Scaling limits of coupled continuous time random walks and residual order statistics through marked point processes

A continuous time random walk (CTRW) is a random walk in which both spatial changes represented by jumps and waiting times between the jumps are random. The CTRW is coupled if a jump and its preceding or following waiting time are dependent random variables, respectively. The aim of this paper is to explain the occurrence of different limit processes for CTRWs with forward- or backward-coupling in Straka and Henry (2011) using marked point processes. We also establish a series representation for the different limits. The methods used also allow us to solve an open problem concerning residual order statistics by LePage (1981).Comment: revised version, to appear in: Stoch. Process. App

Solvable non-Markovian dynamic network

Non-Markovian processes are widespread in natural and human-made systems, yet explicit modeling and analysis of such systems is underdeveloped. We consider a non-Markovian dynamic network with random link activation and deletion (RLAD) and heavy-tailed Mittag-Leffler distribution for the interevent times. We derive an analytically and computationally tractable system of Kolmogorov-like forward equations utilizing the Caputo derivative for the probability of having a given number of active links in the network and solve them. Simulations for the RLAD are also studied for power-law interevent times and we show excellent agreement with the Mittag-Leffler model. This agreement holds even when the RLAD network dynamics is coupled with the susceptible-infected-susceptible spreading dynamics. Thus, the analytically solvable Mittag-Leffler model provides an excellent approximation to the case when the network dynamics is characterized by power-law-distributed interevent times. We further discuss possible generalizations of our result

Continuous-time statistics and generalized relaxation equations

Using two simple examples, the continuous-time random walk as well as a two state Markov chain, the relation between generalized anomalous relaxation equations and semi-Markov processes is illustrated. This relation is then used to discuss continuous-time random statistics in a general setting, for statistics of convolution-type. Two examples are presented in some detail: the sum statistic and the maximum statistic

Financial Data Analysis by means of Coupled Continuous-Time Random Walk in Rachev-Rűschendorf Model

We adapt the continuous-time random walk formalism to describe asset price evolution. We expand the idea proposed by Rachev and Rűschendorf who analyzed the binomial pricing model in the discrete time with randomization of the number of price changes. As a result, in the framework of the proposed model we obtain a mixture of the Gaussian and a generalized arcsine laws as the limiting distribution of log-returns. Moreover, we derive an European-call-option price that is an extension of the Black-Scholes formula. We apply the obtained theoretical results to model actual financial data and try to show that the continuous-time random walk offers alternative tools to deal with several complex issues of financial markets

Financial Data Analysis by means of Coupled Continuous-Time Random Walk in Rachev-Rűschendorf Model

We adapt the continuous-time random walk formalism to describe asset price evolution. We expand the idea proposed by Rachev and Rűschendorf who analyzed the binomial pricing model in the discrete time with randomization of the number of price changes. As a result, in the framework of the proposed model we obtain a mixture of the Gaussian and a generalized arcsine laws as the limiting distribution of log-returns. Moreover, we derive an European-call-option price that is an extension of the Black-Scholes formula. We apply the obtained theoretical results to model actual financial data and try to show that the continuous-time random walk offers alternative tools to deal with several complex issues of financial markets