A generalized transformed path integral approach for stochastic processes

Abstract

In this paper, we present the generalized transformed path integral (GTPI) approach: a grid-based path integral approach for probabilistic description in a large class of stochastic dynamical systems. We showcase the application of our proposed approach to non-singular systems, as well as to singular systems such as second-order (and higher-order) dynamical systems, dynamical systems with zero process noise, and certain dynamical systems with non-white noise excitation. As a part of the approach, we present a novel framework for the description of stochastic dynamical systems in terms of a complementary system—the standard transformed stochastic dynamical system—obtained through a dynamic transformation of the state variables. The state mean and covariance of the transformed system do not change with evolution and the choice of our transformation parameters ensure that they are zero and identity respectively. Thus, the probability density function (PDF) for the state of the transformed system can be evolved in a transformed space where greater numerical accuracy for the distribution can be ensured. A fixed grid in the transformed space coordinates corresponds to an adaptive grid in the original space coordinates; it allows the proposed approach to more efficiently address the challenge of large drift, diffusion, or concentration of PDF in the stochastic dynamical system. In addition, error bounds for distributions in the transformed space can be easily obtained using Chebyshev's inequality. We use an operator splitting–based solution of the Fokker-Planck equation associated with the transformed system to derive a novel short-time propagator and update relations for the evolution of the transformed state PDF in the transformed space. Necessary update relations for the mean and covariance of the original state variables, used in the evolution of the transformed state PDF, are derived from the underlying stochastic models. Illustrative examples were considered to showcase the benefits of the GTPI approach over conventional fixed grid (FG) approaches in a large class of stochastic dynamical systems. In all the cases, results obtained using the GTPI approach show excellent agreement with results from Monte Carlo simulations and available analytical (and stationary) solutions, while results from the FG approach show large errors. The effect of simulation parameters and system parameters on the numerical error in our approach were also studied.