Moment Structure and Growth Rates of a Square-Root Diffusion with Gamma-Distributed Multiplicative Jumps

This paper studies a nonnegative square-root diffusion subject to finite-activity multiplicative jumps whose multipliers follow a Gamma distribution. The analysis develops the mathematical foundation that is needed before moment calculations can be used: existence and pathwise uniqueness of a nonnegative càdlàg solution, absorption at zero, and finiteness of every positive integer moment on finite time intervals. A localization argument then justifies Dynkin’s formula for the unbounded monomials. The resulting moment system is lower triangular. We give both an integral recursion and a matrix-exponential representation, derive stable closed forms for the first three moments using continuous divided differences, and cover all coincident-growth-rate cases without singular expressions. A long-run theorem identifies the dominant exponential rate and the polynomial correction caused by repeated dominant rates. Numerical integration of the moment ordinary differential equations independently confirms the analytic formulas, with maximum relative discrepancies below 10−8 under the coarsest reported tolerance. A multi-parameter analysis further quantifies variance, skewness, and excess kurtosis. The contribution is not the individual use of square-root diffusion, Gamma multipliers, or generator methods; it is a rigorous and computationally stable characterization of their specific multiplicative-jump combination.