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Target achieving portfolio under model misspecification: quadratic optimization framework

Dariusz Zawisza (2012)

Applicationes Mathematicae

We incorporate model uncertainty into a quadratic portfolio optimization framework. We consider an incomplete continuous time market with a non-tradable stochastic factor. Two stochastic game problems are formulated and solved using Hamilton-Jacobi-Bellman-Isaacs equations. The proof of existence and uniqueness of a solution to the resulting semilinear PDE is also provided. The latter can be used to extend many portfolio optimization results.

The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories

Jana Musilová, Stanislav Hronek (2016)

Communications in Mathematics

As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical...

The Dirichlet problem with sublinear nonlinearities

Aleksandra Orpel (2002)

Annales Polonici Mathematici

We investigate the existence of solutions of the Dirichlet problem for the differential inclusion 0 ∈ Δ x ( y ) + ∂ x G ( y , x ( y ) ) for a.e. y ∈ Ω, which is a generalized Euler-Lagrange equation for the functional J ( x ) = ∫ Ω 1 / 2 | ∇ x ( y ) | ² - G ( y , x ( y ) ) d y . We develop a duality theory and formulate the variational principle for this problem. As a consequence of duality, we derive the variational principle for minimizing sequences of J. We consider the case when G is subquadratic at infinity.

The inverse problem in the calculus of variations: new developments

Thoan Do, Geoff Prince (2021)

Communications in Mathematics

We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of n second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas’s famous solution for n = 2 . We then examine a new class of solutions in arbitrary dimension n and give some non-trivial examples in dimension 3.

The linear programming approach to deterministic optimal control problems

Daniel Hernández-Hernández, Onésimo Hernández-Lerma, Michael Taksar (1996)

Applicationes Mathematicae

Given a deterministic optimal control problem (OCP) with value function, say J * , we introduce a linear program ( P ) and its dual ( P * ) whose values satisfy sup ( P * ) ≤ inf ( P ) ≤ J * ( t , x ) . Then we give conditions under which (i) there is no duality gap

The summability of solutions to variational problems since Guido Stampacchia.

Lucio Boccardo (2003)

RACSAM

Inequalities concerning the integral of |∇u|2 on the subsets where |u(x)| is greater than k can be used in order to prove regularity properties of the function u. This method was introduced by Ennio De Giorgi e Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems.

The symmetry reduction of variational integrals

Václav Tryhuk, Veronika Chrastinová (2018)

Mathematica Bohemica

The Routh reduction of cyclic variables in the Lagrange function and the Jacobi-Maupertuis principle of constant energy systems are generalized. The article deals with one-dimensional variational integral subject to differential constraints, the Lagrange variational problem, that admits the Lie group of symmetries. Reduction to the orbit space is investigated in the absolute sense relieved of all accidental structures. In particular, the widest possible coordinate-free approach to the underdetermined...

The symmetry reduction of variational integrals, complement

Veronika Chrastinová, Václav Tryhuk (2018)

Mathematica Bohemica

Some open problems appearing in the primary article on the symmetry reduction are solved. A new and quite simple coordinate-free definition of Poincaré-Cartan forms and the substance of divergence symmetries (quasisymmetries) are clarified. The unbeliavable uniqueness and therefore the global existence of Poincaré-Cartan forms without any uncertain multipliers for the Lagrange variational problems are worth extra mentioning.

The value function representing Hamilton–Jacobi equation with hamiltonian depending on value of solution

A. Misztela (2014)

ESAIM: Control, Optimisation and Calculus of Variations

In the paper we investigate the regularity of the value function representing Hamilton–Jacobi equation: − Ut + H(t, x, U, − Ux) = 0 with a final condition: U(T,x) = g(x). Hamilton–Jacobi equation, in which the Hamiltonian H depends on the value of solution U, is represented by the value function with more complicated structure than the value function in Bolza problem. This function is described with the use of some class of Mayer problems related to the optimal control theory and the calculus of...

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