arXiv subjects

Xiaowen Zhou

Publications and source records attributed to Xiaowen Zhou.

At least 37 records · Page 2Linked to original sources

arXiv2024

Existence of weak solutions to stochastic heat equations driven by truncated $α$-stable white noises with non-Lipschitz coefficients

We consider a class of stochastic heat equations driven by truncated $α$-stable white noises for $1<α<2$ with noise coefficients that are continuous but not necessarily Lipschitz and satisfy globally linear growth conditions. We prove the existence of weak solution, taking values in two different spaces, to such an equation using a weak convergence argument on solutions to the approximating stochastic heat equations. For $1<α<2$ the weak solution is a measure-valued càdlàg process. However, for $1<α<5/3$ the weak solution is a càdlàg process taking function values, and in this case we further show that for $0<p<5/3$ the uniform $p$-th moment for $L^p$-norm of the weak solution is finite, and that the weak solution is uniformly stochastic continuous in $L^p$ sense and satisfies a flow property.

math.PR↗

arXiv2024

Comparison principle for stochastic heat equations driven by $α$-stable white noises

For a class of non-linear stochastic heat equations driven by $α$-stable white noises for $α\in(1,2)$ with Lipschitz coefficients, we first show the existence and pathwise uniqueness of $L^p$-valued càdlàg solutions to such a equation for $p\in(α,2]$ by considering a sequence of approximating stochastic heat equations driven by truncated $α$-stable white noises obtained by removing the big jumps from the original $α$-stable white noises. If the $α$-stable white noise is spectrally one-sided, under additional monotonicity assumption on noise coefficients, we prove a comparison theorem on the $L^2$-valued càdlàg solutions of such a equation. As a consequence, the non-negativity of the $L^2$-valued càdlàg solution is established for the above stochastic heat equation with non-negative initial function.

math.PR↗

arXiv2023

On optimality of barrier dividend control under endogenous regime switching with application to Chapter 11 bankruptcy

Motivated by recent developments in risk management based on the U.S. bankruptcy code, we revisit the De Finetti's optimal dividend problem by incorporating the reorganization process and regulator's intervention documented in Chapter 11 bankruptcy. The resulting surplus process, bearing financial stress towards the more subtle concept of bankruptcy, corresponds to a non-standard spectrally negative Levy process with endogenous regime switching. Some explicit expressions of the expected present values under a barrier strategy, new to the literature, are established in terms of scale functions. With the help of these expressions, when the tail of the Levy measure is log-convex, the optimal dividend control is shown to be of the barrier type and the associated optimal barrier can be identified using scale functions of spectrally negative Levy processes. Some financial implications are also discussed in an illustrative example.

math.OC↗

arXiv2023

Exponential ergodicity of branching processes with immigration and competition

We study the ergodic property of a continuous-state branching process with immigration and competition. The exponential ergodicity in a weighted total variation distance is proved under natural assumptions. The main theorem applies to subcritical, critical and supercritical branching mechanisms, including all those of stable types. The proof is based on the construction of a Markov coupling process and the choice of a nonsymmetric control function for the distance. Those are designed to identify and to take the advantage of the dominating factor from the branching, immigration and competition mechanisms in different parts of the state space. The approach provides a way of finding a lower bound of the ergodicity rate.

math.PR↗

arXiv2023

Quasi-stationary distribution for continuous-state branching processes with competition

We study quasi-stationary distribution of the continuous-state branching process with competition introduced in Berestycki, Fittipaldi and Fontbona\ (Probab. Theory Relat. Fields, 2018). This process is constructed as the unique strong solution to a stochastic integral equation with jumps. An important example is the logistic branching process constructed in Lambert (Ann. Appl. Probab., 2005). We establish the strong Feller property,trajectory Feller property, Lyapunov condition, weak Feller property and irreducibility, respectively. These properties together allow us to prove that when the competition term is strong enough near $+\infty$, then there is a unique quasi-stationary distribution, which attracts all initial distributions with exponential rates.

math.PR↗

arXiv2023

An excursion theoretic approach to Parisian ruin problem

Applying excursion theory, we re-express several well studied fluctuation quantities associated to Parisian ruin problem for Lévy risk processes in terms of integrals with respect to excursion measure for spectrally negative Lévy process. We show that these new expressions reconcile with the previous results on Parisian ruin problem.

math.PR↗

arXiv2023

Exact modulus of continuities for $Λ$-Fleming-Viot processes with Brownian spatial motion

