共查询到20条相似文献,搜索用时 15 毫秒
1.
《高校应用数学学报(英文版)》2020,(3)
The spectrally negative Lévy risk model with random observation times is considered in this paper, in which both dividends and capital injections are made at some independent Poisson observation times. Under the absolute ruin, the expected discounted dividends and the expected discounted capital injections are discussed. We also study the joint Laplace transforms including the absolute ruin time and the total dividends or the total capital injections. All the results are expressed in scale functions. 相似文献
2.
This paper discusses the valuation of credit default swaps, where default is announced when the reference asset price has gone below certain level from the last record maximum, also known as the high-water mark or drawdown. We assume that the protection buyer pays premium at a fixed rate when the asset price is above a pre-specified level and continuously pays whenever the price increases. This payment scheme is in favour of the buyer as she only pays the premium when the market is in good condition for the protection against financial downturn. Under this framework, we look at an embedded option which gives the issuer an opportunity to call back the contract to a new one with reduced premium payment rate and slightly lower default coverage subject to paying a certain cost. We assume that the buyer is risk neutral investor trying to maximize the expected monetary value of the option over a class of stopping time. We discuss optimal solution to the stopping problem when the source of uncertainty of the asset price is modelled by Lévy process with only downward jumps. Using recent development in excursion theory of Lévy process, the results are given explicitly in terms of scale function of the Lévy process. Furthermore, the value function of the stopping problem is shown to satisfy continuous and smooth pasting conditions regardless of regularity of the sample paths of the Lévy process. Optimality and uniqueness of the solution are established using martingale approach for drawdown process and convexity of the scale function under Esscher transform of measure. Some numerical examples are discussed to illustrate the main results. 相似文献
3.
An obvious way to simulate a Lévy process X is to sample its increments over time 1 / n, thus constructing an approximating random walk \(X^{(n)}\). This paper considers the error of such approximation after the two-sided reflection map is applied, with focus on the value of the resulting process Y and regulators L, U at the lower and upper barriers at some fixed time. Under the weak assumption that \(X_\varepsilon /a_\varepsilon \) has a non-trivial weak limit for some scaling function \(a_\varepsilon \) as \(\varepsilon \downarrow 0\), it is proved in particular that \((Y_1-Y^{(n)}_n)/a_{1/n}\) converges weakly to \(\pm \, V\), where the sign depends on the last barrier visited. Here the limit V is the same as in the problem concerning approximation of the supremum as recently described by Ivanovs (Ann Appl Probab, 2018). Some further insight in the distribution of V is provided both theoretically and numerically. 相似文献
4.
In this paper,we study a general Lévy risk process with positive and negative jumps.A renewal equation and an infinite series expression are obtained for the expected discounted penalty function of this risk model.We also examine some asymptotic behaviors for the ruin probability as the initial capital tends to infinity. 相似文献
5.
We consider the threshold dividend strategy where a company’s surplus process is described by the dual Lévy risk model. Namely, the company chooses to pay dividends at a constant rate only when the surplus is above some nonnegative threshold. Classically, such a company is referred to be ruined immediately when the surplus level becomes negative. Recently, researchers investigate the Parisian ruin problem where the company is allowed to operate under negative surplus for a predetermined period known as the Parisian delay. With the help of the fluctuation identities of spectrally negative Lévy processes, we obtain an explicit expression of the expected discounted dividends until Parisian ruin in terms of the relevant scale functions and certain probabilities that need to be evaluated for each specific Lévy process. The optimal threshold level under such a threshold dividend strategy is deduced. Applications and numerical examples are given to illustrate the theoretical results and examine how the expected discounted aggregate dividends and the optimal threshold level change in response to different Parisian delays. 相似文献
6.
This paper considers a Lévy-driven queue (i.e., a Lévy process reflected at 0), and focuses on the distribution of M(t), that is, the minimal value attained in an interval of length t (where it is assumed that the queue is in stationarity at the beginning of the interval). The first contribution is an explicit
characterization of this distribution, in terms of Laplace transforms, for spectrally one-sided Lévy processes (i.e., either
only positive jumps or only negative jumps). The second contribution concerns the asymptotics of ℙ(M(T
u
)>u) (for different classes of functions T
u
and u large); here we have to distinguish between heavy-tailed and light-tailed scenarios. 相似文献
7.
