A PARTIAL-DIFFERENTIAL EQUATION FOR THE CHARACTERISTIC FUNCTION OF THE RESPONSE OF NONLINEAR-SYSTEMS TO ADDITIVE POISSON WHITE-NOISE

A method has been developed for finding the characteristic function ph i(u, t) = E exp iu'X(t) of the solution X(t) of a non-linear stochasti c differential equation driven by Poisson white noise. Poisson white n oise can be viewed as a sequence of indepenent identically distributed pulses arriving at Poisson times. If the drift is a polynomial of X(t ) and the diffusion is independent of this process, phi(u, t) satisfie s a partial differential equation that can be solved numerically. Two approaches are used to establish the partial differential equation of the characteristic function of X(t). The first approach is based on a generalized version of the Ito differentiation formula for stochastic differential equations with Poisson white noise. The second approach u ses elementary arguments to determine the rate of change of phi(u, t) in time. Linear and non-linear systems subjected to Poisson white nois e are used to demonstrate and evaluate the proposed method. The soluti on X(t) of a linear differential equation with Poisson white noise is a filtered Poisson process and the characteristic function for this ty pe of process is available in closed form. It is shown that the charac teristic function of the filtered Poisson process X(t) satisfies the p artial differential equation of phi(u, t). Because the characteristic function of the state X(t) of a non-linear system subjected to Poisson white noise is not known, numerical solutions of the partial differen tial equation of phi(u, t) are compared with estimates of this charact eristic function calculated from independent realization of X(t) obtai ned by Monte Carlo simulation. These comparisons show that the propose d method for calculating the characteristic function of X(t) as the so lution of a partial differential equation is accurate. (C) 1996 Academ ic Press Limited