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Subordinator
About: Subordinator is a research topic. Over the lifetime, 771 publications have been published within this topic receiving 15383 citations.
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TL;DR: In this paper, the problem of conditioning a subordinator to stay in a strip is considered, and it is shown that under mild conditions, conditioning a sub-subordinate to remain in the strip is equivalent to first choosing a point in the potential measure uniformly according to the normalised potential measure, and then conditioning the submissive to approach that point continuously until it is sent to a cemetery state.
Abstract: In this paper, we consider the problem of conditioning a subordinator to stay in a strip. We show that this notion is a straightforward extension of the classical work of Chaumont (1996) and Chaumont and Doney (2005), who developed the theory of L\'evy processes conditioned to stay positive. Moreover, we show that, under mild conditions, conditioning a subordinator to stay in a strip, say $[0,a]$, is equivalent to first choosing a point in $[0,a]$, uniformly according to the normalised potential measure, and then conditioning the subordinator to approach that point continuously, after which it is sent to a cemetery state. Having developed the necessary theory, we describe two applications in the setting of self-similar Markov processes and last passage times of Markov processes.
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TL;DR: In this paper, the generalized fractional Skellam process (GFSP) is considered by time-changing it with an independent inverse stable subordinator, and its distributional properties such as the probability mass function, probability generating function, mean, variance and covariance are obtained.
Abstract: In this paper, we study a Skellam type variant of the generalized counting process (GCP), namely, the generalized Skellam process. Some of its distributional properties such as the probability mass function, probability generating function, mean, variance and covariance are obtained. Its fractional version, namely, the generalized fractional Skellam process (GFSP) is considered by time-changing it with an independent inverse stable subordinator. It is observed that the GFSP is a Skellam type version of the generalized fractional counting process (GFCP) which is a fractional variant of the GCP. It is shown that the one-dimensional distributions of the GFSP are not infinitely divisible. An integral representation for its state probabilities is obtained. We establish its long-range dependence property by using its variance and covariance structure. Also, we consider two time-changed versions of the GFCP. These are obtained by time-changing the GFCP by an independent Levy subordinator and its inverse. Some particular cases of these time-changed processes are discussed by considering specific Levy subordinators.
TL;DR: In this paper , the authors assume the existence of some anticipative information in a market whose risky asset dynamics evolve according to a Brownian motion and a Poisson process, and derive the information drift of the mentioned processes and, both in the pure jump case and in the mixed one, compute the additional expected logarithmic utility.
Abstract: The anticipative information refers to some information about future events that may be disclosed in advance. This information may regard, for example, financial assets and their future trends. In our paper, we assume the existence of some anticipative information in a market whose risky asset dynamics evolve according to a Brownian motion and a Poisson process. Using Malliavin calculus and filtration enlargement techniques, we derive the information drift of the mentioned processes and, both in the pure jump case and in the mixed one, we compute the additional expected logarithmic utility. Many examples are shown, where the anticipative information is related to some conditions that the constituent processes or their running maximum may verify, in particular, we show new examples considering Bernoulli random variables.
01 Jan 2022
TL;DR: In this paper , it was shown that the mean return time in a renewal process with interarrival times is bounded by a factor of 2 when the Lévy exponent of the associated subordinator is a special Bernstein function.
Abstract: Lorden’s inequality asserts that the mean return time in a renewal process with (iid) interarrival times $$Y_1, Y_2,\ldots $$ , is bounded above by $$2\textbf{E}[Y_1]/\textbf{E}[Y_1^2]$$ . We establish this result in the context of regenerative sets, and remove the factor of 2 when the regenerative set enjoys a certain monotonicity property. This property occurs precisely when the Lévy exponent of the associated subordinator is a special Bernstein function. Several equivalent stochastic monotonicity properties of such a regenerative set are demonstrated.
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TL;DR: In this paper, a multivariate rapidly decreasing Levy process is proposed for quanto option pricing, which captures three characteristics observed in real-world market for stock prices and currencies: jumps, heavy tails and skewness.
Abstract: We develop a multivariate Levy model for the pricing of quanto options that captures three characteristics observed in real-world market for stock prices and currencies: jumps, heavy tails and skewness. The model is developed by using a bottom-up approach from a subordinator. We do so by replacing the time of a Brownian motion with a non-decreasing Levy process, rapidly decreasing subordinator. We refer to this model as a multivariate rapidly decreasing Levy process. We consider two benchmarks: Black-Scholes and normal tempered stable process, the later constructed using a classical tempered stable subordinator. We then compare using a time series of daily log-returns and market prices of European-style quanto options the relative performance of the rapidly decreasing Levy process to that of Black-Scholes and the normal tempered stable process. We find that the proposed modeling process is superior to the other two processes for pricing quanto options.