By Hanchao Liu
In this paper, we provide a comprehensive study of how the classical Black–Scholes-type market framework behaves in an infinite-dimensional setting. The asset prices are modeled by stochastic evolution equations in a separable Hilbert space. The evolution of the risk-free asset and the corresponding discounting operators are governed by a strongly continuous group of bounded linear operators. Accordingly, we consider contingent claims with Hilbert-space-valued payoffs, whose values are discounted through the corresponding operators. Within this framework, we investigate the pricing and hedging of contingent claims, deriving the corresponding infinite-dimensional Black–Scholes equations and establishing exact replication results for strong solutions and asymptotic replication results for mild solutions through Yosida approximations. We also study the special case where the infinitesimal generator is given by A=rI, allowing for real-valued contingent claims. We further investigate no-arbitrage and market completeness, introducing a Hilbert lattice structure to formulate arbitrage and characterizing attainability through operator equations associated with martingale representations. Finite-dimensional noise truncations and Yosida approximations are employed to establish approximate and asymptotic approximate completeness results. We introduce geometric Brownian motions in Hilbert spaces and derive sufficient conditions for the existence and uniqueness of an equivalent risk-neutral measure, together with explicit pricing formulas for some tractable examples. These results extend key aspects of the classical Black–Scholes framework to markets with infinite-dimensional asset-price dynamics. As a concrete application, the model can be interpreted as a segmented market consisting of a sequence of one-dimensional geometric Brownian motions.
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