Inference for Variance-Gamma Driven Stochastic Systems

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In this work we present the variance-gamma driven state-space model (VGSSM) - a linear vector stochastic differential equation driven by the variance-gamma (VG) Lévy process, and propose a novel inference framework in such systems. There are closed form expressions for the first four moments of the marginals of the VG process, allowing for more flexible modelling than Brownian motion (BM), retaining BM as a limiting case. The conditionally Gaussian formulation of the variance-gamma process lends itself well to the use of a marginalised particle filter (MPF) which can include the estimation of model parameters as part of the sampling framework. As an example we present a state-space formulation of Langevin dynamics in the VGSSM for estimation of both the observed and the latent first-order dynamics of a system. We apply this specific Langevin formulation to synthetically generated data to validate the results of the MPF, followed by an application to foreign-exchange tick data to demonstrate the method for trend tracking in data sets that are irregularly sampled in time.