Subordinator

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

On the Boundary Theory of Subordinate Killed Lévy Processes

Abstract: Let Z be a subordinate Brownian motion in $\mathbb {R}^{d}$ , d ≥ 2, via a subordinator with Laplace exponent ϕ. We kill the process Z upon exiting a bounded open set $D\subset \mathbb {R}^{d}$ to obtain the killed process ZD, and then we subordinate the process ZD by a subordinator with Laplace exponent ψ. The resulting process is denoted by YD. Both ϕ and ψ are assumed to satisfy certain weak scaling conditions at infinity. We study the potential theory of YD, in particular the boundary theory. First, in case that D is a κ-fat bounded open set, we show that the Harnack inequality holds. If, in addition, D satisfies the local exterior volume condition, then we prove the Carleson estimate. In case D is a smooth open set and the lower weak scaling index of ψ is strictly larger than 1/2, we establish the boundary Harnack principle with explicit decay rate near the boundary of D. On the other hand, when ψ(λ) = λγ with γ ∈ (0, 1/2], we show that the boundary Harnack principle near the boundary of D fails for any bounded C1,1 open set D. Our results give the first example where the Carleson estimate holds true, but the boundary Harnack principle does not. One of the main ingredients in the proofs is the sharp two-sided estimates of the Green function of YD. Under an additional condition on ψ, we establish sharp two-sided estimates of the jumping kernel of YD which exhibit some unexpected boundary behavior. We also prove a boundary Harnack principle for non-negative functions harmonic in a smooth open set E strictly contained in D, showing that the behavior of YD in the interior of D is determined by the composition ψ ∘ ϕ.

Majorization, 4G Theorem and Schrödinger perturbations

Abstract: Schrodinger perturbations of transition densities by singular potentials may fail to be comparable with the original transition density. For instance, this is so for the transition density of a subordinator perturbed by any time-independent unbounded potential. In order to estimate such perturbations, it is convenient to use an auxiliary transition density as a majorant and the 4G inequality for the original transition density and the majorant. We prove the 4G inequality for the 1/2-stable and inverse Gaussian subordinators, discuss the corresponding class of admissible potentials and indicate estimates for the resulting transition densities of Schrodinger operators. The connection of the transition densities to their generators is made via the weak-type notion of fundamental solution.

Information-based models for finance and insurance

Abstract: In financial markets, the information that traders have about an asset is reflected in its price. The arrival of new information then leads to price changes. The `information-based framework' of Brody, Hughston and Macrina (BHM) isolates the emergence of information, and examines its role as a driver of price dynamics. This approach has led to the development of new models that capture a broad range of price behaviour. This thesis extends the work of BHM by introducing a wider class of processes for the generation of the market filtration. In the BHM framework, each asset is associated with a collection of random cash flows. The asset price is the sum of the discounted expectations of the cash flows. Expectations are taken with respect (i) an appropriate measure, and (ii) the filtration generated by a set of so-called information processes that carry noisy or imperfect market information about the cash flows. To model the flow of information, we introduce a class of processes termed L\'evy random bridges (LRBs), generalising the Brownian and gamma information processes of BHM. Conditioned on its terminal value, an LRB is identical in law to a L\'evy bridge. We consider in detail the case where the asset generates a single cash flow $X_T$ at a fixed date $T$. The flow of information about $X_T$ is modelled by an LRB with random terminal value $X_T$. An explicit expression for the price process is found by working out the discounted conditional expectation of $X_T$ with respect to the natural filtration of the LRB. New models are constructed using information processes related to the Poisson process, the Cauchy process, the stable-1/2 subordinator, the variance-gamma process, and the normal inverse-Gaussian process. These are applied to the valuation of credit-risky bonds, vanilla and exotic options, and non-life insurance liabilities.

Some Explicit Krein Representations of Certain Subordinators, Including the Gamma Process

Abstract: 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 [23].