Dynamic optimization and its relation to classical and quantum constrained systems
Introduction
We have recently witnessed the increasing application of ideas from physics to finance and economics, such as path integral techniques applied to the study the Black–Scholes model in its different forms [1], [2], [3], [4], [5], [6], [7]. Some developments have also been used to try to understand the Black–Scholes equation as a quantum mechanical Schrödinger equation [8], [9], [10]. In the last few years, constrained systems techniques, through Dirac’s method [11], [12] have been used to explain some features of stochastic volatility models [13], [14] and the multi-asset Black–Scholes equation [15]. In this paper, we apply these same constrained methods to understand (from a physical point of view) a particular issue: the dynamic optimization problem.
We start by analyzing the dynamic optimization problem for a single-state variable and a control variable . By identifying the state variable as the coordinate of a physical particle and the Lagrange multiplier as its canonical momentum , we can map the theory in the Hamiltonian phase space. Here, the model presents constraints thus, it is necessary (to study the system in a correct way) to use Dirac’s method of constrained systems. The application of this method implies that constraints are of second-class character according to Dirac’s classification. Thus, the dynamic optimization problem can be seen as a second-class physically constrained system.
We also analyze the role of open-loop and closed-loop strategies in the context of Pontryagin’s framework. We explicitly show that the only consistent strategies that permit the Pontryagin equations to be obtained correctly from the optimization of cost functionals are open-loop -strategies (). For closed loop -strategies (), the optimization of the cost functional gives a consistency relation which is related to the Hamilton–Jacobi–Bellman equation.
After that, we explore the quantum side of this classically constrained system. By quantizing it according to the standard rules of quantum mechanics and imposing the constraints as operator equations over the wave function, we arrive at a set of partial differential equations for the wave function. After defining the wave function as , these equations map into some partial differential equations for the function. For right-hand side quantization order, these equations give origin to the Hamilton–Jacobi–Bellman equation of the dynamic programming theory. Thus, Bellman’s maximum principle can be considered as the quantum view of the optimization problem.
To make this paper self-contained for non-physicist readers coming from the optimization field, we start with a brief digression on classical and quantum physics in Section 2.
Section snippets
Hamiltonian quantum and classical mechanics
In physics, quantum-dynamic behavior is defined by the Hamiltonian operator. For the simple case of a nonrelativistic one-dimensional particle subjected to external potential , the Hamiltonian operator reads where is the momentum operator. The wave function at time (given that the wave function at is ) is thus which can be written as a convolution according to where the propagator admits
The Pontryagin approach
Consider an optimal control problem that is commonly used in financial applications (see, for example, [18]). We want to optimize the cost functional where represents a state variable (for example, the production of a certain article) and is a control variable (such as the marketing cost). The state variable must satisfy the market dynamic The problem is to determine how to obtain the production trajectory and the control path to optimize
The optimization problem as a classically constrained system
From a structural point of view, the optimization problem is then characterized completely by the Lagrangian multiplier . For a open-loop -strategy, the optimization of the action (16) gives a system of coupled ordinary differential equations: the Pontryagin equations (18), (19) for both, open or closed-loop optimal -strategies. For a closed-loop -strategy instead, the optimization of the action (16) gives a partial differential equation for : the consistency relation (34), which is
Dynamic optimization and quantum mechanics
Until now, we have studied the dynamic optimization problem from a classical point of view, and we have seen that it is equivalent to a classical physically constrained system. However, what happens at the quantum level? To explore that view, we will quantize our classical system and study its consequences.
Again, consider the classical Hamiltonian Now, we have to quantize the classical Hamiltonian (53). For this purpose, we replace with the appropriate
Conclusions
In this article, we have examined the structure of the dynamic optimization problem from a physical perspective, and we conclude that the correct analysis of the optimization problem must be done either in the phase-space or using the classical Hamiltonian approach. Due to the presence of constraints in the theory, we must apply Dirac’s method for constrained systems. Dirac’s analysis implies that the theory has two second-class constraints. One of these constraints fixes the momentum
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