This paper investigates how continuous-time trading renders complete a financial market in which the underlying risk process is a Brownian motion. A sufficient condition, that the instantaneous dispersion matrix of the relative dividends is non-degenerate, has been established in the literature for single-commodity, pure-exchange economies with many heterogenous agents where the securities’ dividends as well as the agents’ utilities and endowments include flows during the trading horizon which are analytic functions. In sharp contrast, the present analysis is based upon a different mathematical argument that assumes neither analyticity nor a particular underlying economic environment. The novelty of our approach lies in deriving closed-form expressions for the dispersion coefficients of the securities’ prices. To this end, we assume only that the pricing kernels and dividends satisfy standard growth and smoothness restrictions (mild enough to allow even for options). In this sense, our sufficiency conditions apply irrespectively of preferences, endowments or other structural elements (for instance, whether or not the budget constraints include only pure exchange).

本文研究了连续时间交易如何构成一个完整的金融市场，其中潜在的风险过程是布朗运动。文献中已经建立了一个条件，即相对股息的瞬时分散矩阵是不退化的，这种条件适用于具有许多异类代理人的单商品，纯交换经济体，其中证券的股息以及代理人的效用和赋包括交易范围内的流量，这些流量是分析函数。与之形成鲜明对比的是，本分析是基于不同的数学论证，该论证既不假设分析性也不假设特定的潜在经济环境。我们方法的新颖之处在于为证券价格的分散系数得出封闭形式的表达式。为此，我们仅假设定价核心和股息满足标准增长和平滑度限制（适度甚至允许期权）。从这个意义上说，我们的充足条件与偏好，end赋或其他结构要素无关（例如，预算约束条件是否仅包括纯交换）。