Exit Problem Research Articles

Exit problem for Ornstein-Uhlenbeck processes: A random walk approach

In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm so-called Walk on Moving Spheres was already introduced in the Brownian context. The aim is therefore to generalize this numerical approach to the Ornstein-Uhlenbeck process and to describe the efficiency of the method.

Exit time asymptotics for dynamical systems with fast random switching near an unstable equilibrium

We consider the exit problem for a one-dimensional system with random switching near an unstable equilibrium point of the averaged drift. In the infinite switching rate limit, we show that the exit time satisfies a limit theorem with a logarithmic deterministic term and a random correction converging in distribution. Thus, this setting is in the universality class of the unstable equilibrium exit under small white-noise perturbations.

Large deviations for fast transport stochastic RDEs with applications to the exit problem

We study reaction diffusion equations with a deterministic reaction term as well as two random reaction terms, one that acts on the interior of the domain, and another that acts only on the boundary of the domain. We are interested in the regime where the relative sizes of the diffusion and reaction terms are different. Specifically, we consider the case where the diffusion rate is much larger than the rate of reaction, and the deterministic rate of reaction is much larger than either of the random rate of reactions.

Evolution of water column pulled by partially submerged spheres with different velocities and submergence depths

Abstract The evolution of a water column pulled by a partially submerged sphere, known as the water exit problem, is evaluated by numerical simulation. The lattice Boltzmann method was combined with a mass-tracking algorithm. The accuracy of the numerical method was demonstrated by comparisons with previous experimental results. Moreover, the evolution processes of the water column with different Froude and Weber numbers (0.4134 ≤ Fr ≤ 8.2479, 5.15 × 102 ≤ We ≤ 2.06 × 105), as well as different initial submergence depths (Hs/D = 1/2, 1/4, 1/8), were discussed. Spontaneous breakup of water column was observed for higher Weber numbers. Moreover, vertical stretching of the water column approximately followed a power law in which the power exponent depended on the initial setup of the sphere and its time scales. However, the horizontal evolution of the water column followed an approximately linear rule. Variation in the water column diameter appeared to be irregular at different physical times or in non-dimensional time. Furthermore, the initial submergence depth of the sphere was also a significant factor in the pinch-off dynamics and water column evolution. This study of water exit is of a great importance in designing marine vehicles that take off from a free water surface.

Exit Times, Undershoots and Overshoots for Reflected CIR Process with Two-Sided Jumps

In this paper, we investigate the reflected CIR process with two-sided jumps to capture the jump behavior and its non-negativeness. Applying the method of (complex) contour integrals, the closed-form solution to the joint Laplace transform of the first passage time crossing a lower level and the corresponding undershoot is derived. We further extend our arguments to the exit problem from a finite interval and obtain joint Laplace transforms. Our results are expressed in terms of the real and imaginary parts of complex functions by complex matrix. Numerical results are included.

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

AbstractFor 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.

Scaling limit for escapes from unstable equilibria in the vanishing noise limit: Nontrivial Jordan block case

We consider white noise perturbations of a nonlinear dynamical system in the neighborhood of an unstable critical point with linearization given by a Jordan block of full dimension. For the associated exit problem, we study the joint limiting behavior of the exit location and exit time, in the vanishing noise limit. The exit typically happens near one of two special deterministic points associated with the eigendirection, and we obtain several more terms in the expansion for the exit point. The leading correction term is deterministic and logarithmic in the noise magnitude, while the random remainder satisfies a scaling limit.

Excitability inflection for exit problem in Class 1 excitable systems

The path from resting to spiking threshold can generate a typical action potential in neurons. For excitable systems with ubiquitous noise, this process can be investigated in the framework of large deviation theory. In this paper, a special position (named as excitability inflection point) of the exit process in the canonical Class 1 excitable system is identified, which is both the extremal position of the dispersion and the momentum. The role of this point is examined by adding a Gaussian white noise and a pulse with a special position into the deterministic system. Surprisingly, by calculating the firing rate, its extremal position can be made closer to the theoretical excitability inflection point with vanishing pulse instead of vanishing noise. The influences of finite noise and pulse are investigated in detail by fixing one of them and changing the other. They showed distinct impacts on the real extremal positions of the firing rate. Finally, some open problems are given.

