Brownian Perturbation Research Articles

Asymptotics for a diffusion-perturbed risk model with dependence structures, constant interest force, and a random number of delayed claims

During extreme events such as natural or man-made disasters, it is likely that after immediate claims directly caused by the events, there are other delayed claims occurring in a certain period of future. In the classical risk model, each immediate claim is accompanied with a delayed one, but in reality each immediate claim can lead to a random number of delayed ones. To this end, we consider an insurance risk model with a random number of delayed claims as well as Brownian perturbation, dependence structures and constant interest force, and explore asymptotic estimations for an insurer’s ruin-related risks exposed to the extreme events. This article provides a novel insight to model and analyze the potential risks by fully considering the ongoing impacts of heavy-tailed claims, and thus accurately assesses the insurer’s operation risk in the long run.

Asymptotics for the random time ruin probability with non stationary arrivals and Brownian perturbation

In this paper, we consider a risk model with heavy-tailed claims and Brownian perturbation. Assuming that the distribution function of claim-size is subexponential, and the arrival process of claims is a non stationary process satisfying the principle of large deviation, the asymptotic formula for the ruin probability of this risk model at random time is obtained.

The finite-time ruin probability of a risk model with a general counting process and stochastic return

<p style='text-indent:20px;'>This paper considers a general risk model with stochastic return and a Brownian perturbation, where the claim arrival process is a general counting process and the price process of the investment portfolio is expressed as a geometric Lévy process. When the claim sizes are pairwise strong quasi-asymptotically independent random variables with heavy-tailed distributions, the asymptotics of the finite-time ruin probability of this risk model have been obtained.</p>

Finite-time ruin probability of a perturbed risk model with dependent main and delayed claims

This paper considers a delayed claim risk model with stochastic return and Brownian perturbation in which each main claim may be accompanied with a delayed claim occurring after a stochastic period of time, and the price process of the investment portfolio is described as a geometric Lévy process. By means of the asymptotic results for randomly weighted sum of dependent subexponential random variables we obtain some asymptotics for finite-time ruin probability. A simulation study is also performed to check the accuracy of the obtained theoretical result via the crude Monte Carlo method.

Estimates for the Finite-Time Ruin Probability of a Time-Dependent Risk Model with a Brownian Perturbation

In this paper, we consider a time-dependent risk model with a Brownian perturbation. In this model, there is a dependence structure between the claim sizes and their corresponding interarrival times. Assuming the claim sizes have subexponential distributions, we obtain the asymptotic lower bound of the finite-time ruin probability. When the claim sizes have distributions from the class L∩D, the asymptotic upper bound of the finite-time ruin probability has been presented. These results confirm that when the claim sizes are heavy-tailed, the asymptotics of the finite-time ruin probability of this time-dependent model are insensitive to the Brownian perturbation.

The finite-time ruin probability of time-dependent risk model with stochastic return and Brownian perturbation

This paper considers a dependent risk model with stochastic return and Brownian perturbation, where there exists a dependence structure between the claim sizes and the inter-arrival times and the price process of the investment portfolio is a geometric Levy process. When the claim sizes have heavy-tailed distributions, the asymptotic lower and upper bounds of the finite-time ruin probability have been given.

Large deviations of the exit measure through a characteristic boundary for a Poisson driven SDE

Let O be the basin of attraction of a given equilibrium of a dynamical system, whose solution is the law of large numbers limit of the solution of a Poissonian SDE as the size of the population tends to +∞. We consider the law of the exit point from O of that Poissonian SDE. We adapt the approach of Day [J. Math. Anal. Appl. 147 (1990) 134–153] who studied the same problem for an ODE with a small Brownian perturbation. For that purpose, we will use the large deviations principle for the Poissonian SDE reflected at the boundary of O, studied in our recent work Pardoux and Samegni [Stoch. Anal. Appl. 37 (2019) 836–864]. The main motivation of this work is the extension of the results concerning the time of exit from the set O established in Kratz and Pardoux [Vol. 2215 of Lecture Notes in Math.. Springer (2018) 221–327] and Pardoux and Samegni [J. Appl. Probab. 54 (2017) 905–920] to unbounded open sets O. This is done in sections 4.2.5 and 4.2.7 of Britton and Pardoux [Vol. 2255 of Lecture Notes in Math. Springer (2019) 1–120], see also The SIR model with demography subsection below.

