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Xiaowen Zhou

Publications and source records attributed to Xiaowen Zhou.

At least 55 records · Page 3Linked to original sources
arXiv2020

Parisian excursion with capital injection for draw-down reflected Levy insurance risk process

This paper discusses Parisian ruin problem with capital injection for Levy insurance risk process. Capital injection takes place at the draw-down time of the surplus process when it drops below a pre-specified function of its last record maximum. The capital is continuously paid to keep the surplus above the draw-down level until either the surplus process goes above the record high or a Parisian type ruin occurs, which is announced at the first instance the surplus process has undergone an excursion below the record for an independent exponential period of time consecutively since the time the capital was first injected. Some distributional identities concerning the excursion are presented. Firstly, we give the Parisian ruin probability and the joint Laplace transform (possibly killed at the first passage time above a fixed level of the surplus process) of the ruin time, surplus position at ruin, and the total capital injection at ruin. Secondly, we obtain the $q$-potential measure of the surplus process killed at Parisian ruin. Finally, we give expected present value of the total discounted capital payments up to the Parisian ruin time. The results are derived using recent developments in fluctuation and excursion theory of spectrally negative Levy process and are presented semi explicitly in terms of the scale function of the Levy process. Some numerical examples are given to facilitate the analysis of the impact of initial surplus and frequency of observation period to the ruin probability and to the expected total capital injection.

q-fin.MF↗
arXiv2019

A draw-down reflected spectrally negative Lévy process

In this paper we study a spectrally negative Lévy process that is reflected at its draw-down level whenever a draw-down time from the running supremum arrives. Using an excursion-theoretical approach, for such a reflected process we find the Laplace transform of the upper exiting time and an expression of the associated potential measure. When the reflected process is identified as a risk process with capital injections, the expected total amount of discounted capital injections prior to the exiting time and the Laplace transform of the accumulated capital injections until the exiting time are also obtained. The results are expressed in terms of scale functions for spectrally negative Lévy processes.

math.PR↗
arXiv2019

Exit problems for general draw-down times of spectrally negative Lévy processes

For spectrally negative Lévy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functions for the spectrally negative Lévy processes.

math.PR↗
arXiv2019

Draw-down Parisian ruin for spectrally negative Lévy process

In this paper we study the draw-down related Parisian ruin problem for spectrally negative Lévy risk processes. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit time via excursion theory. We also obtain an expression of the potential measure for the process killed at the draw-down Parisian time. As applications, new results are obtained for spectrally negative Lévy risk process with dividend barrier and Parisian ruin.

math.PR↗
arXiv2019

Local times for spectrally negative Lévy processes

For spectrally negative Lévy processes, adapting an approach from \cite{BoLi:sub1} we identify joint Laplace transforms involving local times evaluated at either the first passage times, or independent exponential times, or inverse local times. The Laplace transforms are expressed in terms of the associated scale functions. Connections are made with the permanental process and the Markovian loop soup measure.

math.PR↗
arXiv2018

A general continuous-state nonlinear branching process

In this paper we consider the unique nonnegative solution to the following generalized version of the stochastic differential equation for a continuous-state branching process. \beqnn X_t \ar=\ar x+\int_0^tγ_0(X_s)\dd s+\int_0^t\int_0^{γ_1(X_{s-})} W(\dd s,\dd u)\cr \ar\ar\qquad+\int_0^t\int_{0}^\infty\int_0^{γ_2(X_{s-})} z\tilde{N}(\dd s, \dd z, \dd u), \eeqnn where $W(\dd t,\dd u) $ and $\tilde{N}(\dd s, \dd z, \dd u)$ denote a Gaussian white noise and an independent compensated spectrally positive Poisson random measure, respectively, and $γ_0,γ_1$ and $γ_2$ are functions on $\mbb{R}_+$ with both $γ_1$ and $γ_2$ taking nonnegative values. Intuitively, this process can be identified as a continuous-state branching process with population-size-dependent branching rates and with competition. Using martingale techniques we find rather sharp conditions on extinction, explosion and coming down from infinity behaviors of the process. Some Foster-Lyapunov type criteria are also developed for such a process. More explicit results are obtained when $γ_i, i=0, 1, 2$ are power functions.

math.PR↗
arXiv2018

How long does the surplus stay close to its historical high?

In this paper we find the Laplace transforms of the weighted occupation times for a spectrally negative Lévy surplus process to spend below its running maximum up to the first exit times. The results are expressed in terms of generalized scale functions. For step weight functions, the Laplace transforms can be further expressed in terms of scale functions.

math.PR↗
arXiv2017

On weighted occupation times for refracted spectrally negative Lévy processes

For refracted spectrally negative Lévy processes, we identify expressions of several quantities related to Laplace transforms on their weighted occupation times until first exit times. Such quantities are expressed in terms of unique solutions to integral equations involving weight functions and scale functions for the associated spectrally negative Lévy processes. Previous results on refracted Lévy processes are recovered.

math.PR↗
arXiv2016

On fluctuation theory for spectrally negative Levy processes with Parisian reflection below, and applications

