On Exit Times of a Multivariate Random Walk and Its Embedding in a Quasi Poisson Process

Abstract

We introduce and analyze a delayed renewal process 𝒯 = {τ0,τ1,…} marked by a multivariate random walk (𝒮,𝒯) and its behavior about fixed levels to be crossed by one of the components of (𝒮,𝒯). We derive the joint distribution of first passage time τρ, pre-exit time τρ−1 (i.e., the instant one phase prior to the first passage time), and the respective values of (𝒮,𝒯) at τρ and τρ−1 in a closed form. The results obtained are then applied to a multivariate quasi Poisson process Π, forming a random walk (𝒮(Π),𝒯) embedded in Π over 𝒯. Processes like these can model various phenomena including stock market and option trading. One of the central issues in the investigation of (𝒮(Π),𝒯) is to obtain the information about Π at any moment of time in random vicinities of τρ and τρ−1 previously available only upon 𝒯. The results offer, again, closed form functionals. Numerous examples throughout the paper illustrate introduced constructions and connect the results with real-world applications, most prominently the stock market.