Pricing and Hedging Framework for Infinite-Dimensional Financial Markets

This study extends the classical Black-Scholes market framework to infinite-dimensional scenarios, modeling asset prices with stochastic evolution equations in separable Hilbert spaces, deriving an infinite-dimensional Black-Scholes equation, and establishing exact replication results for strong solutions and asymptotic replication results for weak solutions. The study also introduces Hilbert lattice structures to characterize no-arbitrage conditions, provides sufficient conditions for the existence and uniqueness of equivalent risk-neutral measures, and explicit pricing formulas for some tractable cases; the model can be interpreted as a segmented market composed of a series of one-dimensional geometric Brownian motions.