We incorporate randomness into deterministic theories and compare analytically and numerically some well-known stochastic theories: the Liouville process, the Ornstein-Uhlenbeck process, and a process that is Gaussian and exponentially time correlated Ornstein-Uhlenbeck noise . Different methods of achieving the marginal densities for correlated and uncorrelated noise are discussed. Analytical ...

An α-Wiener bridge is a one-parameter generalization of the usual Wiener bridge, where the parameter α > 0 represents a mean reversion force to zero. We generalize the notion of α-Wiener bridges to continuous functions α : [0, T ) → R. We show that if the limit limt↑T α(t) exists and is positive, then a general α-Wiener bridge is in fact a bridge in the sense that it converges to 0 at time T wi...

We introduce the elliptical Ornstein-Uhlenbeck (OU) process, which is a generalisation of well-known univariate OU process to bivariate time series. This maps out stochastic oscillations over in complex plane, are observed many applications coupled The appeal model that generated using one simple first order SDE, whereas alternative models require more complicated vectorised or higher SDE repre...