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UID/MAT/00297/2020 Regime switching diffusion processes with one or two thresholds and regime switching occurring by a change in the diffusion drift and/or volatility functions parameters of a stochastic differential equation, whose solution defines a continuous time diffusion process, were defined in previous works; the change in regime occurring whenever the trajectory of the process crosses a threshold, possibly with some delay. In this paper we generalise the previous results by allowing the underlying diffusion process to change from one family of diffusions in one regime to an entirely different one in the other regime; these families of diffusions are characterised by specific functional forms for drift and volatility coefficients depending on parameters. We propose an estimation procedure for all the parameters, namely the thresholds, the delay and, for both regimes, diffusion’s parameters and we apply the introduced estimation procedure to both simulated and real data. publishersversion published

This work was done under partial financial support of RFBR (Grant n. 19-01-00451). We present a methodology to study discrete time financial models with one risky asset and a risk free asset that may thought to result as a discretization of a suitable continuous time model. In a numerical example we compare the pricing results, obtained with these models, with results obtained from the related continuous time models. Our approach relies on some known important results describing a particular class of discrete time models – the conditionally Gaussian models – a class that, regardless of its particular definition, contains many interesting instances. We aim at a better understanding of the implications of the discretization procedures which are inevitable, both at the parameter estimation and derivative price computation moments, by reason of the observational and computational limitations. We also present a preliminary study of a a model of stochastic differential equations for commodity spot and futures prices that may be studied with the proposed methodology. For that purpose we summarize a naive theory of Ito integration in Hilbert space. publishersversion published