The discontinuous moulting process in crustaceans poses fundamental challenges for growth modelling and can yield biologically implausible estimates of asymptotic size under traditional continuous-growth frameworks such as the von Bertalanffy curve. We develop a stochastic growth model that jointly characterises the moult increment (MI) and intermoult period (IP) through a unified convolution-based likelihood. Individual growth is represented by a constrained L\'evy subordinator, which enforces monotone but discontinuous size trajectories by modelling MI as a beta-type jump process with biologically realistic support, while IP is described by a gamma generalised linear model. This construction delivers a joint likelihood for MI and IP, permitting maximum likelihood estimation and profile-likelihood inference for parameters governing both the size and timing of moults.

In an application to tank data on \emph{Panulirus ornatus}, we compare several plausible jump and waiting-time distributions and find that the beta-based L\'evy model provides a substantially better fit than gamma or inverse Gaussian alternatives, while yielding von Bertalanffy-type population summaries compatible with current stock-assessment practice. A simulation study confirms that the proposed estimators recover key growth characteristics and reproduce the observed stepwise trajectories under realistic sample sizes. By embedding biologically constrained increments and intermoult timing within a L\'evy-process framework, the model offers a flexible tool for analyzing discontinuous growth in crustaceans and related biological systems, and illustrates how modern jump processes can be adapted to inform fisheries management.