Axiomatic Emergence O; The Physical Origin of Stochastic Analysis Structures

This paper systematically deduces the core mathematical structures of stochas
tic analysis starting from two fundamental principles: the Information Conservation
axiom (A1) and the Finite-Step Computability axiom (A2). Research shows that
continuous-time stochastic processes exemplified by Brownian motion, Itô stochas
tic integrals, stochastic differential equations, and path integrals (Feynman inte
grals) are not ad hoc mathematical tools for describing random phenomena in the
physical world, but rather inevitable mathematical emergences of underlying dis
crete, probabilistic information processes under macroscopic continuous limits. We
prove that the intrinsically generated symmetric Bernoulli branching process within
the A1/A2 framework, via scaling limits (Donsker’s invariance principle), uniquely
converges to standard Brownian motion; in order to define dynamics on this pro
cess, the construction of the Itô integral becomes a necessary choice under the dual
constraints of causality and the continuity of the martingale structure; while the
operation of summing over all possible historical paths according to their infor
mation weights naturally leads to the path integral formulation in the continuous
limit. This paper further proposes quantitative experimental test schemes based
on the ”volatility smile” in financial options markets and the ”sign problem” in
quantum Monte Carlo simulations, thereby establishing a falsifiable bridge between
stochastic mathematical structures and the physical world.