A Simple Characterization of Dynamic Completeness in Continuous Time
This paper examines if (and how) continuous-time trading renders dynamically-complete a financial market in which the underlying risk process is a Brownian motion and the securities pay dividends that are proportional to geometric Brownian motions. A sufficient condition, that the instantaneous dispersion matrix of the relative dividends is non-degenerate, has been established recently in the literature for the case in which the financial market in question is part of a single-commodity, pure-exchange economy with many heterogenous agents, where all intermediate flows of utilities and endowments are analytic functions. The present paper shows that the condition is indeed sufficient, as well as necessary in some important cases, by means of a very different (and more intuitive) mathematical argument that assumes neither analyticity nor a particular economic environment. It requires only that the pricing kernels are continuous and satisfy a standard growth condition. In this sense, dynamic completeness obtains irrespectively of preferences, endowments, and other structural elements (such as whether or not the budget constraints include only pure exchange, whether or not the time horizon is finite with lump-sum dividends available on the terminal date, etc.).
|Date of creation:||20 May 2012|
|Date of revision:||02 Sep 2013|
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