On the well-posedness of classical solutions to non-canonical parabolic equations arising from macroeconomics

Abstract

In this thesis, we study the well-posedness of the classical solution to a non-canonical parabolic equation arising from the optimal savings problem. The model we considered is in the full stochastic setting and also over a finite time horizon. More precisely, to model the problem in a more realistic manner, we add both random noises to the capital dynamics K and to the population dynamics N. The Cobb-Douglas production function F is incorporated in the evolution of the capital dynamics K, while the drift rate in the population dynamics N and the utility functions can be in very general form in contrast to many previous works. Then we study the corresponding Hamilton-Jacobi-Bellman (HJB) equation (derived from the dynamic programming principle (DPP)) for the value function, which is nonlinear and of parabolic type. The unconventional nonlinear HJB equation has R2+ as its spatial domain (since K and N are always assumed to be positive) and cannot be reduced to an equation with just one spatial dimension by introducing new variables compared with some previous works. It has two main features: the Cobb-Douglas production function as the coefficient of the gradient of the value function includes the mismatching of power rates between K and N, this coefficient has exponential growth if we transform the equation to the whole space; moreover, there is a nonlinear singular term, which is essentially a negative power of the gradient, coming from the necessary condition of the optimality of the control. Their resolutions turn out to be the major problem intractable by the classical analysis; to this end, we develop a novel theoretical treatment as a hybrid of weighted Sobolev theory and probabilistic approach for growth rate discovery and energy estimates in face of the unbounded spatial domain. More precisely, we show the global-in-time existence and uniqueness of classical solutions to the underlying model in this thesis. To do so, we first solve the linearized problem by the Schauder estimates and the maximum principle (for the classical solution) together with an approximation technique in the weighted Sobolev space based on the energy estimates (for the weak solution). The nonlinear problem is solved (in the weak sense) by a fixed point argument which relies on the optimal lower and upper growth rate of the solution to the linearized problem and also on the compact embedding of certain weighted Sobolev spaces. We also show that this weak solution to the nonlinear problem is indeed classical by some regularity results. For the optimal growth rate, the traditional maximum principle for parabolic equations is not effective; instead, with some analytic argument together with the use of celebrated Feynman-Kac formula for a pair of partially coupled diffusion processes but with non-Lipschitz driving coefficients, we finally obtain the optimal growth rate. Finally, some empirical studies on the paths of the evolution of capital K and population N are also included.