A Generalization of Stochastic Calculus--A Conjecture

Abstract

Existing models of calculus in finance, such as Ito calculus have an underlying assumption that the parameters are known. This paper relaxes that assumption. The paper proposes the creation of differentiable paths that do not require an expectation to exist. Further, the path minimizes the expected loss from using a predictive calculus and, under mild conditions, stochastically dominates any other path. Further, the path is assured to produce fair gambling odds if used in finance. This method maps to Ito and Stratonovich methods if the parameters are known and dominate them if they are not in the general case.