Spectrum analysis on linear parabolic partial differential equations driven by a multidimensional bounded stochastic process

This paper investigates a general linear parabolic partial differential equation driven by a multidimensional bounded stochastic process. Employing spectral analysis of the associated uniformly elliptic operator, specifically extensions of the Krein–Rutman theorem and maximum principles, we establish the existence and uniqueness (up to scaling) of a strictly positive eigenfunction. We prove that the corresponding simple positive eigenvalue strictly dominates the rest of the spectrum, implying that every strictly positive solution to the parabolic equation is uniquely of separated form. This result generalizes prior univariate Sturm–Liouville approaches to the correlated multivariate setting. On the applications side, we show that this uniqueness theorem yields a rigorous multi-factor version of the Ross recovery theory, uniquely identifying the market price of risk and the Girsanov kernel in bounded diffusion models. Furthermore, we propose a framework for valuing European call options on private equity where the volatility vector is uniquely determined. Finally, our analysis connects to large deviation principles for stochastic PDEs developed by Dupuis et al., offering a robustness perspective on the spectral characterization of long-term risk prices in bounded domains.