In-Suk Wee 《Probability Theory and Related Fields》1992,93(3):359-376
Summary Let {X
t
} be a one-dimensional Lévy process with local timeL(t, x) andL
*(t)=sup{L(t, x): x }. Under an assumption which is more general than being a symmetric stable process with index >1, we obtain a LIL forL*(t). Also with an additional condition of symmetry, a LIL for range is proved.This research is supported by a grant from Korea Science and Engineering Foundations 相似文献
8.
In this work, we present sufficient conditions for the existence of a stationary solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process, and show that these conditions are also necessary if the semigroup is stable, in which case the invariant measure is unique. For typical situations such as the heat equation, we significantly simplify these conditions without assuming any further restrictions on the driving cylindrical Lévy process and demonstrate their application in some examples. 相似文献
9.
We study the optimal dividend problem in the dual model where dividend payments can only be made at the jump times of an independent Poisson process. In this context, Avanzi et al. (2014) solved the case with i.i.d. hyperexponential jumps; they showed the optimality of a (periodic) barrier strategy where dividends are paid at dividend-decision times if and only if the surplus is above some level. In this paper, we generalize the results for a general spectrally positive Lévy process with additional terminal payoff/penalty at ruin, and also solve the case with classical bail-outs so that the surplus is restricted to be nonnegative. The optimal strategies as well as the value functions are concisely written in terms of the scale function. Numerical results are also given. 相似文献
10.
Y. Shimizu 《Mathematical Methods of Statistics》2011,20(2):125-149
We consider a generalized risk process which consists of a subordinator plus a spectrally negative Lévy process. Our interest
is to estimate the expected discounted penalty function (EDPF) from a set of data which is practical in the insurance framework.
We construct an empirical type estimator of the Laplace transform of the EDPF and obtain it by a regularized Laplace inversion.
The asymptotic behavior of the estimator under a high frequency assumption is investigated. 相似文献
11.
In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let $\{Y^{(a)}_{n}:n\ge1\}In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts.
For each a>0, let {Y(a)n:n 3 1}\{Y^{(a)}_{n}:n\ge1\} be a sequence of independent and identically distributed random variables and {X(a)t:t 3 0}\{X^{(a)}_{t}:t\ge0\} be a Lévy process such that X1(a)=dY1(a)X_{1}^{(a)}\stackrel{d}{=}Y_{1}^{(a)},
\mathbbEX1(a) < 0\mathbb{E}X_{1}^{(a)}<0 and
\mathbbEX1(a)-0\mathbb{E}X_{1}^{(a)}\uparrow0 as a↓0. Let S(a)n=?k=1n Y(a)kS^{(a)}_{n}=\sum _{k=1}^{n} Y^{(a)}_{k}. Then, under some mild assumptions, , for some random variable and some function Δ(⋅). We utilize this result to present a number of limit theorems for suprema of Lévy processes in the
heavy-traffic regime. 相似文献
12.
In this paper, we consider the Markov-modulated insurance risk model with tax. We assume that the claim inter-arrivals, claim sizes and premium process are influenced by an external Markovian environment process. The considered tax rule, which is the same as the one considered by Albrecher and Hipp [Blätter DGVFM 28(1):13–28, 2007], is to pay a certain proportion of the premium income, whenever the insurer is in a profitable situation. A system of differential equations of the non-ruin probabilities, given the initial environment state, are established in terms of the ruin probabilities under the Markov-modulated insurance risk model without tax. Furthermore, given the initial state, the differential equations satisfied by the expected accumulated discounted tax until ruin are also derived. We also give the analytical expressions for them by iteration methods. 相似文献
13.
