The L₂ – Structure of Subordinated Solution of Continuous-Time Bilinear Time Series

Abstract

The models of stochastic subordination, or random time indexing, has been recently applied to model financial returns Xtt≥0 exhibiting some characteristic periods of constant values for instance exchange rate. In reality, sharp and large variations for X(t) do occur. These sharp and large variations are linked to information arrivals and/or represent sudden events and hence we have a model with jumps. For this purpose, by substituting the usual deterministic time t as a subordinator Ttt≥0 in a stochastic process Xtt≥0 we obtain a new process XTtt≥0 whose stochastic time is dominated by the subordinator Ttt≥0. Therefore we propose in this paper an alternative approach based on a combination of the continuous-time bilinear (COBL) process subordinated by a Poisson process (that it is a Levy process) which permits us to introduce further randomness for the phenomena which exhibit either a speeded up or slowed down behavior. So, the main probabilistic properties of such models are studied and the explicit expression of the higher-order moments properties are given. Moreover, moments method (MM) is proposed as an estimation issue of the unknown parameters. Simulation studies confirm the theoretical findings and show that the MM method proposal can effectively reduce both the bias and the mean square error of parameter estimates.