Estimating stationary characteristic functions of stochastic systems via semidefinite programming

This paper proposes a methodology to estimate characteristic functions of stochastic differential equations that are defined over polynomials. For such systems, the time evolution of the characteristic function is governed by a partial differential equation; consequently, the stationary characteristic function can be obtained by solving an ordinary differential equation (ODE). However, except for a few special cases, the solution to the ODE consists of unknown coefficients. These coefficients are closely related with the stationary moments of the process, which could be estimated by utilizing the fact that the characteristic function is positive definite. The method is illustrated via examples.