Fundamental theorems of asset pricing for piecewise semimartingales of stochastic dimension

TL;DR

This paper extends fundamental asset pricing theorems to piecewise semimartingales with stochastic dimensions, broadening the mathematical framework for modeling complex financial markets.

Contribution

It introduces the concept of piecewise semimartingales of stochastic dimension and extends key asset pricing theorems to this new setting.

Findings

Extended FTAPs to stochastic dimension processes

Established equivalence between no arbitrage and existence of martingale measures

Provided a framework avoiding infinite-dimensional stochastic integration pitfalls

Abstract

The purpose of this paper is two-fold. First is to extend the notions of an n-dimensional semimartingale and its stochastic integral to a piecewise semimartingale of stochastic dimension. The properties of the former carry over largely intact to the latter, avoiding some of the pitfalls of infinite-dimensional stochastic integration. Second is to extend two fundamental theorems of asset pricing (FTAPs): the equivalence of no free lunch with vanishing risk to the existence of an equivalent sigma-martingale measure for the price process, and the equivalence of no arbitrage of the first kind to the existence of an equivalent local martingale deflator for the set of nonnegative wealth processes.