For a class of $Λ$-Fleming-Viot processes with Brownian spatial motion in $\mathbb{R}^d$ whose associated $Λ$-coalescents come down from infinity, we obtain sharp global and local modulus of continuities for the ancestry processes recovered from the lookdown representations. As applications, we prove both global and local modulus of continuities for the $Λ$-Fleming-Viot support processes. In particular, if the $Λ$-coalescent is the Beta$(2-β,β)$ coalescent for $β\in(1,2]$ with $β=2$ corresponding to Kingman's coalescent, then for $h(t)=\sqrt{t\log (1/t)}$, the global modulus of continuity holds for the support process with modulus function $\sqrt{2β/(β-1)}h(t)$, and both the left and right local modulus of continuities hold for the support process with modulus function $\sqrt{2/(β-1)}h(t)$.

math.PR↗

arXiv2022

Explosion of continuous-state branching processes with competition in Lévy environment

Using the Lyapunov criteria arguments, we find sufficient conditions on explosion/nonexplosion for continuous-state branching processes with competition in Lévy random environment. In particular, we identify the necessary and sufficient conditions on explosion/nonexplosion when the competition function is a power function and the Lévy measure of the associated branching mechanism is stable.

math.PR↗

arXiv2022

On the extinction-extinguishing dichotomy for a stochastic Lotka-Volterra type population dynamical system

We study a two-dimensional process $(X, Y)$ arising as the unique nonnegative solution to a pair of stochastic differential equations driven by independent Brownian motions and compensated spectrally positive Lévy random measures. Both processes $X$ and $Y$ can be identified as continuous-state nonlinear branching processes where the evolution of $Y$ is negatively affected by $X$. Assuming that process $X$ extinguishes, i.e. it converges to $0$ but never reaches $0$ in finite time, and process $Y$ converges to $0$, we identify rather sharp conditions under which the process $Y$ exhibits, respectively, one of the following behaviors: extinction with probability one, extinguishing with probability one or both extinction and extinguishing occurring with strictly positive probabilities.

math.PR↗

arXiv2022

Instantaneous support propagation for $Λ$-Fleming-Viot processes

For a probability-measure-valued neutral Fleming-Viot process $Z_t$ with Lévy mutation and resampling mechanism associated to a general $Λ$-coalescent with multiple collisions, we prove the instantaneous propagation of supports. That is, at any fixed time $t>0$, with probability one the closed support $S(Z_t)$ of the Fleming-Viot process satisfies $S(ν* Z_t) \subseteq S(Z_t)$, where $ν$ is the Lévy measure of the mutation process. To show this result, we apply Donnelly-Kurtz's lookdown particle representation for Fleming-Viot process.

math.PR↗

arXiv2021

Generalized stepping stone model with $Ξ$-resampling mechanism

A generalized stepping stone model with $Ξ$-resampling mechanism is a two dimensional probability-measure-valued stochastic process whose moment dual is similar to that of the classical stepping stone model except that Kingman's coalescent is replaced by $Ξ$-coalescent. We prove the existence of such a process by specifying its moments using the dual function-valued $Ξ$-coalescent process with geographical labels and migration, and then verifying a multidimensional Hausdorff moment problem. We also characterize the stationary distribution of the generalized stepping stone model and show that it is not reversible if the mutation operator is of uniform jump-type.

math.PR↗

arXiv2020

On the explosion of a class of continuous-state nonlinear branching processes

In this paper, we consider a class of generalized continuous-state branching processes obtained by Lamperti type time changes of spectrally positive Lévy processes using different rate functions. When explosion occurs to such a process, we show that the process converges to infinity in finite time asymptotically along a deterministic curve, and identify the speed of explosion for rate function in different regimes. To prove the main theorems, we also establish a new asymptotic result for scale function of spectrally positive Lévy process.

math.PR↗

arXiv2020

On the explosion of the number of fragments in the simple exchangeable fragmentation-coalescence processes

We consider the exchangeable fragmentation-coagulation (EFC) processes, where the coagulations are multiple and not simultaneous, as in a $Λ$-coalescent, and the fragmentations dislocate at finite rate an individual block into sub-blocks of infinite size. Sufficient conditions are found for the block-counting process to explode (i.e. to reach $\infty$) or not and for infinity to be an exit boundary or an entrance boundary. In a case of regularly varying fragmentation and coagulation mechanisms, we find regimes where the boundary $\infty$ can be either an exit, an entrance or a regular boundary. In the latter regular case, the EFC process leaves instantaneously the set of partitions with an infinite number of blocks and returns to it immediately. Proofs are based on a new sufficient condition of explosion for positive continuous-time Markov chains, which is of independent interest.

