There is a growing statistical evidence that many price processes (stock prices, index values, exchange rates, interest rates) exhibit presence of very small, but very frequent jumps. Moreover, this behavior seems to be unstable over time, sometimes the presence of small jumps is very strong, but sometimes almost nonexistent. Traditional decision control analysis uses diffusion models with possible addition of infrequent large jumps to study problems related to the market control.
Because of the statistical evidence of jump presence in many price processes, it is proposed to study: 1) How the market agent (investor, regulator) should react to the presence of jumps? What is the optimal strategy to use? 2) Is the optimal strategy different for the problem without jumps in contrast to the problem with jumps? Is the strategy for the problem without jumps (diffusion model) robust in the sense that if used for the problem with jumps, it performs reasonably well? 3) Given that there is a significant uncertainty about the presence of frequent small jumps in the first place and the strategy found for the diffusion problem is not robust, find a reasonable strategy which would perform well in both models.