Some asymptotic results of the ruin probabilities in a bidimensional renewal risk model with Brownian perturbation

In this paper, a bidimensional renewal risk model with constant force of interest and Brownian perturbation is considered. Assuming that the claim-size distribution function is from the subexponential class, three types of the finite-time ruin probabilities under this model are discussed. We obtain the asymptotic formulas for the three types, which hold uniformly for any finite-time horizon.

The finite-time ruin probability of a risk model with stochastic return and Brownian perturbation

This paper investigates a renewal risk model with stochastic return and Brownian perturbation, where the price process of the investment portfolio is described as a geometric Levy process. When the claim sizes have a subexponential distribution, we derive the asymptotics for the finite-time ruin probability of the above risk model. The obtained result confirms that the asymptotics for the finite-time ruin probability of the risk model with heavy-tailed claim sizes are insensitive to the Brownian perturbation.

On fair reinsurance premiums; Capital injections in a perturbed risk model

We consider a risk model where deficits after ruin are covered by a new type of reinsurance contract that provides capital injections. To allow the insurance company’s survival after ruin, the reinsurer injects capital only at ruin times caused by jumps larger than a chosen retention level. Otherwise capital must be raised from the shareholders for small deficits. The problem here is to determine adequate reinsurance premiums. It seems fair to base the net reinsurance premium on the discounted expected value of any future capital injections. Inspired by the results of Huzak et al. (2004) and Ben Salah (2014) on successive ruin events, we show that an explicit formula for these reinsurance premiums exists in a setting where aggregate claims are modeled by a subordinator and a Brownian perturbation. Here ruin events are due either to Brownian oscillations or jumps and reinsurance capital injections only apply in the latter case. The results are illustrated explicitly for two specific risk models and in some numerical examples.

Stabilization and destabilization of nonlinear systems via intermittent stochastic noise with application to memristor-based system

This paper considers the stabilization and destabilization of a given nonlinear system by an intermittent Brownian noise perturbation. We give some distinct conditions and conclusions on almost sure exponential stability and instability, which are related to the control period T and the noise width δ. These results are then exploited to examine stabilization and destabilization via intermittent stochastic perturbation and applied to the stabilization of a memristor-based chaotic system. Two numerical examples are presented to illustrate the theoretical results.

A note on the finite-time ruin probability of a renewal risk model with Brownian perturbation

In this note, we consider a renewal risk model with constant force of interest and Brownian perturbation. Assuming that the claim-size distribution function is from the subexponential class, we derive for the finite-time ruin probability a precise asymptotic expansion, which holds uniformly for any finite time horizon. Our result confirms the intuition that the asymptotic ruin probabilities of risk models with heavy-tailed claims are insensitive to the Brownian perturbation.

Stochastic Dynamic Trapping in Robotic Manipulation of Micro-Objects Using Optical Tweezers

Various automatic manipulation techniques have been developed for manipulating micro-objects using optical tweezers. Because of the small trapping force of optical traps and increase in kinetic energy during manipulation, a trapped object may not remain trappable, especially in the presence of random Brownian perturbation. However, there is no theoretical analysis so far to help understand the effects of dynamic motion and Brownian forces on the trappability problem of optical tweezers. This paper investigates the optical manipulation of micro-objects under random perturbations. Here, we provide for the first time a theoretical and experimental analysis of the dynamic trapping problem from stochastic perspectives. We derive the relationship between trapping probability and maximum manipulation velocity. A controller with appropriate velocity bound is then proposed to ensure that the system is bound and stable. The experimental results confirm the accuracy of our theoretical analysis and illustrate the necessity and usefulness of the proposed controller.

On matrix exponential approximations of ruin probabilities for the classic and Brownian perturbed Cramér–Lundberg processes

Padé rational approximations are a very convenient approximation tool, due to the easiness of obtaining them, as solutions of linear systems. Not surprisingly, many matrix exponential approximations used in applied probability are particular cases of the first and second order “admissible Padé approximations” of a Laplace transform, where admissible stands for nonnegative in the case of a density, and for nonincreasing in the case of a ccdf (survival function).Our first contribution below is the observation that for Cramér–Lundberg processes and Brownian perturbed Cramér–Lundberg processes there are three distinct rational approximations of the Pollaczek–Khinchine transform, corresponding to approximating (a) the claims transform, (b) the stationary excess transform, and (c) the aggregate loss transform.A second contribution is providing three new always admissible second order approximations for the ruin probabilities of the Cramér–Lundberg process with Brownian perturbation, one of which reduces in the absence of perturbation to De Vylder’s approximation.Our third contribution is a method for comparing the resulting approximations, based on the concept of largest weak-admissibility interval of the compounding/traffic intensity parameter ρ.

