Operator-Theoretic Probability Framework in Morrey Spaces With Applications to Option Price Dynamics

We develop a unified operator-theoretic and probabilistic framework for a class of fractional and jump-type evolution equations in the generalized Morrey spaces. The analysis is based on the semigroup theory and subordination principles which allow us for the treatment of nonlocal temporal dynamics and discontinuous effects. We establish existence, uniqueness, and stability of mild solutions, together with finding the sharp bounds for the associated solution operators. A probabilistic representation, via time-changed processes and jump structures, provides an explicit interpretation of the evolution operators in terms of transition mechanisms. Furthermore, a moment-generating function approach is introduced to characterize solution operators and to control their norms, revealing a structural link between analytic and stochastic properties. Applications in the context of the fractional Black–Scholes–type models with jumps demonstrate the effectiveness of the framework in capturing anomalous diffusion, heavy-tailed behavior, and volatility clustering.