1,721,187 research outputs found
Bounds for mixing times for finite semi-Markov processes with heavy-tail jump distribution
Full text linkConsider a Markov chain with finite state space and suppose you wish to change time replacing the integer step index with a random counting process . What happens to the mixing time of the Markov chain? We present a partial reply in a particular case of interest in which is a counting renewal process with power-law distributed inter-arrival times of index . We then focus on , leading to infinite expectation for inter-arrival times and further study the situation in which inter-arrival times follow the Mittag-Leffler distribution of order
Limit theorems for prices of options written on semi-Markov processes
Full text linkWe consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator’s Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes
A note on intraday option pricing
No full textCompound renewal processes can be used as an approximate
phenomenological model of tick-by-tick price fluctuations. An exact and
explicit general formula is derived for the martingale price of a European call
option written on a compound renewal process. The option price is obtained
using the direct method of indicator functions. The applicability of this result is
discussed
Random exchange models and the distribution of wealth
No full textI am presenting my personal point of view on what is interesting in Econophysics. In particular, I focus on random exchange models for the distribution of wealth in order to illustrate the concept of statistical equilibrium in Economics
Continuous-time statistics and generalized relaxation equations
Full text linkUsing 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
A functional limit theorem for stochastic integrals driven by a time-changed symmetric sigma-stable Levy process
No full textUnder proper scaling and distributional assumptions, we prove the convergence in the Skorokhod space endowed with the -topology of a sequence of stochastic integrals of a deterministic function driven by a time-changed symmetric Œ±-stable Lévy process. The time change is given by the inverse Œ≤-stable subordinator
A fractional generalization of the Dirichlet distribution and related distributions
Full text linkThis paper is devoted to a fractional generalization of the Dirichlet distribution. The form of the multivariate distribution is derived assuming that the n partitions of the interval [0, Wn] are independent and identically distributed random variables following the generalized Mittag-Leffler distribution. The expected value and variance of the one-dimensional marginal are derived as well as the form of its probability density function. A related generalized Dirichlet distribution is studied that provides a reasonable approximation for some values of the parameters. The relation between this distribution and other generalizations of the Dirichlet distribution is discussed. Monte Carlo simulations of the one-dimensional marginals for both distributions are presented
Solvable non-Markovian dynamic network
Full text linkNon-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
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