Abstract. We consider parameter estimation for a stochastic thermionic emission model generated by a diffusion process with multiplicative noise and temperature- and electric field-dependent drift. Since the transition density is not available in closed form, standard likelihood based inference is infeasible. We construct a quasi-likelihood using a Hermite type parametric expansion, yielding an explicit high order approximation without solving the Kolmogorov equation. This leads to a quasi-maximum likelihood estimator for discretely observed emission currents. Least squares, least-absolute-error, and one step estimators are included for comparison. Monte-Carlo experiments under physically structured PDE generated inputs show that fully iterated estimators, especially Hermite based QML and selected second order contrast schemes, provide more stable inference than one step corrections, which are sensitive to the input complexity.

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