Large deviation principle for dynamical systems coupled with diffusion–transmutation processes

In this paper, we introduce a mathematical apparatus that is relevant for understanding a dynamical system with small random perturbations and coupled with the so-called transmutation process — where the latter jumps from one mode to another, and thus modifying the dynamics of the system. In particular, we study the exit problem, i.e., an asymptotic estimate for the exit probabilities with which the corresponding processes exit from a given bounded open domain, and then formally prove a large deviation principle for the exit position joint with the type occupation times as the random perturbation vanishes. Moreover, under certain conditions, we also determine the exit place and the type of distribution at the exit time and, as a consequence of this, such information also give the limit of the Dirichlet problem for the corresponding partial differential equation systems with a vanishing small parameter.

Exit Problems as the Generalized Solutions of Dirichlet Problems

This paper investigates sufficient conditions for a Feynman--Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the...

The first exit problem of reaction-diffusion equations for small multiplicative Lévy noise

The first exit problem of reaction-diffusion equations for small multiplicative Lévy noise

Continuous-time Random Walks for the Numerical Solution of Stochastic Differential Equations

This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be realized exactly using a Monte Carlo method. In this construction the spatial increment of the approximation can be bounded uniformly in space, which guarantees that the schemes are numerically stable for both finite and long time simulation of SDEs. By directly analyzing the generator of the approximation, we prove that the approximation has a sharp stochastic Lyapunov function when applied to an SDE with a drift field that is locally Lipschitz continuous and weakly dissipative. We use this stochastic Lyapunov function to extend a local semimartingale representation of the approximation. This extension permits to analyze the complexity of the approximation. Using the theory of semigroups of linear operators on Banach spaces, we show that the approximation is (weakly) accurate in representing finite and infinite-time statistics, with an order of accuracy identical to that of its generator. The proofs are carried out in the context of both fixed and variable spatial step sizes. Theoretical and numerical studies confirm these statements, and provide evidence that these schemes have several advantages over standard methods based on time-discretization. In particular, they are accurate, eliminate nonphysical moves in simulating SDEs with boundaries (or confined domains), prevent exploding trajectories from occurring when simulating stiff SDEs, and solve first exit problems without time-interpolation errors. 2010 Mathematics Subject Classification. Primary 65C30; Secondary, 60J25, 60J75.

Generalized refracted Lévy process and its application to exit problem

Generalizing Kyprianou–Loeffen’s refracted Lévy processes, we define a new refracted Lévy process which is a Markov process whose positive and negative motions are Lévy processes different from each other. To construct it we utilize the excursion theory. We study its exit problem and the potential measures of the killed processes. We also discuss approximation problem.

Inflationary universe in deformed phase space scenario

We consider a noncommutative (NC) inflationary model with a homogeneous scalar field minimally coupled to gravity. The particular NC inflationary setting herein proposed, produces entirely new consequences as summarized in what follows. We first analyze the free field case and subsequently examine the situation where the scalar field is subjected to a polynomial and exponential potentials. We propose to use a canonical deformation between momenta, in a spatially flat Friedmann–Lemaî tre–Robertson–Walker (FLRW) universe, and while the Friedmann equation (Hamiltonian constraint) remains unaffected the Friedmann acceleration equation (and thus the Klein–Gordon equation) is modified by an extra term linear in the NC parameter. This concrete noncommutativity on the momenta allows interesting dynamics that other NC models seem not to allow. Let us be more precise. This extra term behaves as the sole explicit pressure that under the right circumstances implies a period of accelerated expansion of the universe. We find that in the absence of the scalar field potential, and in contrast with the commutative case, in which the scale factor always decelerates, we obtain an inflationary phase for small negative values of the NC parameter. Subsequently, the period of accelerated expansion is smoothly replaced by an appropriate deceleration phase providing an interesting model regarding the graceful exit problem in inflationary models. This last property is present either in the free field case or under the influence of the scalar field potentials considered here. Moreover, in the case of the free scalar field, we show that not only the horizon problem is solved but also there is some resemblance between the evolution equation of the scale factor associated to our model and that for the R2 (Starobinsky) inflationary model. Therefore, our herein NC model not only can be taken as an appropriate scenario to get a successful kinetic inflation, but also is a convenient setting to obtain inflationary universe possessing the graceful exit when scalar field potentials are present.

