3,833 research outputs found
"Itô's Lemma" and the Bellman equation: An applied view
Rare and randomly occurring events are important features of the economic world. In continuous time they can easily be modeled by Poisson processes. Analyzing optimal behavior in such a setup requires the appropriate version of the change of variables formula and the Hamilton-Jacobi-Bellman equation. This paper provides examples for the application of both tools in economic modeling. It accompanies the proofs in Sennewald (2005), who shows, under milder conditions than before, that the Hamilton-Jacobi-Bellman equation is both a necessary and sufficient criterion for optimality. The main example here consists of a consumption-investment problem with labor income. It is shown how the Hamilton-Jacobi-Bellman equation can be used to derive both a Keynes-Ramsey rule and a closed form solution. We also provide a new result. --Stochastic differential equation,Poisson process,Bellman equation,Portfolio optimization,Consump
"Ito's Lemma" and the Bellman equation for Poisson processes: An applied view
Rare and randomly occurring events are important features of the economic world. In continuous time they can easily be modeled by Poisson processes. Analyzing optimal behavior in such a setup requires the appropriate version of the change of variables formula and the Hamilton-Jacobi-Bellman equation. This paper provides examples for the application of both tools in economic modeling. It accompanies the proofs in Sennewald (2005), who shows, under milder conditions than before, that the Hamilton-Jacobi-Bellman equation is both a necessary and sufficient criterion for optimality. The main example here consists of a consumption-investment problem with labor income. It is shown how the Hamilton-Jacobi-Bellman equation can be used to derive both a Keynes-Ramsey rule and a closed form solution. We also provide a new result. --Stochastic differential equation,Poisson process,Bellman equation,Portfolio optimization,Consumption optimization
On the Fragility of the Basis on the Hamilton-Jacobi-Bellman Equation in Economic Dynamics
In this paper, we provide an example of the optimal growth model in which
there exists infinitely many solutions to the Hamilton-Jacobi-Bellman equation
but the value function does not satisfy this equation. We consider the cause of
this phenomenon, and find that the lack of a solution to the original problem
is crucial. We show that under several conditions, there exists a solution to
the original problem if and only if the value function solves the
Hamilton-Jacobi-Bellman equation. Moreover, in this case, the value function is
the unique nondecreasing concave solution to the Hamilton-Jacobi-Bellman
equation. We also show that without our conditions, this uniqueness result does
not hold
“Itô’s Lemma“ and the Bellman Equation for Poisson Processes: An Applied View
Using the Hamilton-Jacobi-Bellman equation, we derive both a Keynes-Ramsey rule and a closed form solution for an optimal consumption-investment problem with labor income. The utility function is unbounded and uncertainty stems from a Poisson process. Our results can be derived because of the proofs presented in the accompanying paper by Sennewald (2006). Additional examples are given which highlight the correct use of the Hamilton-Jacobi-Bellman equation and the change-of-variables formula (sometimes referred to as “Ito’s-Lemma”) under Poisson uncertainty.stochastic differential equation, Poisson process, Bellman equation, portfolio optimization, consumption optimization
Optimal consumption of the finite time horizon Ramsey problem
AbstractIn this paper, we study the stochastic Ramsey problem related to an economic growth model with the CES production function in a finite time horizon. By changing variables, the Hamilton–Jacobi–Bellman equation associated with this optimization problem is transformed. By the viscosity solution technique, we show the existence of a classical solution of the transformed Hamilton–Jacobi–Bellman equation, and then give an optimal consumption policy of the original problem
Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation
We consider a utility maximization problem for an investment-consumption
portfolio when the current utility depends also on the wealth process. Such
kind of problems arise, e.g., in portfolio optimization with random horizon or
with random trading times. To overcome the difficulties of the problem we use
the dual approach. We define a dual problem and treat it by means of dynamic
programming, showing that the viscosity solutions of the associated
Hamilton-Jacobi-Bellman equation belong to a suitable class of smooth
functions. This allows to define a smooth solution of the primal
Hamilton-Jacobi-Bellman equation, proving that this solution is indeed unique
in a suitable class and coincides with the value function of the primal
problem. Some financial applications of the results are provided
A Model for Optimal Human Navigation with Stochastic Effects
We present a method for optimal path planning of human walking paths in
mountainous terrain, using a control theoretic formulation and a
Hamilton-Jacobi-Bellman equation. Previous models for human navigation were
entirely deterministic, assuming perfect knowledge of the ambient elevation
data and human walking velocity as a function of local slope of the terrain.
Our model includes a stochastic component which can account for uncertainty in
the problem, and thus includes a Hamilton-Jacobi-Bellman equation with
viscosity. We discuss the model in the presence and absence of stochastic
effects, and suggest numerical methods for simulating the model. We discuss two
different notions of an optimal path when there is uncertainty in the problem.
Finally, we compare the optimal paths suggested by the model at different
levels of uncertainty, and observe that as the size of the uncertainty tends to
zero (and thus the viscosity in the equation tends to zero), the optimal path
tends toward the deterministic optimal path
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