This study investigates the effects of varying sampling intervals on the long memory characteristics of certain stochastic processes. We find that although different sampling intervals do not affect the decay rate of discrete time long memory autocorrelation functions in large lags, the autocorrelation functions in short lags are affected significantly. The level of the autocorrelation function...

The initial time-dependence of a state in circumstances where it makes transitions to, or decay second has been investigated. In classical stochastic processes, the observed time dependence transition proportional to $t^2$ is attributed noise with memory. contrast quantum mechanics, quadratic form unable decelerate evolution system.

We introduce an extended version of the fractional Ornstein-Uhlenbeck (FOU) process where the integrand is replaced by the exponential of an independent Lévy process. We call the process the generalized fractional Ornstein-Uhlenbeck (GFOU) process. Alternatively, the process can be constructed from a generalized Ornstein-Uhlenbeck (GOU) process using an independent fractional Brownian motion (F...

It is known that a stochastic differential equation (SDE) induces two probabilistic objects, namely a difusion process and a stochastic flow. While the diffusion process is determined by the innitesimal mean and variance given by the coefficients of the SDE, this is not the case for the stochastic flow induced by the SDE. In order to characterize the stochastic flow uniquely the innitesimal cov...

The article proves the separation theorem for optimal control of stochastic systems in the case when an observed continuous-discrete-time process possess memory of arbitrary ration relating to a state vector.

this paper studies a maintenance policy for a system composed of two components, which are subject to continuous deterioration and consequently stochastic failure. the failure of each component results in the failure of the system. the components are inspected periodically and their deterioration degrees are monitored. the components can be maintained using different maintenance actions (repair...

Initially, process mining focused on discovering models from event data, but in recent years the use and importance of conformance checking has increased. Conformance aims to uncover differences between a model an log. Many techniques measures have been proposed. Typically, these take into account frequencies traces log, do not consider probabilities model. This asymmetry leads various complica...

The origin of the long-range memory in non-equilibrium systems is still an open problem as the phenomenon can be reproduced using models based on Markov processes. In these cases, the notion of spurious memory is introduced. A good example of Markov processes with spurious memory is a stochastic process driven by a non-linear stochastic differential equation (SDE). This example is at odds with ...

Many of the problems considered in optimization and learning assume that solutions exist in a dynamic. Hence, algorithms are required that dynamically adapt with the problem’s conditions and search new conditions. Mostly, utilization of information from the past allows to quickly adapting changes after. This is the idea underlining the use of memory in this field, what involves key design issue...