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. Copyright Kluwer Academic Publishers 2003