Showing papers on "Subordinator published in 2002"

Stochastic solution of space-time fractional diffusion equations.

Abstract: Classical and anomalous diffusion equations employ integer derivatives, fractional derivatives, and other pseudodifferential operators in space. In this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are Mittag-Leffler type. This leads to explicit solutions for space-time fractional diffusion equations with multiscaling space-fractional derivatives, and additional insight into the meaning of these equations.

Stochastic volatility, jumps and hidden time changes

Abstract: Stochastic volatility and jumps are viewed as arising from Brownian subordination given here by an independent purely discontinuous process and we inquire into the relation between the realized variance or quadratic variation of the process and the time change. The class of models considered encompasses a wide range of models employed in practical financial modeling. It is shown that in general the time change cannot be recovered from the composite process and we obtain its conditional distribution in a variety of cases. The implications of our results for working with stochastic volatility models in general is also described. We solve the recovery problem, i.e. the identification the conditional law for a variety of cases, the simplest solution being for the gamma time change when this conditional law is that of the first hitting time process of Brownian motion with drift attaining the level of the variation of the time changed process. We also introduce and solve in certain cases the problem of stochastic scaling. A stochastic scalar is a subordinator that recovers the law of a given subordinator when evaluated at an independent and time scaled copy of the given subordinator. These results are of importance in comparing price quality delivered by alternate exchanges.

Right Inverses of Nonsymmetric Lévy Processes

Abstract: We analyze the existence and properties of right inverses K for nonsymmetric Levy processes X, extending recent work of Evans [7] in the symmetric setting. First, both X and −X have right inverses if and only if X is recurrent and has a nontrivial Gaussian component. Our main result is then a description of the excursion measure n Z of the strong Markov process Z = X − L (reflected process) where Lt = inf{ x> 0 : Kx >t }. Specifically, n Z is essentially the restriction of n X to the “excursions starting negative.” Second, when only asking for right inverses of X, a certain “strength of asymmetry” is needed. Millar’s [9] notion of creeping turns out necessary but not sufficient for the existence of right inverses. We analyze this both in the bounded and unbounded variation case with a particular emphasis on results in terms of the Levy–Khintchine characteristics. 1. Introduction. Consider a real-valued Levy process, that is, a continuous time process with stationary independent increments and cadlag paths. Right inverses of Levy processes have first been studied in the symmetric setting by Evans [7]. He defines an increasing process K to be a right inverse of a Levy process X if it satisfies X(Kx ) = x for all x ≥ 0. The minimal such K turns out to be a subordinator (i.e., an increasing Levy process). Specifically, Evans characterized the symmetric Levy processes that have right inverses. He introduced Lt = inf{x ≥ 0 : Kx >t } and the reflected process Z = X − L and showed that Z is strong Markov with local time L at zero, he gave some fluctuation type identities for Z and L and proved formulae for the entrance laws to the excursion measure n Z of Z. Evans also showed that K is distributed like a linear time change of the inverse local time of X at zero. In this work we basically answer three questions left open by Evans. First, what happens when symmetry fails? The only result of Evans that is clearly based on symmetry is the coincidence of the laws of K and the inverse local time of X in zero. Also, the class of processes possessing inverses is not restricted to those having a positive Gaussian component (in the symmetric setting, this is Evans’s characterization result), but those form again an important class. We provide a further characterization in the bounded variation case and we also point

Advection and dispersion in time and space

Abstract: Previous work showed how moving particles that rest along their trajectory lead to time-nonlocal advection–dispersion equations. If the waiting times have infinite mean, the model equation contains a fractional time derivative of order between 0 and 1. In this article, we develop a new advection–dispersion equation with an additional fractional time derivative of order between 1 and 2. Solutions to the equation are obtained by subordination. The form of the time derivative is related to the probability distribution of particle waiting times and the subordinator is given as the first passage time density of the waiting time process which is computed explicitly.