1. General aspects. The main purpose of this paper is to describe propagation of financial distress in a network of firms linked by business relationships. Once the model for financial contagion has been described, we quantify the impact of contagion on the losses suffered by a financial institution
Exact Solutions and Dynamics of Globally Coupled Phase Oscillators
"... . We analyze meanfield models of synchronization of phase oscillators with singular couplings and subject to external random forces. They are related to the KuramotoSakaguchi model. Their probability densities satisfy local partial di#erential equations similar to the Porous Medium, Burgers and ex ..."
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. We analyze meanfield models of synchronization of phase oscillators with singular couplings and subject to external random forces. They are related to the KuramotoSakaguchi model. Their probability densities satisfy local partial di#erential equations similar to the Porous Medium, Burgers and extended Burgers equations depending on the degree of singularity of the coupling. We show that Porous Medium oscillators (the most singularly coupled) do not synchronize and that (transient) synchronization is possible only at zero temperature for Burgers oscillators. The extended Burgers oscillators have a nonlocal coupling first introduced by Daido and they may synchronize at any temperature. Exact expressions for their synchronized phases and for Daido's order function are given in terms of elliptic functions. 1. Introduction Collective synchronization of large populations of nonlinearly coupled phase oscillators has been intensely studied since Winfree realized its importance for biologic...
© Institute of Mathematical Statistics, 2009 LARGE PORTFOLIO LOSSES: A DYNAMIC CONTAGION MODEL
"... Using particle system methodologies we study the propagation of financial distress in a network of firms facing credit risk. We investigate the phenomenon of a credit crisis and quantify the losses that a bank may suffer in a large credit portfolio. Applying a large deviation principle we compute th ..."
Abstract
 Add to MetaCart
Using particle system methodologies we study the propagation of financial distress in a network of firms facing credit risk. We investigate the phenomenon of a credit crisis and quantify the losses that a bank may suffer in a large credit portfolio. Applying a large deviation principle we compute the limiting distributions of the system and determine the time evolution of the credit quality indicators of the firms, deriving moreover the dynamics of a global financial health indicator. We finally describe a suitable version of the “Central Limit Theorem ” useful to study large portfolio losses. Simulation results are provided as well as applications to portfolio loss distribution analysis. 1. Introduction. 1.1. General aspects. The main purpose of this paper is to describe propagation of financial distress in a network of firms linked by business relationships. Once the model for financial contagion has been described, we quantify the impact of contagion on the losses suffered by a financial institution holding a large