A Continuous Time Financial Market with a Poisson Process as Transaction Timer

Abstract

We study the influence of transactions on investors' portfolio values under the assumption that the investors' transaction times are determined by Poisson point processes, whose intensity measures can naturally be interpreted as transaction frequencies. We give lower and upper bounds on the expectations of portfolio values in terms of transaction intensities, and prove that there exist a sequence of portfolio fractions and a transaction frequency which maximize the expectation of the portfolio value for a finite horizon. We also give bounds on transaction frequencies for preventing the investors from losing money. Then the optimal transaction strategies for finite and infinite time horizons and the asymptotic effects of making transactions are discussed based on the concept of a benefit function of transactions. Finally, we investigate the influence of transactions on financial markets, with the market mean rates of return and volatilities being connected with the transaction frequency. We show that the market becomes unprofitable in a finite time if an overwhelming amount of transactions is involved and the market is suitable for some limited transactions when its trade capacity does not increase beyond any limit at a relatively high speed. Our models and simulations illustrate how the investors' collective action affects the financial market.