Stochastic Capital Theory

Abstract

Many problems in capital theory — particularly ‘Austrian’ capital theory —take the following form: an asset has an intrinsic value X(t) at time t. If he takes a particular action at time T, then the asset’s owner gets X(T) at T. In anticipation of future usage we shall call the action taken at T stopping and refer to T as a stopping time. This set-up raises two natural, and related, questions. When should the intrinsic process be stopped? What is the present value of the asset? The standard examples are when to drink the wine whose quality at t is given by X(t) or when to cut down the tree which contains lumber with a value of X(t). If the discount rate is r then these questions may be simply answered. The optimal stopping time T* maximizes e^-rT X(T) and the present value of the tree is its discounted value

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(1)

To distinguish from intrinsic value, call this latter quantity the market value of the asset.

This is a revision of NBER technical paper 23 (May 1982). The research reported here is part of the NBER’s research program in Financial Markets and Monetary Economics. We are grateful to the National Science Foundation and the University of Wisconsin Graduate School for Research support. Rothschild and Stiglitz held the Oskar Morgenstern Dis­tinguished Fellowship, Mathematica and Brock was Sherman Fairchild Distinguished Scho­lar, California Institute of Technology while some of the work on this paper was done.