A Nearly Exact Reformulation of the Girsanov Linearization for Stochastically Driven Nonlinear Oscillators

Abstract

The Girsanov linearization method (GLM), proposed earlier in Saha, N., and Roy, D., 2007, "The Girsanov Linearisation Method for Stochastically Driven Nonlinear Oscillators," J. Appl. Mech.,74, pp. 885-897, is reformulated to arrive at a nearly exact, semianalytical, weak and explicit scheme for nonlinear mechanical oscillators under additive stochastic excitations. At the heart of the reformulated linearization is a temporally localized rejection sampling strategy that, combined with a resampling scheme, enables selecting from and appropriately modifying an ensemble of locally linearized trajectories while weakly applying the Girsanov correction (the Radon-Nikodym derivative) for the linearization errors. The semianalyticity is due to an explicit linearization of the nonlinear drift terms and it plays a crucial role in keeping the Radon-Nikodym derivative "nearly bounded" above by the inverse of the linearization time step (which means that only a subset of linearized trajectories with low, yet finite, probability exceeds this bound). Drift linearization is conveniently accomplished via the first few (lower order) terms in the associated stochastic (¡to) Taylor expansion to exclude (multiple) stochastic integrals from the numerical treatment. Similarly, the Radon-Nikodym derivative, which is a strictly positive, exponential (super-) martingale, is converted to a canonical form and evaluated over each time step without directly computing the stochastic integrals appearing in its argument. Through their numeric implementations for a few lowdimensional nonlinear oscillators, the proposed variants of the scheme, presently referred to as the Girsanov corrected linearization method (GCLM), are shown to exhibit remarkably higher numerical accuracy over a much larger range of the time step size than is possible with the local drift-linearization schemes on their own.

Iterated stochastic filters with additive updates for dynamic system identification: Annealing-type iterations and the filter bank

“…(only in law, implying that weak stochastic solutions are admissible) as 1 0 ii tt  . In view of the fact that the error due to local linearization may be weakly corrected via a Girsanov change of measure [20,21] and in order to maintain notational simplicity, we will henceforth refer to the linearized (predicted) solution ()…”

A change of measure enhanced near exact Euler Maruyama scheme for the solution to nonlinear stochastic dynamical systems

“…Another numerical approach for solving non-linear systems under additive stochastic excitation is the Girsanov based formulation of the local transversal linearization (LTL) schemes [17,18]. In this family of methods, the errors in the non-linearity approximation by local linearization schemes are absorbed inside the stochastic diffusion term.…”

Generalized weakly corrected Milstein solutions to stochastic differential equations

“…The correction in this study is additive in nature [16] and idealized as a Radon-Nikodym derivative, which is also a solution to a scalar SDE in exponential form [17,18]. It defines the distribution of weights among the ensembles of the linearized system responses and performs the sampling of linearized solution based on the weights [16,19]. The stochastic exponentials arising due to the MSIs involved in exponential Radon-Nikodym derivative tend to hamper the estimation of the proper weight/fitness of the ensembles [19,20], thus the correction is incorporated into the Milstein solution as an additive term in the current framework [16].…”

“…It defines the distribution of weights among the ensembles of the linearized system responses and performs the sampling of linearized solution based on the weights [16,19]. The stochastic exponentials arising due to the MSIs involved in exponential Radon-Nikodym derivative tend to hamper the estimation of the proper weight/fitness of the ensembles [19,20], thus the correction is incorporated into the Milstein solution as an additive term in the current framework [16]. To derive an appropriate additive correction term for the Milstein schemes, the central idea is the concept of change of measure, such that the Milstein approximated integrated process is measurable with respect to the filtration generated by the error, that will in turn improve the convergence of the Milstein schemes [21,22].…”