Computing Higher Moments In The Linear-Quadratic-Exponential-Gaussian Optimal Control Problem

Consider the discrete-time state equation and feedback control rule (1) x(t) = Fx(t-1) + Gu(t) + e(t), (2) u(t) = Px(t-1), where x is an nx1 state vector, u is an mx1 control variable, and e is an nx1 disturbance distributed NIID(0,S), F and G are nxn and nxm parameter matrices, and P is an mxn feedback control matrix. The ordinary linear-quadratic (LQ) performance index is (3) v(x(t-1),N) = [x(t-1)'Qx(t-1) + u(t)'Ru(t)] + d[x(t)'Qx(t) + u(t+1)'Ru(t+1)] + ... + d^N[x(t+N-1)'Qx(t+N-1) + u(t+N)'Ru(t+N)], where N is a finite or infinite planning horizon, 0 0, respectively, indicates risk "preference, neutrality, or avoidance"). The discrete-time LQEG problem is: given x(t-1), N, and the parameters, minimize (4) with respect to P, subject to (1)-(2).Jacobson (1973) proved that, for N finite, the optimal P is obtained by iterating on a discrete-time recursive Riccati equation. As N approaches infinity, the equation converges to a nonrecursive or algebraic Riccati equation, which can be solved quickly and accurately using the Schur-decomposition method (Laub, 1979). Karp (1985) addressed the problem of determining the contribution of higher moments (> 2) of e to the value of the optimized LQEG performance index, J*(x(t-1),N). Expanding J*(x(t-1),N) in a Taylor series in v(x(t-1),N) and using some matrix differentiation rules of MacRae (1974), Karp derived an algorithm for computing a two-term approximation of J*(x(t-1),N) based on the 2nd and 4th moments of e (odd moments of e are zero). The complexity of MacRae's differentiation rules apparently dissuaded Karp from attempting to derive equations for computing higher-order (> 2) terms based on higher moments (> 4) of e.The present paper extends Karp's results in three ways. (1) Using a much simpler approach to matrix differentiation based on total-differential rather than partial-derivative forms of matrix derivatives (Magnus and Neudecker, 1988), the paper derives a simple recursion for computing any moments of a Gaussian random vector. (2) Using this result, the paper derives and applies an algorithm for computing any k-term Taylor approximation of J*(x(t-1),N), for finite N, based on moments 2, ..., 2k of e. (3) Using the perturbation method (Judd, 1998, chs. 13-14), the paper obtains corresponding results for infinite N. Result (3) illustrates the role of higher Gaussian moments in accurate perturbation solution of nonlinear dynamic stochastic economic models (cf., Chen and Zadrozny, 2000).