Subordinator

About: Subordinator is a research topic. Over the lifetime, 771 publications have been published within this topic receiving 15383 citations.

A fractional counting process and its connection with the Poisson process

Abstract: We consider a fractional counting process with jumps of amplitude $1,2,\ldots,k$, with $k\in \mathbb{N}$, whose probabilities satisfy a suitable system of fractional difference-differential equations. We obtain the moment generating function and the probability law of the resulting process in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When $k=2$ we also express the distribution of the first passage time of the fractional counting process in an integral form. Finally, we show that the ratios given by the powers of the fractional Poisson process and of the counting process over their means tend to 1 in probability.

Power-type derivatives for rough volatility with jumps

Abstract: In this paper we propose an efficient pricing-hedging framework for volatility derivatives which simultaneously takes into account path roughness and jumps. Instead of dealing with log-volatility, we directly model the instantaneous variance of a risky asset in terms of a fractional Ornstein-Uhlenbeck process driven by an infinite-activity Levy subordinator, which is shown to exhibit roughness under suitable conditions and also eludes the need for an independent Brownian component. This structure renders the characteristic function of forward variance obtainable at least in semi-closed form, subject to a generic integrable kernel. To analyze financial derivatives, primarily swaps and European-style options, on average forward volatility, we introduce a general class of power-type derivatives on the average forward variance, which also provide a way of adjusting the option investor's risk exposure. Pricing formulae are based on numerical inverse Fourier transform and, as illustrated by an empirical study on VIX options, permit stable and efficient model calibration once specified.

Asymptotic behaviour of first passage time distributions for subordinators

Abstract: In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt in 1987. In the particular case of a subordinator in the domain of attraction of a stable distribution the results are the analogue of the results obtained by the authors for non-monotone Levy processes. For subordinators an approach different to that used for non-monotone Levy processes is necessary because the excursion techniques are not available and also because typically in the non-monotone case the tail distribution of the first passage time has polynomial decrease, while in the subordinator case it is exponential.

Non-markovian limits of additive functionals of Markov processes

Abstract: In this paper we consider an additive functional of an observable $V(x)$ of a Markov jump process We assume that the law of the expected jump time $t(x)$ under the invariant probability measure $\pi$ of the skeleton chain belongs to the domain of attraction of a subordinator Then, the scaled limit of the functional is a Mittag-Leffler proces, provided that $\Psi(x):=V(x)t(x)$ is square integrable wrt $\pi$ When the law of $\Psi(x)$ belongs to a domain of attraction of a stable law the resulting process can be described by a composition of a stable process and the inverse of a subordinator and these processes are not necessarily independent On the other hand when the singularities of $\Psi(x)$ and $t(x)$ do not overlap with large probability the law of the resulting process has some scaling invariance property We provide an application of the results to a process that arises in quantum transport theory