Search

For skew Brownian motion with two-valued drift, adopting a perturbation approach we find expressions of its potential densities. As applications, we recover its transition density and study its long-time asymptotic behaviors. We also compare with previous results on transition densities for skew Brownian motions. We propose two approaches for generating quasi-random samples by approximating the cumulative distribution function and discussing their risk measurement application., Comment: 26 pages, 3 figures, 2 tables

We consider a class of stochastic control problems which has been widely used in optimal foraging theory. The state processes have two distinct dynamics, characterized by two pairs of drift and diffusion coefficients, depending on whether it takes values bigger or smaller than a threshold value. Adopting a perturbation type approach, we find an expression for potential measure of the optimal state process. We then obtain an expression for the transition density of the optimal state process by inverting the associated Laplace transform. Properties including the stationary distribution of the optimal state process are discussed. Finally, the expression of the value function is given for this class stochastic control problems.

We study the explosion phenomenon of nonlinear continuous-state branching processes (nonlinear CSBPs). First an explicit integral test for explosion is designed when the rate function does not increase too fast. We then exhibit three different regimes of explosion when the branching mechanism and the rate function are regularly varying respectively at $0$ and $\infty$ with indices $\alpha$ and $\beta$ such that $0\leq\alpha\leq \beta$ and $\beta>0$. If $\alpha>0$ then the renormalisation of the process before its explosion is linear. When moreover $\alpha\neq \beta$, the limiting distribution is that of a ratio of two independent random variables whose laws are identified. When $\alpha=\beta$, the limiting random variable shrinks to a constant. Last, when $\alpha=0$, i.e. the branching mechanism is slowly varying at $0$, the process is studied with the help of a nonlinear renormalisation. The limiting distribution becomes the inverse uniform distribution. This complements results known in the case of finite mean and provides new insight for the classical explosive continuous-state branching processes (for which $\beta=1$).

In this paper we propose a skew Brownian motion with a two-valued drift as a risk model with endogenous regime switching. We solve its two-sided exit problem and consider an optimal control problem for the skew Brownian risk model. In particular, we identify sufficient conditions for either a barrier dividend strategy or a band dividend strategy to be optimal.

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.

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

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

We consider a class of stochastic heat equations driven by truncated $\alpha$-stable white noises for $1<\alpha<2$ 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 $1<\alpha<2$ the weak solution is a measure-valued c\`{a}dl\`{a}g process. However, for $1<\alpha<5/3$ the weak solution is a c\`{a}dl\`{a}g process taking function values, and in this case we further show that for $0

For a class of $\Lambda$-Fleming-Viot processes with Brownian spatial motion in $\mathbb{R}^d$ whose associated $\Lambda$-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 $\Lambda$-Fleming-Viot support processes. In particular, if the $\Lambda$-coalescent is the Beta$(2-\beta,\beta)$ coalescent for $\beta\in(1,2]$ with $\beta=2$ 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 $\sqrt{2\beta/(\beta-1)}h(t)$, and both the left and right local modulus of continuities hold for the support process with modulus function $\sqrt{2/(\beta-1)}h(t)$., Comment: 27 pages

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.

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

For a probability-measure-valued neutral Fleming-Viot process $Z_t$ with L\'evy mutation and resampling mechanism associated to a general $\Lambda$-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 $S(\nu * Z_t) \subseteq S(Z_t)$, where $\nu$ is the L\'evy measure of the mutation process. To show this result, we apply Donnelly-Kurtz's lookdown particle representation for Fleming-Viot process., Comment: 23 pages