Two-side exit problems for taxed Lévy risk process involving the general draw-down time

The literature has been witnessing an aroused interest in the study of the two-side exit problems for various models. Motivated by Kyprianou and Zhou (2009) and Li et al. (2018), the present paper concerns the two-side exit problems of the taxed spectrally negative Lévy risk process involving the general draw-down time. Our two-side exit problem is separated into two sub-problems: one being the Laplace transform of the up-exiting time of a certain level on the event that the taxed risk process up-crosses that level before the general draw-down time; the other being the Laplace transform of the draw-down time on the event that draw-down of the taxed risk process occurs before it up-crosses a certain level. Using a modified approximating method of Li et al. (2018) together with the excursion theory, solutions for the aforementioned two-side exit problems are obtained.

Robustness and the Paradox of Bridging Organizations: The Exit Problem in Regional Water Governance Networks in Central America

ABSTRACTBridging organizations facilitate a range of governance processes, including cooperation and social learning, and are theorized to be a key component of robust governance systems. In this article, we use node removal simulations to test structural hypotheses of robustness in a regional water governance network in Central America. We investigate the response of network measures supporting core governance processes to the targeted removal of bridging organizations and other actors, which we compare to random and centrality-based simulations. The results indicate removing bridging organizations has a greater impact on the network than any other type of actor, suggesting bridging organizations are critical to the robustness of the governance system. Furthermore, network structures supporting cooperation may be less robust than structures facilitating social learning. We conclude with policy implications of the research findings as they relate to the exit problem in governance systems with a large presence of international development actors.

Probability evolution method for exit location distribution

The exit problem in the framework of the large deviation theory has been a hot topic in the past few decades. The most probable escape path in the weak-noise limit has been clarified by the Freidlin–Wentzell action functional. However, noise in real physical systems cannot be arbitrarily small while noise with finite strength may induce nontrivial phenomena, such as noise-induced shift and noise-induced saddle-point avoidance. Traditional Monte Carlo simulation of noise-induced escape will take exponentially large time as noise approaches zero. The majority of the time is wasted on the uninteresting wandering around the attractors. In this paper, a new method is proposed to decrease the escape simulation time by an exponentially large factor by introducing a series of interfaces and by applying the reinjection on them. This method can be used to calculate the exit location distribution. It is verified by examining two classical examples and is compared with theoretical predictions. The results show that the method performs well for weak noise while may induce certain deviations for large noise. Finally, some possible ways to improve our method are discussed.

On the effective Stefan-Boltzmann law and the thermodynamic origin of the initial radiation density in warm inflation

In this presentation, we are going to explain the thermodynamic origin of warm inflation scenarios by using the effetive Stefan-Boltzmann law. In the warm inflation scenarios, radiation always exists to avoid the graceful exit problem, for which the radiation energy density should be assumed to be finite at the starting point of the warm inflation. To find out the origin of the non-vanishing initial radiation energy density, we derive an effective Stefan-Boltzmann law by considering the non-vanishing trace of the total energy-momentum tensors. The effective Stefan-Boltzmann law successfully shows where the initial radiation energy density is thermodynamically originated from. And by using the above effective Stefan-Boltzmann law, we also study the cosmological scalar perturbation, and obtain the sufficient radiation energy density in order for GUT baryogenesis at the end of inflation. This proceeding is based on Ref. [1]

Pricing insurance drawdown-type contracts with underlying Lévy assets

In this paper we consider some insurance policies related to drawdown and drawup events of log-returns for an underlying asset modeled by a spectrally negative geometric Lévy process. We consider four contracts, three of which were introduced in Zhang (2013) for a geometric Brownian motion. The first one is an insurance contract where the protection buyer pays a constant premium until the drawdown of fixed size of log-returns occurs. In return he/she receives a certain insured amount at the drawdown epoch. The next insurance contract provides protection from any specified drawdown with a drawup contingency. This contract expires early if a certain fixed drawup event occurs prior to the fixed drawdown. The last two contracts are extensions of the previous ones by an additional cancellation feature which allows the investor to terminate the contract earlier. We focus on two problems: calculating the fair premium p for the basic contracts and identifying the optimal stopping rule for the policies with the cancellation feature. To do this we solve some two-sided exit problems related to drawdown and drawup of spectrally negative Lévy processes, which is of independent mathematical interest. We also heavily rely on the theory of optimal stopping.

Two-Sided Exit Problems in the Ordered Risk Model

The insurance risk model in the presence of two horizontal absorbing barriers is considered. The lower barrier is the usual ruin barrier while the upper one corresponds to the dividend barrier. The distribution of two first-exit times of the risk process from the strip between the two horizontal lines is under study. The claim arrival process is governed by an Order Statistic Point Process (OSPP) which enables the derivation of formulas in terms of the joint distribution of the order statistics of a sample of uniform random variables.