Regime-Switching Pearson Diffusion Processes: Exact Conditional-Moment Representations and Applications to Stochastic Systems with Changing Environments

We develop a conditional-moment framework for time-dependent Pearson diffusions modulated by a finite-state continuous-time Markov chain. Using established polynomial preservation, we derive an explicit regime-indexed coefficient hierarchy for the full quadratic Pearson diffusion coefficient. The resulting finite-dimensional representation is exact, although its time-dependent coefficients may require numerical integration. Stochastic verification identifies the polynomial candidate with the conditional moment, including admissible boundary initial states. Time-homogeneous coefficients yield a block matrix exponential, and two-regime spectral formulas for the first two moments accommodate eigenvalues shared by different coefficient matrices. Specializations cover Ornstein–Uhlenbeck, Cox–Ingersoll–Ross, Jacobi, Fisher–Snedecor, reciprocal-gamma, and Student diffusions. These provide illustrative mathematical models rather than empirical calibrations. Numerical experiments examine calendar-time effects, regime switching, and bounded-state tail sensitivity, with sampling uncertainty and discretization effects assessed separately. A deterministic stability test demonstrates the importance of evaluating divided differences without subtractive cancellation. Higher moments supply shape measures and conservative terminal-probability bounds; the framework does not provide a general transition- density formula.