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In various application domains, there is a certain `null cell', inside a multinomial setup, where observations are recorded for the other cells, but where one cannot count the number of occurrences for the null cell. I develop inference theory for assessing such unknown numbers, counting the uncounted, in situations where counts are available for the other cells, via parametric modelling. The methods are used to estimate the number of persons killed in Guatemala during the Genocidio guatemalteco years 1978--1995. There are three carefully curated lists of killed people, where the information can be mapped to a Venn diagram with $2^3=8$ cells. Summing over the seven observed cells, $R=\hbox{47,803}$ killed individuals can be identified, but how big is $N_{0,0,0}$, and hence $N=N_{0,0,0}+R$?

Expectations of path integrals of killed stochastic processes play a central role in several applications across physics, chemistry, and finance. Simulation-based evaluation of these functionals is often biased and numerically expensive due to the need to explicitly approximate stochastic paths and the challenge of correctly modeling them in the neighborhood of the killing boundary. We consider Itô processes killed at the boundary of some set in the $n$-dimensional space and introduce a novel stochastic method with negligible bias and lower computational cost to evaluate path integrals without simulated paths. Our approach draws a connection between stochastic bridges and killed processes to sample only exit times and locations instead of the full path. We apply it to a Wiener process killed in the $n$-ball and explicitly derive the density of the Brownian bridge confined to the $n$-ball for $n = 1, 2, 3$. Finally, we present two numerical examples that demonstrate the efficiency and negligible bias of the novel procedure compared to an evaluation using the standard Euler-Maruyama method.

In this paper, we study asymptotic behaviors of a subcritical branching killed Brownian motion with drift $-ρ$ and offspring distribution $\{p_k:k\ge 0\}$. Let $\widetildeζ^{-ρ}$ be the extinction time of this subcritical branching killed Brownian motion, $\widetilde{M}_t^{-ρ}$ the maximal position of all the particles alive at time $t$ and $\widetilde{M}^{-ρ}:=\max_{t\ge 0}\widetilde{M}_t^{-ρ}$ the all time maximal position. Let $\mathbb{P}_x$ be the law of this subcritical branching killed Brownian motion when the initial particle is located at $x\in (0,\infty)$. Under the assumption $\sum_{k=1}^\infty k (\log k) p_k <\infty$, we establish the decay rates of $\mathbb{P}_x(\widetildeζ^{-ρ}>t)$ and $\mathbb{P}_x(\widetilde{M}^{-ρ}>y)$ as $t$ and $y$ tend to $\infty$ respectively. We also establish the decay rate of $\mathbb{P}_x(\widetilde{M}_t^{-ρ}>z(t,ρ))$ as $t\to\infty$, where $z(t,ρ)=\sqrt{t}z-ρt$ for $ρ\leq 0$ and $z(t,ρ)=z$ for $ρ>0$. As a consequence, we obtain a Yaglom-type limit theorem.

By establishing a local version of Bismut formula for Dirichlet semigroups on a regular domain, gradient estimates are derived for killed SDEs with singular drifts. As an application, the total variation distance between two solutions of killed DDSDEs is bounded above by the truncated $1$-Wasserstein distance of initial distributions, in the regular and singular cases respectively.

