Optimal consumption of the finite time horizon Ramsey problem
The economic growth model mathematically is presented by the Ramsey problem. In the Introduction the concept of the stochastic Ramsey problem with a finite horizon time \(T > 0\) is considered. The aim of this consideration is to choose a consumption rate processes \(c = \{c(t)\}\) maximizing the discounted expected utilities with a constant discount rate \(\alpha \geq 0.\) The associated Hamilton-Jacobi-Bellman (HJB) equation is given by a parabolic partial differential equation. By changing variables the HJB equation is transformed to a partial differential equation with respect to the viscosity \(V.\) In Section 2 the existence of the viscosity solution \(V\) with \(T = \infty\) is shown. The stochastic control problem \(V(t,x)\) is considered. The auxiliary control problems \(V_{n}(t,x)\) are introduced and in Proposition 2.3 for each \(t,x \geq 0\) the convergence of \(V_{n}(t,x)\) to \(V(t,x)\) as \(n \to \infty\) is proved. In Theorem 2.5 the dynamic programming principle for \(V_{n}\) is established. Theorem 2.8 shows that \(V\) is a viscosity solution to the associated HJB equation with given boundary conditions. In Section 3 the \(C^{1,2}-\)regularity of the continuous viscosity solution \(V\) is studied. A synthesis of the optimal consumption \(c^{*}\) for the stochastic Ramsey problem is presented. The function \(u(t,x) = V(T-t,x)\) is defined and in Theorem 3.2 it is shown that this function is a solution of the associated HJB equation. In Section 4 an application of the obtained results to the infinite time horizon problem is made. The concept of the discounted expected utilities with a discount rate \(\alpha \geq 0\) is developed. A function \(\widehat{V}(x),\) \(x \geq 0\) is defined. Theorem 4.1 shows that the function \(\widehat{V}(x)\) is a solution of a equation of type of HJB equation. Theorem 4.2 shows the form of the optimal policy for the discounted expected utilities problem. Two examples are considered.
scientific article; zbMATH DE number 5577092
scientific article