Risk-neutral dynamics for the ARIMA-GARCH random process with errors distributed according to the Johnson's $S_U$ law

Risk-neutral world is one of the fundamental principles of financial mathematics, for definition of a fair value of derivative financial instruments. The article deals with the construction of risk-neutral dynamics for the ARIMA-GARCH (Autoregressive Integrated Moving Average, Generalized AutoRegressive Conditional Heteroskedasticity) random process with errors distributed according to the Johnson's $S_U$ law. Methods for finding risk-neutral coefficients require the existence of a generating function of moments (examples of such transformations are the Escher transformation, the extended Girsanov principle). A generating function of moments is not known for Student and Johnson's $S_U$ distributions. The authors form a generating function of moments for the Johnson's $S_U$ distribution and prove that a modification of the extended Girsanov principle may obtain a risk-neutral measure with respect to the chosen distribution.