In this paper, we study asymptotic behaviors of the tails of extinction time and maximal displacement of a critical branching killed Lévy process $(Z_t^{(0,\infty)})_{t\ge 0}$ in $\mathbb{R}$, in which all particles (and their descendants) are killed upon exiting $(0, \infty)$. Let $ζ^{(0,\infty)}$ and $M_t^{(0,\infty)}$ be the extinction time and maximal position of all the particles alive at time $t$ of this branching killed Lévy process and define $M^{(0,\infty)}: = \sup_{t\geq 0} M_t^{(0,\infty)}$. Under the assumption that the offspring distribution belongs to the domain of attraction of an $α$-stable distribution, $α\in (1, 2]$, and some moment conditions on the spatial motion, we give the decay rates of the survival probabilities $$ \mathbb{P}_{y}(ζ^{(0,\infty)}>t), \quad \mathbb{P}_{\sqrt{t}y}(ζ^{(0,\infty)}>t) $$ and the tail probabilities $$ \mathbb{P}_{y}(M^{(0,\infty)}\geq x), \quad \mathbb{P}_{xy}(M^{(0,\infty)}\geq x). $$ We also study the scaling limits of $M_t^{(0,\infty)}$ and the point process $Z_t^{(0,\infty)}$ under $\mathbb{P}_{\sqrt{t}y}(\cdot |ζ^{(0,\infty)}>t)$ and $\mathbb{P}_y(\cdot |ζ^{(0,\infty)}>t)$. The scaling limits under $\mathbb{P}_{\sq

We consider the simple random walk on $\mathbb{Z}^d$ killed with probability $p(|x|)$ at site $x$ for a function $p$ decaying at infinity. Due to recurrence in dimension $d=2$, the killed random walk (KRW) dies almost surely if $p$ is positive, while in dimension $d \geq 3$ it is known that the KRW dies almost surely if and only if $\int_0^{\infty}rp(r)dr = \infty$, under mild technical assumptions on $p$. In this paper we consider, for any $d \geq 2$, functions $p$ for which the KRW dies almost surely and we ask ourselves if the KRW conditioned to survive is well-defined. More precisely, given an exhaustion $(Λ_R)_{R \in \mathbb{N}}$ of $\mathbb{Z}^d$, does the KRW conditioned to leave $Λ_R$ before dying converges in distribution towards a limit which does not depend on the exhaustion? We first prove that this conditioning is well-defined for $p(r) = o(r^{-2})$, and that it is not for $p(r) = \min(1, r^{-α})$ for $α\in (14/9,2)$. This question is connected to branching random walks and the infinite snake. More precisely, in dimension $d=4$, the infinite snake is related to the KRW with $p(r) \asymp (r^2\log(r))^{-1}$, therefore our results imply that the infinite snake conditioned

We provide a probabilistic representation for the derivative of the semigroup corresponding to a diffusion process killed at the boundary of a half interval. In particular, we show that the derivative of the semi-group can be expressed as the expected value of a functional of a reflected diffusion process. Furthermore, as an application, we obtain a Bismut-Elworthy-Li formula which is also valid at the boundary.

In this paper, we obtain the exact asymptotic behavior of Green functions of homogeneous random walks in $\Z^d$ killed at the first exit from and open cone of $\R^d$. Our approach combines methods of functional equations, integral representations of the Green function and Woess' approach for the case of homogeneous random walks in $\Z^d$.

Consider a branching random walk on the real line with a killing barrier at zero: starting from a nonnegative point, particles reproduce and move independently, but are killed when they touch the negative half-line. The population of the killed branching random walk dies out almost surely in both critical and subcritical cases, where by subcritical case we mean that the rightmost particle of the branching random walk without killing has a negative speed, and by critical case, when this speed is zero. We investigate the total progeny of the killed branching random walk and give their precise tail distribution both in the critical and subcritical cases, which solves an open problem of Aldous [Power laws and killed branching random walks, http://www.stat.berkeley.edu/~aldous/Research/OP/brw.html].

We prove explicit and sharp two-sided estimates for the transition density of the Langevin process with quadratic potential, killed outside of the position interval (0,1). The long-time asymptotics of this transition density are also obtained. In particular, this allows us to show that the killed semigroup is uniformly conditionally ergodic.

We prove the Harnack inequality and boundary Harnack principle for the absolute value of a one-dimensional recurrent subordinate Brownian motion killed upon hitting $0$, when $0$ is regular for itself and the Laplace exponent of the subordinator satisfies certain global scaling conditions. Using the conditional gauge theorem for symmetric Hunt processes we prove that the Green function of this process killed outside of some interval $(a,b)$ is comparable to the Green function of the corresponding killed subordinate Brownian motion. We also consider several properties of the compensated resolvent kernel $h$, which is harmonic for our process on $(0,1)$.

