Stochastic programming: Difference between revisions

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These questions are not independent. For example, the number of scenarios constructed will affect both the tractability of the deterministic equivalent and the quality of the obtained solutions.

== Stochastic linear programprogramming==

A stochastic [[linear program]] is a specific instance of the classical two-stage stochastic program. A stochastic LP is built from a collection of multi-period linear programs (LPs), each having the same structure but somewhat different data. The <math>k^{th}</math> two-period LP, representing the <math>k^{th}</math> scenario, may be regarded as having the following form:

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is the sample variance estimate of <math>\sigma^2(x)</math>. That is, the error of estimation of <math>g(x)</math> is (stochastically) of order <math> O(\sqrt{N})</math>.

== Applications and Examples==

== Multistage portfolio optimization==

=== Biological applications ===

[[Stochastic dynamic programming]] is frequently used to model [[ethology|animal behaviour]] in such fields as [[behavioural ecology]].<ref>Mangel, M. & Clark, C. W. 1988. ''Dynamic modeling in behavioral ecology.'' Princeton University Press {{ISBN|0-691-08506-4}}</ref><ref>Houston, A. I & McNamara, J. M. 1999. ''Models of adaptive behaviour: an approach based on state''. Cambridge University Press {{ISBN|0-521-65539-0}}</ref> Empirical tests of models of [[Optimal foraging theory|optimal foraging]], [[Biological life cycle|life-history]] transitions such as [[Fledge|fledging in birds]] and egg laying in [[parasitoid]] wasps have shown the value of this modelling technique in explaining the evolution of behavioural decision making. These models are typically many-staged, rather than two-staged.

===Economic applications===

[[Stochastic dynamic programming]] is a useful tool in understanding decision making under uncertainty. The accumulation of capital stock under uncertainty is one example; often it is used by resource economists to analyze [[Nicholas Georgescu-Roegen#Man.27s economic struggle and the social evolution of mankind .28bioeconomics.29|bioeconomic problems]]<ref>Howitt, R., Msangi, S., Reynaud, A and K. Knapp. 2002. [http://www.agecon.ucdavis.edu/aredepart/facultydocs/Howitt/Polyapprox3a.pdf "Using Polynomial Approximations to Solve Stochastic Dynamic Programming Problems: or A "Betty Crocker " Approach to SDP."] University of California, Davis, Department of Agricultural and Resource Economics Working Paper.</ref> where the uncertainty enters in such as weather, etc.

=== MultistageExample: multistage portfolio optimization ===

{{Main|Intertemporal portfolio choice}}

{{See also|Merton's portfolio problem}}

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</math>

==== Stagewise independent random process ====

For a general distribution of the process <math>\xi_t</math>, it may be hard to solve these dynamic programming equations. The situation simplifies dramatically if the process <math>\xi_t</math> is stagewise independent, i.e., <math>\xi_t</math> is (stochastically) independent of <math>\xi_1,\dots,\xi_{t-1}</math> for <math>t=2,\dots,T</math>. In this case, the corresponding conditional expectations become unconditional expectations, and the function <math>Q_t(W_t)</math>, <math>t=1,\dots,T-1</math> does not depend on <math>\xi_{[t]}</math>. That is, <math>Q_{T-1}(W_{T-1})</math> is the optimal value of the problem

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for <math>t=T-2,\dots,1</math>.

==Applications==

=== Biological applications ===

[[Stochastic dynamic programming]] is frequently used to model [[ethology|animal behaviour]] in such fields as [[behavioural ecology]].<ref>Mangel, M. & Clark, C. W. 1988. ''Dynamic modeling in behavioral ecology.'' Princeton University Press {{ISBN|0-691-08506-4}}</ref><ref>Houston, A. I & McNamara, J. M. 1999. ''Models of adaptive behaviour: an approach based on state''. Cambridge University Press {{ISBN|0-521-65539-0}}</ref> Empirical tests of models of [[Optimal foraging theory|optimal foraging]], [[Biological life cycle|life-history]] transitions such as [[Fledge|fledging in birds]] and egg laying in [[parasitoid]] wasps have shown the value of this modelling technique in explaining the evolution of behavioural decision making. These models are typically many-staged, rather than two-staged.

===Economic applications===

[[Stochastic dynamic programming]] is a useful tool in understanding decision making under uncertainty. The accumulation of capital stock under uncertainty is one example; often it is used by resource economists to analyze [[Nicholas Georgescu-Roegen#Man.27s economic struggle and the social evolution of mankind .28bioeconomics.29|bioeconomic problems]]<ref>Howitt, R., Msangi, S., Reynaud, A and K. Knapp. 2002. [http://www.agecon.ucdavis.edu/aredepart/facultydocs/Howitt/Polyapprox3a.pdf "Using Polynomial Approximations to Solve Stochastic Dynamic Programming Problems: or A "Betty Crocker " Approach to SDP."] University of California, Davis, Department of Agricultural and Resource Economics Working Paper.</ref> where the uncertainty enters in such as weather, etc.

==Software tools==