math.PR↗

arXiv2020

On the boundary classification of $Λ$-Wright-Fisher processes with frequency-dependent selection

We construct extensions of the pure-jump $Λ$-Wright-Fisher processes with frequency-dependent selection ($Λ$-WF processes with selection) beyond their first passage time at the boundary $1$. We show that they satisfy some duality relationships with the block counting process of simple exchangeable fragmentation-coalescence processes (EFC). One-to-one correspondences between the nature of the boundary $1$ of the $Λ$-WF process with selection and the boundary $\infty$ of the block counting process are established. New properties for the $Λ$-WF processes with selection and the block counting processes of the simple EFC processes are deduced from these correspondences. Some conditions are provided for the selection to be either weak enough for boundary $1$ to be an exit boundary or strong enough for $1$ to be an entrance boundary. When the measure $Λ$ and the selection mechanism satisfy some regular variation properties, conditions are found in order that the extended $Λ$-WF process with selection makes excursions out from the boundary $1$ before getting absorbed at $0$. In the latter process, $1$ is a transient regular reflecting boundary. This corresponds to a new phenomenon for the deleterious allele which can spread into the population in a set of times of zero Lebesgue measure, before vanishing in finite time almost surely.

math.PR↗

arXiv2020

Boundary behaviors for a class of continuous-state nonlinear branching processes in critical cases

Using Foster-Lyapunov techniques we establish new conditions on non-extinction, non-explosion, coming down from infinity and staying infinite, respectively, for the general continuous-state nonlinear branching processes introduced in Li et al. (2019). These results can be applied to identify boundary behaviors for the critical cases of the above nonlinear branching processes with power rate functions driven by Brownian motion and (or) stable Poisson random measure, which was left open in Li et al. (2019). In particular, we show that even in the critical cases, a phase transition happens between coming down from infinity and staying infinite.

math.PR↗

arXiv2020

Time-changed spectrally positive Lévy processes starting from infinity

Consider a spectrally positive Lévy process $Z$ with log-Laplace exponent $Ψ$ and a positive continuous function $R$ on $(0,\infty)$. We investigate the entrance from $\infty$ of the process $X$ obtained by changing time in $Z$ with the inverse of the additive functional $η(t)=\int_{0}^{t}\frac{{\rm d} s}{R(Z_s)}$. We provide a necessary and sufficient condition for infinity to be an entrance boundary of the process $X$. Under this condition, the process can start from infinity and we study its speed of coming down from infinity. When the Lévy process has a negative drift $δ:=-γ<0$, sufficient conditions over $R$ and $Ψ$ are found for the process to come down from infinity along the deterministic function $(x_t,t\geq 0)$ solution to ${\rm d} x_t=-γR(x_t) {\rm d} t$, with $x_0=\infty$. When $Ψ(λ)\sim cλ^α$, with $λ\rightarrow 0$, $α\in (1,2]$, $c>0$ and $R$ is regularly varying at $\infty$ with index $θ>α$, the process comes down from infinity and we find a renormalisation in law of its running infimum at small times.

math.PR↗

arXiv2020

On the entrance at infinity of Feller processes with no negative jumps

Consider a non-explosive positive Feller process with no negative jumps. It is shown in this note that when infinity is an entrance boundary, in the sense that the entrance times of the process remain bounded when the initial value tends to infinity, the process admits a Feller extension on the compactified state space $[0,\infty]$. Moreover, when started from infinity, the extended Markov process on $[0,\infty]$ leaves infinity instantaneously and stays finite, almost-surely. Arguments are adapted from a proof given by O. Kallenberg for diffusions. We also show that the process started from $x$ converges weakly towards that started from infinity in the Skorokhod space, when $x$ goes to infinity.

math.PR↗

arXiv2020

Integral functionals for spectrally positive Levy processes

We find necessary and sufficient conditions for almost sure finiteness of integral functionals of spectrally positive Lévy processes. Via Lamperti type transforms, these results can be applied to obtain new integral tests on extinction and explosion behaviors for a class of continuous-state nonlinear branching processes.

math.PR↗