On the Mathematics and Economic Assumptions of Continuous-Time Models (Classic Reprint) - Hardcover

Robert C. Merton

Hardcover

ISBN 10: 0260265756 ISBN 13: 9780260265753

Publisher: Forgotten Books, 2020

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Excerpt from On the Mathematics and Economic Assumptions of Continuous-Time Models

The second component of cost is that the mathematical tools required for the formal manipulations used to derive the continuous-time equations are somewhat specialized and, therefore, may not be familiar. For example, the sample paths for stochastic variables generated by diffusion processes, while continuous, are almost nowhere differentiable in the usual sense, and therefore a more general type of differential equation is required to ex press the dynamics of such processes. While there is a substantial mathematics literature on these generalized stochastic equations, g/ the derivations, although elegant, are often cryptic and difficult to follow. Moreover, these derivations provide little insight into the relationships between the formal amathematical assumptions and the corresponding economic assumptions. Hence, this paper attempts to bridge this gap by using only elementary probability methods to derive the basic mathematical theorems required for continuous time analysis and as part of the derivations to make explicit the economic assumptions implicitly imbedded in the mathematical assumptions.

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