My bibliography  Save this paper

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

## Author

Listed:
• Baoline Chen, Peter A. Zadrozny

(Bureau of Labor Statistics)

## Abstract

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).

## Suggested Citation

• Baoline Chen, Peter A. Zadrozny, 2000. "Computing Higher Moments In The Linear-Quadratic-Exponential-Gaussian Optimal Control Problem," Computing in Economics and Finance 2000 310, Society for Computational Economics.
• Handle: RePEc:sce:scecf0:310
as

To our knowledge, this item is not available for download. To find whether it is available, there are three options:
1. Check below whether another version of this item is available online.
2. Check on the provider's web page whether it is in fact available.
3. Perform a search for a similarly titled item that would be available.

## Corrections

All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:sce:scecf0:310. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Christopher F. Baum). General contact details of provider: http://edirc.repec.org/data/sceeeea.html .

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

We have no references for this item. You can help adding them by using this form .

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.