Raw Moments of Asset Returns Under a Non-Homogeneous Poisson Bid Arrival Process with Beta-Distributed Percentage Bids

Suppose that bids, or offers, to purchase an asset for sale occur according to a non-homogeneous Poisson process. Bid amounts, measured in percentages, are assumed to be independent and identically distributed random variables. Faced with the pressure of having to sell the asset by a certain time period (or else risk getting nothing at all for it), this article analyzes the probability distribution of the obtained return on the asset under a specific type of selling strategy, advancing the initial results found in Karlin’s seminal paper from 1962. In particular, the main result derived is an explicit expression for the raw moments of the return on the asset. Such moments are widely used in probability theory and stochastic modeling, since, in most instances, having all the raw moments fully determines the underlying probability distribution. The special case when bid amounts have a beta distribution with integer shape parameters is examined in depth. Numerical results are presented and discussed for a variety of different selling strategies, time-varying offer rates, and bid amount distributions.