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CTF-Chapter 3 - On the Mathematics and Economics...

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On the Mathematics and Economics Assumptions of Continuous-Time Models BaoheWang BaoheWang [email protected] [email protected]
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3.1 Introduction This chapter attempts to (1) bridge the gap by using only elementary probability theory and calculus to derive the basic theorems required for continuous time analysis. (2) make explicit the economics assumptions implicitly embedded in the mathematical assumption.
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The general approach is to keep the assumption as weak as possible. But we need make a choice between the losses in generality and the reduction in mathematical complexity.
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The substantive contributions of continuous time analysis to financial economic theory: (1) trading take place continuously in time (2) the underlying stochastic variables follow diffusion type motions with continuous sample paths The twin assumptions lead to a set of behavioral equations for intertemporal portfolio selection that are both simpler and rich than those derived from the corresponding discrete trading model.
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The continuous trading is an abstraction from physical reality. If the length of time between revisions is very short, the continuous trading solution will be a approximation to the discrete trading solution. The application of continuous time analysis in the empirical study of financial economic data is more recent and less developed.
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In early studies, we assume that the logarithm of the ratio of successive prices had a Gaussian distribution. But the sample characteristics of the time series were frequently inconsistent with these assumed population properties. Attempts to resolve these discrepancies proceeded along two separate paths.
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The first maintains the independent increments and stationarity assumptions but replaces the Gaussian with a more general stable (Parato- Levy) distribution. The stable family frequently fit the tails of the empirical distributions better than the Gaussian. But there is little empirical evidence to support adoption of the stable Paretian hypothesis over that of any leptokurtotic distribution.
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Moreover, the infinite variance property of the non-Gaussian stable distributions implis : (1) most of our statistical tools, which are based upon finite-moment assumptions are useless. (2) the first-moment or expected value of the arithmetic price change does not exist.
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The second, Cootner (1964) consider the alternative path of finite-moment processes whose distributions are nonstationary . The general continuous time framework requires that the underlying process be a mixture of diffusion and Poisson-directed processes. The general continuous time framework can accommodate a wide range of specific hypotheses including the “ reflecting barrier” model.
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Rosenberg (1972) shows that a Gaussion model with a changing variance rate appear to explain the observed fat-tail characteristics of return.
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