Showing papers by "J. Michael Harrison published in 1978"

Speculative Investor Behavior in a Stock Market with Heterogeneous Expectations

Abstract: Consider a common stock that pays dividends at a discrete sequence of future times: t = 1,2, Taking all other prices and the random process that determines future dividends as exogenously given, we can ask what will be the price ofthe stock? In a world with a complete set of contingency claims markets, in which every investor can buy and sell without restriction, the answer is given by arbitrage. Let dtixt) denote the dividend that will be paid at time t if contingency Xj prevails, and let Ptixt) denote the current {t = 0) price ofa one dollar claim payable at time t if contingency Xt prevails. Then the current stock price must be 2(2;t,i3t(x«)dt(xf). Furthermore, in such a world it makes no difference whether markets reopen after initial trading. If markets were to reopen, investors would be content to maintain the positions they obtained initially (cf. Arrow, 1968). The situation becomes more complicated if markets are imperfect or incomplete or both. Ownership ofthe stock implies not only ownership of a dividend stream but also the right to sell that dividend stream at a future date. Investors may be unable initially to achieve positions with which they will be forever content, and thus the current stock price may be affected by whether or not markets will reopen in the future. If they do reopen, a speculative phenomenon may appear. An investor may buy the stock now so as to sell it later for more than he thinks it is actually worth, thereby reaping capital gains. This possibility of speculative profits will then be reflected in the current price. Keynes (1931, Ch. 12) attributes primary importance to this phenomenon (and goes on to suggest that it might be better if markets never reopened).

The diffusion approximation for tandem queues in heavy traffic

Abstract: Consider a pair of single server queues arranged in series. (This is the simplest example of a queuing network.) In an earlier paper [2], a limit theorem was proved to justify a heavy traffic approximation for the (two-dimensional) equilibrium waiting-time distribution. Specifically the waiting-time distribution was shown to be approximated by the limit distribution F of a certain vector stochastic process Z. The process Z was defined as an explicit, but relatively complicated, transformation of vector Brownian motion, and the general problem of determining F was left unsolved. It is shown in this paper that Z is a diffusion process (continuous strong Markov process) whose state space S is the non-negative quadrant. On the interior of S, the process behaves as an ordinary vector Brownian motion, and it reflects instantaneously at each boundary surface (axis). At one axis, the reflection is normal, but at the other axis it has a tangential component as well. The generator of Z is calculated. It is shown that the limit distribution F is the solution of a first-passage problem for a certain dual diffusion process Z ∗. The generator of Z ∗ is calculated, and the analytical theory of Markov processes is used to derive a partial differential equation (with boundary conditions) for the density f of F. Necessary and sufficient conditions are found for f to be separable (for the limit distribution to have independent components). This extends slightly the class of explicit solutions found previously in [2]. Another special case is solved explicitly, showing that the density is not in general separable.

The Recurrence Classification of Risk and Storage Processes

Abstract: Let X = {X(t), t ≥ 0} be a Markov storage process with compound Poisson input A = {A(t), t ≥ 0} and general release rule r(·). In a previous paper, a necessary and sufficient condition for the positive recurrence of X was obtained, and its stationary distribution was computed. Here we complete the recurrence classification of X, determining the conditions for null recurrence and transience. Closely related to X is a Markov process X0 = {X0(t), t ≥ 0}. Its paths are absolutely continuous and increasing between downward jumps, the instantaneous rate of increase at time t being r(X0(t)). The jumps are generated by A but are truncated as necessary to keep X nonnegative. It is shown that X0 is positive recurrent iff X is transient, null recurrent iff X is null recurrent, and transient iff X is positive recurrent. Furthermore, in the case where X0 is transient, the probability that X0 ever hits zero (viewed as a function of the initial state) has the same density as the stationary distribution of X. A similar d...