Approximate formulas for estimating rare event probabilities in non-Gaussian non-stationary stochastic dynamics Jun Shao 1 , David Clair 1 , Michel Fogli 1 , Frédéric Bernardin 2 1 Institut Pascal, UMR CNRS/UBP 6602, Campus des Cézeaux, 63171, Aubière CEDEX, France 2 CEREMA, Direction territoriale Centre-Est, Département Laboratoire de Clermont-Ferrand, 63017, Clermont-Ferrand CEDEX, France email: jun.shao@ifma.fr, david.clair@univ-bpclermont.fr, michel.fogli@polytech.univ-bpclermont.fr, frederic.bernardin@cerema.fr ABSTRACT: In their book "Level sets and extrema of random processes and fields", J.M. Azaïs and M. Wschebor give results enabling the construction, under some conditions, of approximations and bounds for the distribution tails of the extrema of general random processes. The authors of the present paper have recently conducted a study to numerically analyze these approximations and bounds in cases where the random processes are the responses of stochastic oscillators. They present here some results of this analysis, which show the accuracy and relevance of such approximations. This validation work is the first step in a broader study being carried by the authors, aiming to implement efficient numerical tools for the practical use of these different approximations. KEY WORDS: Extrema of processes; stochastic oscillator response; Rice series; Monte Carlo simulation. 1 INTRODUCTION The probabilistic analysis of the extrema of stochastic processes is a classic topic which is of great interest in many scientific fields and which has been the source of much research work for more than seventy years. Many of these studies are con- cerned with the problem of computing the distribution function F M T (u)= P(M T ≤ u), u ∈ R, of a scalar random variable M T defined as M T = sup t ∈J X (t ), where X =(X (t ), t ∈ I ) is a given R-valued stochastic process indexed on an interval I of R (i.e. t ∈ I ) and J ⊂ I is a subinterval of I of the form J =[0, T ], with 0 < T < +∞. Other works focus on the related problem of cal- culating the probability P(M T > u)= 1 - P(M T ≤ u) for a large enough value of u, that is to say on the calculation of the tail of the distribution function of M T . Exact solutions to these problems by means of closed formu- las have been found for a very restricted number of stochas- tic processes [1], among which Brownian motion or the Wiener process W =( W (t ), t ≥ 0), the Brownian bridge B =(B(t ), 0 ≤ t ≤ 1), B(t ) := W (t ) - tW (1), Brownian motion with linear drift ([2],[3]), and the processes Y =( Y (t ), 0 ≤ t ≤ 1), Y (t ) := B(t ) - R 1 0 B(s)ds [4] and Z =(Z(t ), 0 ≤ t ≤ T ), Z(t ) := R t 0 W (s)ds + yt ([[5],[6],[7]). Explicit formulas for the exact solutions to these problems also exist for some stationary Gaussian pro- cesses having a particular covariance function ([8]-[17]). The methods for finding these formulas are ad hoc and therefore non-transposable to more general processes, even in the Gaus- sian case. In the latter case, many results are available, all based on the pioneering works of S.O.Rice ([19],[20],[21]) and M.R.Leadbetter ([22]-[25]). They consist of either inequalities for P(M T > u), with a large enough u, or asymptotic approxi- mations of P(M T > u) for u tending to infinity and fixed T . In each case, the main difference between these results comes from the assumptions adopted for the Gaussian process X (stationar- ity, regularity of its trajectories, ··· ). Also in the Gaussian case, asymptotic approximations of P(M T > u) can be found in the literature for T tending to infinity and fixed u. In the general case and under some conditions, that is to say if process X is neither stationary nor Gaussian and satisfies some assumptions to be specified, an exact formula can be found for the probability P(M T > u). It expresses this probability as the sum of a Rice series defined in terms of factorial moments of the random number of upcrossings of level u by process X on the interval J =[0, T ]. The fact that this formula is exact is obvi- ously attractive. However, its use in practical applications is not easy because on the one hand the assumptions under which it is valid (and which relate to process X ) are often difficult to verify and, on the other hand, the factorial moments that compose it (and which are given by the Rice formulas) cannot be analyti- cally calculated and are difficult, indeed impossible, to estimate numerically from the Rice formulas. An interesting feature of the Rice series is its enveloping property resulting from the fact that it is an alternating series. Hence, replacing the total sum of this series by a partial sum gives upper and lower bounds for the probability P(M T > u). The truncation error can thus be bounded by the absolute value of the first neglected term. So, if all the factorial moments of the partial sum can be calculated, this enables the target probability to be estimated with some ef- ficiency. J.M.Azaïs and M.Wschebor’s book [1] brings together all the recent results focused on the calculation of the tails of distri- bution functions of the extrema of random processes and fields in Gaussian and non-Gaussian contexts. Recently the authors of the present paper have experimented some of these results numerically through examples derived from the reliability anal- ysis of random dynamic systems. The purpose of this paper is to present some results of this work. The problem which was considered to perform this study is exposed in the next section. 2 PROBLEM STATEMENT Note first that in what follows it is supposed that all the random quantities considered (random variables, stochastic processes) are defined on the same probability space (A , F , P). Let I be an interval of R (with possibly I = R), J ⊂ I a com- pact subinterval of I of the form J =[0, T ], 0 < T < +∞, X = (X (t ), t ∈ I ) a second-order R-valued stochastic process defined on (A , F , P) and indexed on I (i.e. t ∈ I ), and M * T the second- Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4 2797