On a generalization of the expected discounted penalty function to include deficits at and beyond ruin

In this paper we propose an extended concept of the expected discounted penalty function (EDPF) that takes into account new ruin-related random variables. In the well-known EDPF introduced in seminal papers by Gerber and Shiu (Insur Math Econ 21:129–137, 1997, N Am Actuar J 2(1):48–78, 1998) and Gerber and Landry (Insur Math Econ 22:263–276, 1998), we consider the expectation of a sequence of discounted penalty functions of new record minima reached by a claim of the risk process after ruin (and before recovery). Inspired by results of Huzak et al. (Ann Appl Probab 14(3):1378–1397, 2004) and developments in fluctuation theory for spectrally negative Levy processes, we provide a characterization for this extended EDPF in a setting involving a cumulative claims modeled by a subordinator, and Brownian perturbation. We illustrate how the extended EDPF can be used to compute the expected discounted value of capital injections for the Brownian perturbed risk model.

Large deviations for multi-scale Markovian switching systems with a small diffusion

This work develops asymptotic properties of randomly switching and time inhomogeneous dynamic systems under Brownian perturbation with a small diffusion. The switching process is modeled by a continuous-time Markov chain, which portraits discrete events that cannot be modeled by a diffusion process. In the model, there are two small parameters. One of them is e associated with the generator of the continuous-time, inhomogeneous Markov chain, and the other is δ = δe signifies the small intensity of the diffusion. Assume e → 0a ndδe → 0a se → 0. This paper focuses on large deviations type of estimates for such Markovian switching systems with small diffusions. The ratio e/δe can be a nonzero constant, or equal to 0, or ∞. These three different cases yield three different outcomes. This paper analyzes the three cases and present the corresponding asymptotic properties.

Control of Multiagent Systems: A Stochastic Pinning Viewpoint

A stochastic pinning approach for multiagent systems is developed, which guarantees such systems being almost surely stable. It is seen that the pinning is closely related to being a Bernoulli variable. It has been proved for the first time that a series of systems can be stabilized by a Brownian noise perturbation in terms of a pinning scheme. A new terminology named “stochastic pinning control” is introduced to describe the given pinning algorithm. Additionally, two general cases that the expectation of the Bernoulli variable with bounded uncertainty or being unknown are studied. Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed methods.

Semimartingale decomposition of convex functions of continuous semimartingales by Brownian perturbation

In this note we prove that the local martingale part of a convex function f of a d -dimensional semimartingale X = M + A can be written in terms of an Ito stochastic integral ∫ H ( X )d M , where H ( x ) is some particular measurable choice of subgradient of f at x , and M is the martingale part of X . This result was first proved by Bouleau in [N. Bouleau, C. R. Acad. Sci. Paris Ser. I Math. 292 (1981) 87–90]. Here we present a new treatment of the problem. We first prove the result for , ϵ > 0, where B is a standard Brownian motion, and then pass to the limit as ϵ → 0, using results in [M.T. Barlow and P. Protter, On convergence of semimartingales. In Seminaire de Probabilites, XXIV, 1988/89 , Lect. Notes Math., vol. 1426. Springer, Berlin (1990) 188–193; E. Carlen and P. Protter, Illinois J. Math. 36 (1992) 420–427]. The former paper concerns convergence of semimartingale decompositions of semimartingales, while the latter studies a special case of converging convex functions of semimartingales.

Adaptive stochastic methods for sampling driven molecular systems

Thermostatting methods are discussed in the context of canonical sampling in the presence of driving stochastic forces. Generalisations of the Nosé-Hoover method and Langevin dynamics are introduced which are able to dissipate excess heat introduced by steady Brownian perturbation (without a priori knowledge of its strength) while preserving ergodicity. Implementation and parameter selection are considered. It is demonstrated using numerical experiments that the methods derived can adaptively control the target canonical ensemble in the presence of nonlinear driving perturbations.

Analysis of Noise under Regime Switching

In this paper we consider a stochastic nonlinear system under regime switching. Given a system x(t)=f(x(t),r(t),t) in which f satisfies so-called one-side polynomial growth condition. We introduce two Brownian noise feedbacks and stochastically perturb this system into dx(t)=(x(t),r(t),t)dt+ σ (r(t))|x(t)|βx(t)dW1(t)+q(r(t))x(t)dW2(t) . It can be proved that appropriate noise intensity may suppress the potentially explode in a finite time and ensure that this system is almost surely exponentially stable although the corresponding system without Brownian noise perturbation may be unstable system.