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On the existence of recurrent extensions of selfsimilar markov processes
 Electronic Communications in Probability, 11:230–241
, 2006
"... Abstract. Let X = (Xt)t≥0 be a selfsimilar Markov process with values in [0,∞[, such that the state 0 is a trap. We present a necessary and sufficient condition for the existence of a selfsimilar recurrent extension of X that leaves 0 continuously. This condition is expressed in terms of the Lévy ..."
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Abstract. Let X = (Xt)t≥0 be a selfsimilar Markov process with values in [0,∞[, such that the state 0 is a trap. We present a necessary and sufficient condition for the existence of a selfsimilar recurrent extension of X that leaves 0 continuously. This condition is expressed in terms of the Lévy process associated with X by the Lamperti transformation. 1.
Model Checking for a Class of Performance Properties of Fluid Stochastic Models
"... Abstract. Recently, there is an explosive development of fluid approaches to computer and distributed systems. These approaches are inherently stochastic and generate continuous state space models. Usually, the performance measures for these systems are defined using probabilities of reaching certai ..."
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Abstract. Recently, there is an explosive development of fluid approaches to computer and distributed systems. These approaches are inherently stochastic and generate continuous state space models. Usually, the performance measures for these systems are defined using probabilities of reaching certain sets of the state space. These measures are well understood in the discrete context and many efficient model checking procedures have been developed for specifications involving them. The continuous case is far more complicated and new methods are necessary. In this paper we propose a general model checking strategy founded on advanced concepts and results of stochastic analysis. Due to the problem complexity, in this paper, we achieve the first necessary step of characterizing mathematically the problem. We construct upper bounds for the performance measures using Martin capacities. We introduce a concept of bisimulation that preserves the performance measures and a metric that characterizes the bisimulation.
ELECTRONIC COMMUNICATIONS in PROBABILITY ON THE EXISTENCE OF RECURRENT EXTENSIONS OF SELFSIMILAR MARKOV PROCESSES
, 2006
"... condition, excursion Let X = (Xt)t≥0 be a selfsimilar Markov process with values in [0,∞[, such that the state 0 is a trap. We present a necessary and sufficient condition for the existence of a selfsimilar recurrent extension of X that leaves 0 continuously. This condition is expressed in terms o ..."
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condition, excursion Let X = (Xt)t≥0 be a selfsimilar Markov process with values in [0,∞[, such that the state 0 is a trap. We present a necessary and sufficient condition for the existence of a selfsimilar recurrent extension of X that leaves 0 continuously. This condition is expressed in terms of the Lévy process associated with X by the Lamperti transformation. 1