Higher-Moments in Perturbation Solution of the Linear-Quadratic Exponential Gaussian Optimal Control Problem

The paper obtains two principal results. First, using a new definition ofhigher-order (>2) matrix derivatives, the paper derives a recursion forcomputing any Gaussian multivariate moment. Second, the paper uses this resultin a perturbation method to derive equations for computing the 4th-orderTaylor-series approximation of the objective function of the linear-quadraticexponential Gaussian (LQEG) optimal control problem. Previously, Karp (1985)formulated the 4th multivariate Gaussian moment in terms of MacRae'sdefinition of a matrix derivative. His approach extends with difficulty to anyhigher (>4) multivariate Gaussian moment. The present recursionstraightforwardly computes any multivariate Gaussian moment. Karp used hisformulation of the Gaussian 4th moment to compute a 2nd-order approximationof the finite-horizon LQEG objective function. Using the simpler formulation,the present paper applies the perturbation method to derive equations forcomputing a 4th-order approximation of the infinite-horizon LQEG objectivefunction. By illustrating a convenient definition of matrix derivatives in thenumerical solution of the LQEG problem with the perturbation method, the papercontributes to the computational economist's toolbox for solving stochasticnonlinear dynamic optimization problems.