Lagrange approach to the optimal control of diffusions
type: Item
instance of: scholarly article
title: Lagrange approach to the optimal control of diffusions
zbMATH Open document ID: 0794.93108
DOI: 10.1007/BF00998149
author: Michele Pavon
author: Peter Kosmol
published in: Acta Applicandae Mathematicae
publication date: 1994-03-14T00:00:00Z
review text: The constrained stochastic optimal control with complete observations is studied by taking advantage of the great generality of Lagrange's lemma: \(J(y_ 0)=\min_ S J(y)\) if \(y_ 0\in S\) is a global minimizer of the real function \(J+\Lambda\) on a set \(Y\supset S\) and if \(\Lambda\) is a Lagrange functional on the restrictive set \(S\) in the sense that \(\Lambda\) is constant and finite over \(S\). In the case of controlled diffusions, \(S\) is defined by the requirement that both \(x\) and \(u\) be adapted to a fixed filtration and by the stochastic differential equation described diffusions. Two classes of Lagrange functionals are introduced. The first class is defined when the adjoint process \(\lambda_ t\) is pathwise differentiable and then the functionals are seen to correspond to certain linear functionals on the space of primal variables. The authors proceed with step one where both optimality conditions and heuristics are used to compute candidate multipliers and possibly optimal solutions. In the second step, they check sufficient optimal conditions for the global minimization of \(J+\Lambda\) corresponding to chosen multipliers and solutions. The method allows to obtain optimality conditions without any resort to stochastic calculus and functional analysis. The linear quadratic problem with state and control dependent diffusion coefficient is considered as an example. A second class, which requires Ito's rule, allows to establish optimality in a larger class of problems. The second method does not require separated structure nonconvexity with respect to \(x\) and accomodates as special cases the calculations in Pontryagin's principle (by using the function \(\langle \lambda_ t,t\rangle\)) and in the dynamic programming (by using the value function). This method is illustrated by solving both deterministic and stochastic versions of a simple investment model and a stochastic model of optimal portfolio selection.
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 49K25
zbMATH DE Number: 508792
zbMATH Keywords: diffusion process
zbMATH Keywords: Lagrange functionals
zbMATH Keywords: structure nonconvexity
MaRDI profile type: Publication
cites work: Q3036161
cites work: Optimal control theory
cites work: Conjugate convex functions in optimal stochastic control
cites work: An Introductory Approach to Duality in Optimal Stochastic Control
cites work: A stochastic control approach to reciprocal diffusion processes
cites work: Q3350415
cites work: Dynamic Programming Conditions for Partially Observable Stochastic Systems
cites work: Schrödinger processes and large deviations
cites work: The optimal control of diffusions
cites work: Exit probabilities and optimal stochastic control
cites work: Q4086303
cites work: Convex Duality Approach to the Optimal Control of Diffusions
cites work: Some Examples of Optimal Stochastic Controls OR: The Stochastic Maximum Principle at Work
cites work: Q3962903
cites work: Q5631335
cites work: A course on optimization and best approximation
cites work: On a stochastic representation for the principal eigenvalue of a second-order differential equation
cites work: A decomposition of the Brownian path
cites work: Optimization Problems in the Theory of Continuous Trading
cites work: Q3998777
cites work: Necessary Conditions for Continuous Parameter Stochastic Optimization Problems
cites work: Transformations of diffusion and Schrödinger processes
cites work: Stochastic control and nonequilibrium thermodynamical systems
cites work: Q3358623
cites work: A simplified variational characterization of Schrödinger processes
description: scientific article; zbMATH DE number 508792