The physical mechanism of stochastic calculus in random walks

Abstract Stochastic differential equations (SDEs) play an important role in fields ranging from physics and biology to economics. The interpretation of stochastic calculus in the presence of multiplicative noise continues to be an open question. Commonly, the choice of stochastic calculus rules is largely based on empirical knowledge and lacks quantitative substantiation. In this study, we introduce a functional method that quantitatively links stochastic calculus rules to the underlying physical mechanisms in random walks. For a given diffusion coefficient, we construct three models to exemplify the physical features of conventional stochastic calculus. Our work provides a new perspective for quantitatively addressing state-dependent noise and aims to contribute to the understanding of the physical factors underlying uncertainty in SDEs.