Jesús Marín-Solano

Consumption and portfolio rules for time-inconsistent investors

This paper extends the classical consumption and portfolio rules model in continuous time [Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: The continuous time case. Review of Economics and Statistics 51, 247-257, Merton, R.C., 1971. Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory 3, 373-413] to the framework of decision-makers with time-inconsistent preferences. The model is solved for different utility functions for both, naive and sophisticated agents, and the results are compared. In order to solve the problem for sophisticated agents, we derive a modified HJB (Hamilton-Jacobi-Bellman) equation. It is illustrated how for CRRA functions within the family of HARA functions (logarithmic and power utilities) the optimal portfolio rule does not depend on the discount rate, but this is not the case for a general utility function, such as the exponential (CARA) utility function.

A geometrical approach to Classical Field Theories: a constraint algorithm for singular theories

We construct a geometrical formulation for first order classical field theories in terms of fibered manifolds and connections. Using this formulation, a constraint algorithm for singular field theories is developed. This algorithm extends the constraint algorithm in mechanics.

The constraint algorithm in the jet formalism

Abstract A constraint algorithm for degenerate non-autonomous Lagrangians is developed by using the jet formalism on fibered manifolds. This algorithm extends the one of Gotay and Nester for the autonomous Lagrangians and the one of Chinea, de Leon and Marrero for non-autonomous Lagrangians on trivial fibered manifolds.

Some geometric aspects of variational calculus in constrained systems

Abstract We give a geometric description of variational principles in constrained mechanics. For the general case of nonholonomic constraints, a unified variational approach to both vakonomic and nonholonomic frameworks is given, and the corresponding equations of motion are recovered. Special attention is paid to the existence of infinitesimal variations in both cases, and it is proved that these variations coincide when the constraints are integrable. As examples, we give geometric formulations of the equations of motion for the case of optimal control and for vakonomic and nonholonomic mechanics with constraints linear in the velocities.

Lagrangian-Hamiltonian unified formalism for field theory

The Rusk–Skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems. It incorporates all the characteristics of Lagrangian and Hamiltonian descriptions of these systems (including dynamical equations and solutions, constraints, Legendre map, evolution operators, equivalence, etc.). In this work we extend this unified framework to first-order classical field theories, and show how this description comprises the main features of the Lagrangian and Hamiltonian formalisms, both for the regular and singular cases. This formulation is a first step toward further applications in optimal control theory for partial differential equations.

Geometric reduction in optimal control theory with symmetries

Abstract A general study of symmetries in optimal control theory is given, starting from the presymplectic description of this kind of systems. Then, Noethers theorem, as well as the corresponding reduction procedure (based on the application of the Marsden-Weinstein theorem adapted to the presymplectic case) are stated both in the regular and singular cases, which are previously described.

Non-constant discounting and differential games with random time horizon

Previous results on non-constant discounting in continuous time are extended to the field of deterministic differential games with a stochastic terminal time. A dynamic programming equation is derived for problems with general time inconsistent preferences and random duration. Different cooperative and non-cooperative solution concepts for differential games with random duration are analyzed. The results are illustrated by solving the cake-eating problem describing the classical model of management of a nonrenewable resource.

Singular Lagrangian Systems on Jet Bundles

The jet bundle description of time-dependent mechanics is revisited. The constraint algorithm for singular Lagrangians is discussed and an exhaustive description of the constraint functions is given. By means of auxiliary connections we give a basis of constraint functions in the Lagrangian and Hamiltonian sides. An additional description of constraints is also given considering at the same time compatibility, stability and second order condition problems. Finally, a classification of the constraints in first and second class is obtained using a cosymplectic geometry setting. Using the second class constraints, a Dirac bracket is introduced, extending the well-known construction by Dirac.

PRE-MULTISYMPLECTIC CONSTRAINT ALGORITHM FOR FIELD THEORIES

We present a geometric algorithm for obtaining consistent solutions to systems of partial differential equations, mainly arising from singular covariant first-order classical field theories. This algorithm gives an intrinsic description of all the constraint submanifolds. The field equations are stated geometrically, either representing their solutions by integrable connections or, what is equivalent, by certain kinds of integrable m-vector fields. First, we consider the problem of finding connections or multivector fields solutions to the field equations in a general framework: a pre-multisymplectic fiber bundle (which will be identified with the first-order jet bundle and the multi-momentum bundle when Lagrangian and Hamiltonian field theories are considered). Then, the problem is stated and solved in a linear context, and a pointwise application of the results leads to the algorithm for the general case. In a second step, the integrability of the solutions is also studied. Finally, the method is applied to Lagrangian and Hamiltonian field theories and, for the former, the problem of finding holonomic solutions is also analyzed.

On the computation of the aggregate claims distribution in the individual life model with bivariate dependencies

Abstract In this paper, we focus on the computation of the aggregate claims distribution in the individual life model when the portfolio is composed of independent pairs of dependent risks. We prove that the bivariate probabilities associated to each couple under any intermediate positive (negative) dependency hypothesis about their mortality can always be written as a convex linear combination between the independent and the comonotonic (mutually exclusive) ones. Considering this structure for the bivariate probabilities, we then obtain two recursive schemes for computing the distribution of the aggregate claims of the portfolio. These recursions greatly facilitate the computation of the aggregate claims distribution of a life insurance (sub)portfolio exclusively composed of dependent couples, and ease the interpretation of the impact of the dependence on the associated stop-loss premiums. Numerical results are given to demonstrate the applicability and efficiency of the method.