Generalized dimensions, large deviations and the distribution of rare events

When a subsystem goes into an infrequent state, how does the remainder of the system behave? We show how to calculate the relevant distributions using the notions of reversed time for Markov processes and large deviations. For ease of exposition, most of the work deals with a specific queueing model due to Flatto, Hahn, and Wright. However, we show how the theorems may be applied to much more general jump-Markov systems. We also show how the tools of time-reversal and large deviations complement each other to yield general theorems. We show that the way a constant coefficient process approaches a rare event is roughly by following the path of another constant coefficient process. We also obtain some properties, including a priori bounds, for the change of measure associated with some large deviations functionals; these are useful for accelerating simulations.

Download Full-text

Rare Events, Large Deviations

Springer Finance - Mathematical Finance — Bachelier Congress 2000 ◽

10.1007/978-3-662-12429-1_5 ◽

2002 ◽

pp. 85-92

Author(s):

S. R. S. Varadhan

Keyword(s):

Large Deviations ◽

Rare Events

Download Full-text

Large deviations and rare events in the study of processes describing the spread of traffic congestions in urban network systems

2021 55th Annual Conference on Information Sciences and Systems (CISS) ◽

10.1109/ciss50987.2021.9400254 ◽

2021 ◽

Author(s):

Getachew K. Befekadu

Keyword(s):

Large Deviations ◽

Rare Events ◽

Network Systems ◽

Urban Network

Download Full-text

Numerical Methods for Monitoring Rare Events in Nonlinear Stochastic Systems

Mekhatronika Avtomatizatsiya Upravlenie ◽

10.17587/mau.22.291-297 ◽

2021 ◽

Vol 22 (6) ◽

pp. 291-297

Author(s):

A. A. Kabanov ◽

S. A. Dubovik

Keyword(s):

Optimal Control ◽

Numerical Methods ◽

Large Deviations ◽

Stochastic Systems ◽

Rare Events ◽

Rare Event ◽

Stable State ◽

The State ◽

Nonlinear Stochastic Systems ◽

State Dependent

In this article, we consider the development of numerical methods of large deviations analysis for rare events in nonlinear stochastic systems. The large deviations of the controlled process from a certain stable state are the basis for predicting the occurrenceof a critical situation (a rare event). The rare event forecasting problem is reduced to the Lagrange-Pontryagin optimal control problem.The presented approach for solving the Lagrange-Pontryagin problem differs from the approach used earlier for linear systems in that it uses feedback control. In the nonlinear case, approximate methods based on the representation of the system model in the state-space form with state-dependent coefficients (SDC) matrixes are used: the state-dependent Riccati equation (SDRE) and the asymptotic sequence of Riccati equations (ASRE). The considered optimal control problem allow us to obtain a numerical-analytical solutionthat is convenient for real-time implementation. Based on the developed methods of large deviations analysis, algorithms for estimating the probability of occurrence of a rare event in a dynamical systemare presented. The numerical applicability of the developed methods is shown by the example of the FitzHugh-Nagumo model for the analysis of switching between excitable modes. The simulation results revealed an additional problem related to the so-called parameterization problem of the SDC matrices. Since the use of different representations for SDC matrices gives different results in terms of the system trajectory, the choice of matrices is proposed to be carried out at each algorithm iteration so as to provide conditions for the solvability of the Lagrange-Pontryagin problem.

Download Full-text