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Abstract. We extend the notions of irreducibility and periodicity of a stochastic matrix to a unital positive linear map on a finite-dimensional * -algebra and discuss the non-commutative version of the Perron-Frobenius theorem. For a completely positive linear map with ( ) = ∑ ℓ ℓ * ℓ , we give conditions on the ℓ 's equivalent to irreducibility or periodicity of . As an example, positive linear maps on 2 (ℂ) are analyzed.

We study the structure of a simple dynamic optimization problem consisting of one state and one control variable, from a physicist's point of view. By using an analogy to a physical model, we study this system in the classical and quantum frameworks. Classically, the dynamic optimization problem is equivalent to a classical mechanics constrained system, so we must use the Dirac method to analyze it in a correct way. We find that there are two second-class constraints in the model: one fix the momenta associated with the control variables, and the other is a reminder of the optimal control law. The dynamic evolution of this constrained system is given by the Dirac's bracket of the canonical variables with the Hamiltonian. This dynamic results to be identical to the unconstrained one given by the Pontryagin equations, which are the correct classical equations of motion for our physical optimization problem. In the same Pontryagin scheme, by imposing a closed-loop λ-strategy, the optimality condition for the action gives a consistency relation, which is associated to the Hamilton-Jacobi-Bellman equation of the dynamic programming method. A similar result is achieved by quantizing the classical model. By setting the wave function Ψ(x, t) = e iS(x,t) in the quantum Schrödinger equation, a non-linear partial equation is obtained for the S function. For the right-hand side quantization, this is the Hamilton-Jacobi-Bellman equation, when S(x, t) is identified with the optimal value function. Thus, the Hamilton-Jacobi-Bellman equation in Bellman's maximum principle, can be interpreted as the quantum approach of the optimization problem.

An interacting Black-Scholes model for option pricing, where the usual constant interest rate r is replaced by a stochastic time dependent rate r(t) of the form r(t) = r + f (t)Ẇ (t), accounting for market imperfections and prices non-alignment, was developed in [1]. The white noise amplitude f (t), called arbitrage bubble, generates a time dependent potential U (t) which changes the usual equilibrium dynamics of the traditional Black-Scholes model. The purpose of this article is to tackle the inverse problem, that is, is it possible to extract the time dependent potential U (t) and its associated bubble shape f (t) from the real empirical financial data? In order to give an answer to this question, the interacting Black-Scholes equation must be interpreted as a quantum Schrödinger equation with hamiltonian operator H = H 0 + U (t), where H 0 is the equilibrium Black-Scholes hamiltonian and U (t) is the interaction term. If the U (t) term is small enough, the interaction potential can be thought as a perturbation, so one can compute the solution of the interacting Black-Scholes equation in an approximate form by perturbation theory. In [2] by applying the semi-classical considerations, an approximate solution of the non equilibrium Black-Scholes equation for an arbitrary bubble shape f (t) was developed. Using this semi-classical solution and the knowledge about the mispricing of the financial data, one can determinate an equation, which solutions permit obtain the functional form of the potential term U (t) and its associated bubble f (t). In all the studied cases, the non equilibrium model performs a better estimation of the real data than the usual equilibrium model. It is expected that this new and simple methodology for calibrating and simulating option pricing solutions in the presence of market imperfections, could help to improve option pricing estimations.