For two independent Lévy processes $ξ$ and $η$ and an exponentially distributed random variable $τ$ with parameter $q>0$ that is independent of $ξ$ and $η$, the killed exponential functional is given by $V_{q,ξ,η} := \int_0^τ\mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$. With the killed exponential functional arising as the stationary distribution of a Markov process, we calculate the infinitesimal generator of the process and use it to derive different distributional equations describing the law of $V_{q,ξ,η}$, as well as functional equations for its Lebesgue density in the absolutely continuous case. Various special cases and examples are considered, yielding more explicit information on the law of the killed exponential functional and illustrating the applications of the equations obtained. Interpreting the case $q=0$ as $τ=\infty$ leads to the classical exponential functional $\int_0^\infty \mathrm{e}^{-ξ_{s-}} \, \mathrm{d} η_s$, allowing to extend many previous results to include killing.

Reinforced processes are known to provide a stochastic representation for the quasi-stationary distribution of a given killed Markov process - describing the killed Markov process at fixed time instants. In this paper we shall adapt the construction to provide a pathwise description. We also obtain a stochastic approximation for the quasi-limiting distributions of reducible killed Markov processes as a corollary.

To characterize nonlinear Dirichlet problems in an open domain, we investigate killed distribution dependent SDEs. By constructing the coupling by projection and using the Zvonkin/Girsanov transforms, the well-posedness is proved for three different situations: 1) monotone case with distribution dependent noise (possibly degenerate), 2) singular case with non-degenerate distribution dependent noise, and 3) singular case with non-degenerate distribution independent noise. In the first two cases the domain is $C^2$ smooth such that the Lipschitz continuity in initial distributions is also derived, and in the last case the domain is arbitrary.

Parameter estimation in diffusion processes from discrete observations up to a first-hitting time is clearly of practical relevance, but does not seem to have been studied so far. In neuroscience, many models for the membrane potential evolution involve the presence of an upper threshold. Data are modeled as discretely observed diffusions which are killed when the threshold is reached. Statistical inference is often based on the misspecified likelihood ignoring the presence of the threshold causing severe bias, e.g. the bias incurred in the drift parameters of the Ornstein-Uhlenbeck model for biological relevant parameters can be up to 25-100%. We calculate or approximate the likelihood function of the killed process. When estimating from a single trajectory, considerable bias may still be present, and the distribution of the estimates can be heavily skewed and with a huge variance. Parametric bootstrap is effective in correcting the bias. Standard asymptotic results do not apply, but consistency and asymptotic normality may be recovered when multiple trajectories are observed, if the mean first-passage time through the threshold is finite. Numerical examples illustrate the results

We investigate Martin boundary for a non-centered random walk on ${\mathbb Z}^d$ killed up on the time $τ_\vartheta$ of the first exit from a convex cone with a vertex at $0$. The approach combines large deviation estimates, the ratio limite theorem and the ladder height process. The results are applied to identify the Martin boundary for a random walk killed upon the first exit from a convex cone having $C^1$ boundary.

The long time behavior of an absorbed Markov process is well described by the limiting distribution of the process conditioned to not be killed when it is observed. Our aim is to give an approximation's method of this limit, when the process is a 1-dimensional Itô diffusion whose drift is allowed to explode at the boundary. In a first step, we show how to restrict the study to the case of a diffusion with values in a bounded interval and whose drift is bounded. In a second step, we show an approximation method of the limiting conditional distribution of such diffusions, based on a Fleming-Viot type interacting particle system. We end the paper with two numerical applications : to the logistic Feller diffusion and to the Wright-Fisher diffusion with values in $]0,1[$ conditioned to be killed at 0.