Lagrange approach to the optimal control of diffusions
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.
scientific article; zbMATH DE number 508792
scientific article