A renewal theorem and supremum of a perturbed random walk

带干扰的更新风险过程的破产问题

On the ruin problem in the renewal risk processes perturbed by diffusion

本文考虑带扰动的更新风险过程。分别建立了破产时Gerber-Shiu函数在索赔和振动情况下的积分微分方程。得到了Gerber-Shiu函数的显式Laplace变换，并在索赔额分布为有理族时，得到了Gerber-Shiu函数的封闭形式表达式.最后，我们提出的数值例子来说明的主要结果。

Renewal theory for iterated perturbed random walks on a general branching process tree: intermediate generations

Let $(ξ_k,η_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $(T_k)_{k\in\mathbb{N}}$ defined by $T_k:=ξ_1+\cdots+ξ_{k-1}+η_k$ for $k\in\mathbb{N}$. Further, by an iterated perturbed random walk is meant the sequence of point processes defining the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. For $j\in\mathbb{N}$ and $t\…

Simple random walk on Z^2 perturbed on the axis (renewal case)

We study a simple random walk on Z^2 with constraints on the axis. Motivation comes from physics when particles (a gas for example, see [Dal88]) are submitted to a local field. In our case we assume that the particle evolves freely in the cones but when touching the axis a force pushes it back progressively to the origin. The main result proves that this force can be parametrized in such a way that a renewal structure appears in the trajectory of the random walk. This implies the existence of an ergodic result for the parts of the trajectory restricted to the axis.

Renewal theory for iterated perturbed random walks on a general branching process tree: early generations

Let $(ξ_k,η_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T:=(T_k)_{k\in\mathbb{N}}$ defined by $T_k:=ξ_1+\ldots+ξ_{k-1}+η_k$ for $k\in\mathbb{N}$. Consider a general branching process generated by $T$ and denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $\leq t$. We treat early generations, that is, fixed generations $j$ which do not depend on $t$. In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell …

Renewal structure and local time for diffusions in random environment

We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W\_κ$, with $ 0\textless{}κ\textless{}1$, and focus on the behavior of the local times $(\mathcal{L}(t,x),x)$ of $X$ before time $t\textgreater{}0$.In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite sites. These limits can be written explicitly from a two dimensional stable L{é}vy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of $X$.

运行上确界泛函扰动的谱负Levy过程

Spectrally negative Levy processes perturbed by functionals of their running supremum

在经典的Cramer-Lundberg风险保险模型的背景下，Albrecher和Hipp（2007）引入了纳税的概念。更准确地说，如果$X = \{X_t : t\geq 0\}$代表Cramer-Lundberg过程，并且对于所有$t\geq 0$，$S_t = \sup_{s\leq t}X_s$，那么Albrecher和Hipp（2007）研究了$X_t - γS_t$，$t\geq 0$，其中$γ\in(0,1)$是纳税率。Albrecher等人在\cite{albr_ren_zhou}中将该模型推广到$X$是一个谱负Lévy过程的情况。最后，Kyprianou和Zhou（2009）进一步扩展了该模型，允许过程$S = \{S_t : t\geq 0\}$的税率随$S$的当前值而变化。具体地说，他们考虑了所谓的扰动谱负Levy过程，\[ U_t=X_t-\int_{(0,t]}γ(S_u)\,{\rm d} S_u,\qquad t\geq 0, \]下的假设$γ:[0,\infty)\rightarrow [0,1)$和$\int_0^\infty (1-γ(s)){\rm d}s =\infty$。在这篇文章中，我们证明了Kyprianou和Zhou（2009）中的一些恒等式对于更一般的速率函数类$γ:[0,\infty)\rightarrow \mathbb{R}$仍…