Fluctuations of Recurrent Processes and Their Applications to the Stock Market

Abstract

We study a class of stochastic processes describing the evolution of stocks or mutual funds. A stock price protocol is often restricted to particular random observation epochs making price variations look like a jump process. The results include the random time when the price drops for the first time, the price itself, the highest price and the epoch before the fall, as well as all of these in the opposite direction, in terms of easily tractable functionals, from which one can derive probability distributions and all kind of moments. The methodology is based on the fluctuation analysis of marked recurrent processes (with position dependent marking) and the adherence of an auxiliary nonnegative discrete-valued component that is to observe the stock price variations. A particular case of the model under investigation also includes a variant of the marked Poisson process with arbitrarily continuously distributed marks and observation times at one of which a decision to sell or buy a stock can be made.

Key Words:

• Fluctuation analysis
• Stock market
• Mutual funds
• Commodities
• Options
• Multivariate marked point process
• Marked Poisson process
• First excess level theory
• First passage time
• Ramdom observations

AMS Subject Classification:

• 60K10
• 60K15
• 60K25