As well known, all functionals of a Markov process may be expressed in terms of the generator operator, modulo some analytic work. In the case of spectrally negative Markov processes however, it is conjectured that everything can be expressed in a more direct way using the $W$ scale function which intervenes in the two-sided first passage problem, modulo performing various integrals. This conjecture arises from work on Levy processes \cite{AKP,Pispot,APP,Iva,IP, ivanovs2013potential,AIZ,APY}, where the $W$ scale function has explicit Laplace transform, and is therefore easily computable; furthermore it was found in the papers above that a second scale function $Z$ introduced in \cite{AKP} greatly simplifies first passage laws, especially for reflected processes. This paper gathers a collection of first passage formulas for spectrally negative Parisian Lévy processes, expressed in terms of $W,Z$ which may serve as an "instruction kit" for computing quantities of interest in applications, for example in risk theory and mathematical finance. To illustrate the usefulness of our list, we construct a new index for the valuation of financial companies modeled by spectrally negative Lévy processes, based on a Dickson-Waters modifications of the de Finetti optimal expected discounted dividends objective. We offer as well an index for the valuation of conglomerates of financial companies. An implicit question arising is to investigate analog results for other classes of spectrally negative Markovian processes.

math.PR↗
arXiv2016

Pathwise uniqueness for a SPDE with Hölder continuous coefficient driven by α-stable noise

In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with Hölder continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} ΔX_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L}_t(x),~~~ t>0, ~x\in\mathbb{R}, \end{eqnarray*} where $\dot{L}$ denotes an $α$-stable white noise on $\mathbb{R}_+\times \mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $β$-Hölder continuous for $1<α<2$ and $0<β<1$. For $G\equiv0$ and $H(x)=x^β$, in Mytnik (2002) a weak solution to the above SPDE was constructedand the pathwise uniqueness of the solution was left as an open problem. In this paper we give an affirmative answer to this problem for certain values of $α$ and $β$. In particular, for $αβ=1$, where the solution to the above equation is the density of a super-Brownian motion with $α$-stable branching (see also Mytnik (2002)), our result leads to its pathwise uniqueness for $1<α<\sqrt{5}-1$. The local Hölder continuity of the solution is also obtained in this paper for fixed time $t>0$.

math.PR↗
arXiv2016

A distribution-function-valued SPDE and its applications

In this paper we further study the stochastic partial differential equation first proposed by Xiong (2013). Under localized conditions on the coefficients we show that the solution is in fact distribution-function-valued and we establish the pathwise uniqueness of the solution. As applications we obtain the well-posedness of the martingale problems for two classes of measure-valued diffusions: interacting super-Brownian motions and interacting Fleming-Viot processes. Properties of the two superprocesses such as the existence of density fields and the extinction behaviors are also studied.

math.PR↗
arXiv2016

On the last exit times for spectrally negative Lévy processes

Using a new approach, for spectrally negative Lévy processes we find joint Laplace transforms involving the last exit time (from a semi-infinite interval), the value of the process at the last exit time and the associated occupation time, which generalizes some previous results.

math.PR↗
arXiv2016

Exit identities for Lévy processes observed at Poisson arrival times

For a spectrally one-sided Lévy process, we extend various two-sided exit identities to the situation when the process is only observed at arrival epochs of an independent Poisson process. In addition, we consider exit problems of this type for processes reflected either from above or from below. The resulting Laplace transforms of the main quantities of interest are in terms of scale functions and turn out to be simple analogues of the classical formulas.

math.PR↗
arXiv2013

Some support properties for a class of Lambda-Fleming-Viot processes

For a class of $Λ$-Fleming-Viot processes with underlying Brownian motion whose associated $Λ$-coalescents come down from infinity, we prove a one-sided modulus of continuity result for their ancestry processes recovered from the lookdown construction of Donnelly and Kurtz. As applications, we first show that such a $Λ$-Fleming-Viot support process has one-sided modulus of continuity (with modulus function $C\sqrt{t\log(1/t)}$) at any fixed time. We also show that the support is compact simultaneously at all positive times, and given the initial compactness, its range is uniformly compact over any finite time interval. In addition, under a mild condition on the $Λ$-coalescence rates, we find a uniform upper bound on Hausdorff dimension of the support and an upper bound on Hausdorff dimension of the range.

math.PR↗
arXiv2012

The reversibility and an SPDE for the generalized Fleming-Viot Processes with mutation

The (Ξ, A)-Fleming-Viot process with mutation is a probability-measure-valued process whose moment dual is similar to that of the classical Fleming-Viot process except that the Kingman's coalescent is replaced by the Ξ-coalescent, the coalescent with simultaneous multiple collisions. We first prove the existence of such a process for general mutation generator A. We then investigate its reversibility. We also study both the weak and strong uniqueness of solution to the associated stochastic partial differential equation.

math.PR↗
arXiv2012

Occupation times of intervals until first passage times for spectrally negative Lévy processes

In this paper, we identify Laplace transforms of occupation times of intervals until first passage times for spectrally negative Lévy processes. New analytical identities for scale functions are derived and therefore the results are explicitly stated in terms of the scale functions of the process. Applications to option pricing and insurance risk models are also presented.

math.PR↗