There are several examples of where banach fixed point theorem can be used in economics for more detail you can check oks. The functional equation 1 is called a bellman equation. To see that tis a contraction mapping, take two bounded functions j 1 and j 2 on x. Now apply the contraction mapping theorem with x viewed as a parameter. Moreover, the banach fixedpoint theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in m the iterated function sequence x, f x, f f x, f f f x. By contraction mapping theorem, we get a unique esuch that te e, or equivalently, fx. Wealsowantto try to establish whether equation 2 can be thought of as the limit of the dp problem. Stochastic estimation and control spring 2007 lecture. Sketch of the proof the contraction mapping theorem. Just consider k is capital, fk is the production function, fk y.
Mathematical analysis of reinforcement learning bellman. The contraction mapping theorem admits a converse 3, 16, pp. Recursive metho ds introduction to dynamic optimization nr. Continuoustimemethodsbellmanequation, brownianmotion, itoprocess, and itos lemma. Discrete time methods bellman equation, contraction mapping theorem, and blackwells su. Keywords stochastic dynamic programming, contraction mapping, bellman equation. The goal is to obtain some results on the solution to the bellman equation or xed points of the bellman operator without topological assumptions in the dynamic economic model based on recursive utility. Continuoustimemethodsbellmanequation, brownianmotion, itoprocess. Markov decision processes and bellman equations emo todorov. Macroeconomics sequence, block i the contraction mapping. The use of the contraction mapping theorem is that it can be applied to any metric space, so in particular to the space of functions.
Laibson matthew basilico spring, 20 course outline. Contraction mapping principle approach to differential equations bishnu p. Several mathematical theorems the contraction mapping the orem also called the banach fixed point theorem, the theorem of the maxi mum or berges. Contraction mapping, inverse and implicit function theorems 1 the contraction mapping theorem denition 1. These questions are resolved by contraction mapping theorem. Dynamic programming, bellmens equation, contraction mapping theorem, blackwells sufficiency conditions. We must of course check that the hypotheses are satis. The contraction mapping principle is one of the most useful tools in the study of nonlinear equations, be they algebraic equations, integral or di. In this video, i am proving banach fixed point theorem which states that every contraction mapping on a complete metric space has unique fixed point. This is because bellmans equation defines a contraction.
Banach fixed point theorem contraction mapping theorem. The operator that is the rhs of the bellman equation operates on functions, and the solution is a fixed point in some space of functions. We can hence apply the contraction mapping theorem and show that the. Then gx,z is a function of z only and one may use the contraction mapping theorem to solve z gx,z. Bellmans equation study of bellman equation with bounded f contraction mapping and theorem of the maximum. Dynamic programming through bellman operators stanford. Building upon all of this, we mathematically proved the unique optimality of the bellman optimality equation for every mdp. Dhungana department of mathematics, mahendra ratna campus tribhuvan university, kathmandu nepal abstract using an extension of the contraction mapping principle, a new approach has been proposed in proving the existence of unique solutions of some differential equations. It writes the value of a decision problem at a certain point in time in terms of the payoff from some initial choices and the value of the remaining decision problem. Lectures notes on deterministic dynamic programming.
Method of contraction map iliang chern department of applied mathematics. In this approach, the bellman operator is shown to be some type of contraction, and a variant of the contraction mapping theorem is used to show that the bellman operator has a unique xed point in the space of continuous functions with a suitable topology. X is a contraction with constant c, any iterate fn is a contraction with constant cn. V in b s, k v wk kv wk contraction mapping theorem. Suppose that x is a complete metric space and that f. The bellman equation florian scheuer 1 plan prove properties of the bellman equation in particular, existence and uniqueness of solution. The main theorem theorem if v is a solution of the bellman equation, v lv, then v v furthermore, under the model assumptions stated above a unique solution of the bellman equation always exists. Applications to growth, search, consumption, asset pricing 2.
The following theorem shows that for etd0, the projected bellman operator. Morhaim 2002 are based on variants of the contraction mapping theorem. A liey ingredient of the formulation is the abstraction of three widely shared. This article formulates and analyzes a broad class of optimi zation problems including many, but not all, dynamic programming problems. The banach fixed point theorem is also called the contraction mapping theorem, and it is in general use to prove that an unique solution to a given equation exists. Contraction mapping theorem below is a statement and proof of the contraction mapping theorem. I the proof of this theorem is based on the property that l is a contraction mapping. We study an endogenous growth model as well as the lucas asset pricing model in an exchange economy, signi. If you look at the equation, its pretty clear that the solution has to be a fixed point of the operator on the rhs of the bellman equation. Contraction mapping, inverse and implicit function. A careful explanation of these properties with an introduction to markov chains. We used the envelope theorem in order to derive the euler equation, and consequently to characterize the solution to the problem closer. Understanding exact dynamic programming through bellman.
Preliminaries dynamic programming dynamic programming is a mathematical optimization method. Contraction mappings in the theory underlying dynamic. Elementary results on solutions to the bellman equation of. From contraction mapping theorem, knowing that blackwells suf. Contraction mappings in the theory underlying dynamic programming. Several mathematical theorems the contraction mapping the orem also called the banach fixed point theorem, the theorem of the maximum or berges maximum theorem, and blackwells su ciency conditions. Bellman equation gives recursive decomposition, value function stores and reuses solutions.
The method proposed exploits the well known fact that the value function is a solution to the bellman equation stokey et al. Thinking of the bellman equation as a functional equation helps. We do not present the most general or strongest version of this theorem, but a version that admits a straightforward proof using the contraction mapping. Optimality, the bellman equation, and presents three methods for solving the bellman equation with example. This equation is called the bellman functional equation in dynamic programming.
Applying it to equation 1 will establish the existence of a unique value function v, greatly facilitating the analysis of such dynamic models. Chapter 4 existence and uniqueness of solutions for. Notes on discrete time stochastic dynamic programming. From the contraction mapping theorem, it follows that bellmans equation has a unique fixed point. As in kamihigashis 2014 approach, we use a xed point theorem of an ordered space instead of a variant of the contraction.
Contraction mapping theorem the bellman equation is v. We also present a number of applications of the theorem. X is a contraction mapping, then t has one and only one. Their argument does not depend on a contraction mapping technique but rather builds on important insights derived from the assumptions imposed on the return functions. Contraction mapping principle approach to differential. From the contraction mapping theorem, it follows that bellmans equation has a unique xed point. Notice that the value function is the expected discounted present value of the. Discrete time methods a bellman equation, contraction mapping theorem, blackwells su cient conditions, numerical methods i. Dynamic programming assumes full knowledge of the mdp. As a consequence, we use the contraction mapping theorem to show there exist one unique solution to the bellman equation. Recursive utility and the solution to the bellman equation. We learned some basic mathematical tools like metric spaces, complete metric spaces, cauchy sequences, contraction mapping and the banach fixed point theorem.
In mathematics, the banachcaccioppoli fixedpoint theorem also known as the contraction mapping theorem or contractive mapping theorem is an important tool in the theory of metric spaces. Chapter 4 existence and uniqueness of solutions for nonlinear. In class we used the contraction mapping theorem to prove the existence and uniqueness of solutions to a first order not necessarily linear ode on some interval 0,h. In mathematics, a contraction mapping, or contraction or contractor, on a metric space m,d is a function f from m to itself, with the property that there is some nonnegative real number. It not only tells you that there is a unique solution, it tells you how to find it. So we can invoke contraction mapping theorem to claim fixed point. A solution to this problem is a function v that satisfies the equation above.
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