Further examples of explicit Krein representations of certain subordinators

Some explicit Krein representations of certain subordinators, including the Gamma process

We give a representation of the Gamma subordinator as a Krein functional of Brownian motion, using the known representations for stable subordinators and Esscher transforms. In particular, we have obtained Krein representations of the subordinators which govern the two parameter Poisson-Dirichlet family of distributions.

关于整数值次ordinator的某些积分泛函

On certain integral functionals of integer-valued subordinators

本文推导了任意整数值次ordinator指数泛函密度的显式表达式及纯跳跃次ordinator的极限表示，通过分段基函数处理带正漂移情况，推进Lévy泛函分布解析研究并给出逆幂泛函密度的极限级数表示，数值性能良好。

Weak Convergence of Subordinators to Extremal Processes

For certain subordinators $(X_t)_{t\ge 0}$ it is shown that the process $(-t\log X_{ts})_{s>0}$ tends to an extremal process $(\hatη_s)_{s>0}$ in the sense of convergence of the finite dimensional distributions. Additionally it is also shown that $(z\wedge(-t\log X_{ts}))_{s\ge 0}$ converges weakly to $(z\wedge\hatη_s)_{s\ge0}$ in $\mathcal{D}[0,\infty)$, the space of càdlàg functions equipped with Skorohod's $J_1$ metric.

New families of subordinators with explicit transition probability semigroup

There exist only a few known examples of subordinators for which the transition probability density can be computed explicitly along side an expression for its Lévy measure and Laplace exponent. Such examples are useful in several areas of applied probability, for example, they are used in mathematical finance for modeling stochastic time change, they appear in combinatorial probability to construct sampling formulae, which in turn is related to a variety of issues in the theory of coalescence models, moreover, they have also been extensively used in the potential analysis of subordinated Brow…

Occupation Times for Time-changed Processes with Applications to Parisian Options

Stochastic processes time-changed by an inverse subordinator have been suggested as a way to model the price of assets in illiquid markets, where the jumps of the subordinator correspond to periods of time where one is unable to sell an asset. We develop an excursion theory for time-changed reflected Brownian motion and use this to express the price of certain European options with Parisian barrier condition in terms of solutions of a time-fractional PDE. We provide a general description of the occupation measures of time-changed processes and use this to prove a Ray-Knight theorem for the occ…

Point process simulation of generalised inverse Gaussian processes and estimation of the Jaeger integral

In this paper novel simulation methods are provided for the generalised inverse Gaussian (GIG) Lévy process. Such processes are intractable for simulation except in certain special edge cases, since the Lévy density associated with the GIG process is expressed as an integral involving certain Bessel Functions, known as the Jaeger integral in diffusive transport applications. We here show for the first time how to solve the problem indirectly, using generalised shot-noise methods to simulate the underlying point processes and constructing an auxiliary variables approach that avoids any direct c…