Subordinator

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

Boundary Harnack principle for the absolute value of a one-dimensional subordinate Brownian motion killed at $0$

Abstract: We prove the Harnack inequality and boundary Harnack principle for the absolute value of a one-dimensional recurrent subordinate Brownian motion killed upon hitting $0$, when $0$ is regular for itself and the Laplace exponent of the subordinator satisfies certain global scaling conditions. Using the conditional gauge theorem for symmetric Hunt processes we prove that the Green function of this process killed outside of some interval $(a,b)$ is comparable to the Green function of the corresponding killed subordinate Brownian motion. We also consider several properties of the compensated resolvent kernel $h$, which is harmonic for our process on $(0,1)$.

Necessity of weak subordination for some strongly subordinated L\'evy processes

Abstract: Consider the strong subordination of a multivariate Levy process with a multivariate subordinator. If the subordinate is a stack of independent Levy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a Levy process, otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a Levy process even when strong subordination does not. Here, we prove that strong and weak subordination are equal in law under the aforementioned condition. In addition, we prove that if strong subordination is a Levy process, then it is necessarily equal in law to weak subordination in two cases: firstly, when the subordinator is deterministic and secondly, when it is pure-jump with finite activity.

Stochastic models with mixtures of tempered stable subordinators

Abstract: In this article, we introduce mixtures of tempered stable subordinators (TSS). These mixtures define a class of subordinators which generalize tempered stable subordinators. The main properties like probability density function (pdf), L´evy density, moments, governing Fokker-Planck-Kolmogorov (FPK) type equations, asymptotic form of potential density and asymptotic form of the renewal function for the corresponding inverse subordinator are discussed. We generalize these results to n-th order mixtures of TSS. The governing fractional difference and differential equations of time-changed Poisson process and Brownian motion are also discussed.

Functional limit theorems for random walks perturbed by positive alpha-stable jumps.

Abstract: Let $\xi_1$, $\xi_2,\ldots$ be i.i.d. random variables of zero mean and finite variance and $\eta_1$, $\eta_2,\ldots$ positive i.i.d. random variables whose distribution belongs to the domain of attraction of an $\alpha$-stable distribution, $\alpha\in (0,1)$. The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump $\xi_k$ occurs; if the present position of the Markov chain is nonpositive, then its next position is $\eta_j$. We prove a functional limit theorem for this Markov chain under Donsker's scaling. The weak limit is a nonnegative process $(X(t))_{t\geq 0}$ satisfying a stochastic equation ${\rm d}X(t)={\rm d}W(t)+ {\rm d}U_\alpha(L_X^{(0)}(t))$, where $W$ is a Brownian motion, $U_\alpha$ is an $\alpha$-stable subordinator which is independent of $W$, and $L_X^{(0)}$ is a local time of $X$ at $0$. Also, we explain that $X$ is a Feller Brownian motion with a `jump-type' exit from $0$.

Weak Subordination of Multivariate L\'evy Processes and Variance Generalised Gamma Convolutions

Abstract: Subordinating a multivariate Levy process, the subordinate, with a univariate subordinator gives rise to a pathwise construction of a new Levy process, provided the subordinator and the subordinate are independent processes. The variance-gamma model in finance was generated accordingly from a Brownian motion and a gamma process. Alternatively, multivariate subordination can be used to create Levy processes, but this requires the subordinate to have independent components. In this paper, we show that there exists another operation acting on pairs $(T,X)$ of Levy processes which creates a Levy process $X\odot T$. Here, $T$ is a subordinator, but $X$ is an arbitrary Levy process with possibly dependent components. We show that this method is an extension of both univariate and multivariate subordination and provide two applications. We illustrate our methods giving a weak formulation of the variance-$\alpha$-gamma process that exhibits a wider range of dependence than using traditional subordination. Also, the variance generalised gamma convolution class of Levy processes formed by subordinating Brownian motion with Thorin subordinators is further extended using weak subordination.