New Results of Itˆo’s Formula Using q-Calculus

Abstract

In this paper, we extend the classical Itˆo formula by employing the framework of quantum calculus to derive its q-analogue, applicable to both classical Brownian motion and q-Itˆo processes via the quantum Taylor expansion. We also determine the q-infinitesimal generators associated with stochastic q-differential equations. To illustrate the applicability of our results, we present the q-Black-Scholes equation as a concrete example. Throughout the study, we assume 0 < q < 1, and we show that the classical results of Itˆo calculus are recovered in the limit as q → 1.

Keywords: q-calculus; Itˆo calculus; semigroup; stochastic differential equations.

2010 Mathematics Subject Classification. 05A30; 60H05; 60H10; 60J65