For an arbitrary Lévy process X which is not a compound Poisson process, we are interested in its occupation times. We use a quite novel and useful approach to derive formulas for the Laplace transform of the joint distribution of X and its occupation times. Our formulas are compact, and more importantly, the forms of the formulas clearly demonstrate the essential quantities for the calculation of occupation times of X. It is believed that our results are important not only for the study of stochastic processes, but also for financial applications. 相似文献
14.
We study for a class of symmetric Lévy processes with state space R n the transition density pt(x) in terms of two one-parameter families of metrics, (dt)t>0 and (δt)t>0. The first family of metrics describes the diagonal term pt(0); it is induced by the characteristic exponent ψ of the Lévy process by dt(x, y) = 1/2tψ(x-y). The second and new family of metrics δt relates to 1/2tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: pt(x) = pt(0)e- δ2t (x,0) where pt(0) corresponds to a volume term related to tψ and where an "exponential" decay is governed by δ2t . This gives a complete and new geometric, intrinsic interpretation of pt(x). 相似文献
15.
In this paper, we study the Jarque–Bera test for a class of univariate parametric stochastic differential equations (SDE) dX t = b(X t , α)dt + dZ t , constructed based on the sample observed at discrete time points \({t^{n}_{i}=ih_{n}}\) , i = 1, 2, . . . , n, where Z is a nondegenerate Lévy process with finite moments and h is a sequence of positive real numbers with nh n → ∞ and \({nh_{n}^{2} \to 0}\) as n → ∞. It is shown that under proper conditions, the Jarque–Bera test statistic based on the Euler residuals can be used to test for the normality of the unobserved Z and the proposed test is consistent against the presence of any nontrivial jump components. Our result indicates that the Jarque–Bera test is easy to implement and asymptotically distribution-free with no fine-tuning parameters. Simulation results to validate the test are given for illustration. 相似文献
16.
This paper extends the model and analysis in that of Vandaele and Vanmaele [Insurance: Mathematics and Economics, 2008, 42:
1128–1137]. We assume that parameters of the Lévy process which models the dynamic of risky asset in the financial market
depend on a finite state Markov chain. The state of the Markov chain can be interpreted as the state of the economy. Under
the regime switching Lévy model, we obtain the locally risk-minimizing hedging strategies for some unit-linked life insurance
products, including both the pure endowment policy and the term insurance contract. 相似文献
17.
《Stochastic Processes and their Applications》2003,103(2):311-349
The Euler scheme is a well-known method of approximation of solutions of stochastic differential equations (SDEs). A lot of results are now available concerning the precision of this approximation in case of equations driven by a drift and a Brownian motion. More recently, people got interested in the approximation of solutions of SDEs driven by a general Lévy process. One of the problem when we use Lévy processes is that we cannot simulate them in general and so we cannot apply the Euler scheme. We propose here a new method of approximation based on the cutoff of the small jumps of the Lévy process involved. In order to find the speed of convergence of our approximation, we will use results about stability of the solutions of SDEs. 相似文献
18.
19.
Vladimir I. Chebotarev 《Journal of Theoretical Probability》1999,12(1):13-25
In 1984, Bentkus,(2, 3) and independently,Nagaev and Chebotarev,(11) introduced somemodifications of the well-known one-dimensional Cramér conditionlim sup|t| |E
e
it| < 1 for the case of Hilbert space (denote it by H). Although many H-valued random variables satisfy these conditions, there remained open the question if these conditionshold for the random variable X
0 = I
{ < x007D;–x ( is uniformly distributed on [0, 1]), whichtakes its values in H = L
2[0,1], and in a sense generates the so-called2-statistic. It is shown in the present paper thatX
0 does not satisfy the condition of Nagaev andChebotarev,(11) and also, at least partly, thecondition of Bentkus.(2, 3)
One more versionof the Cramér condition in the case of Hilbert space is suggestedin the paper not for individual summands, but for the whole sum. Ageneralization of the estimate of the characteristic function of squarednorm of the sum of independent truncations is obtained in terms ofquantities, which define this modification. It is shown thatX
0 satisfies this version of the Cramércondition. 相似文献
20.
Given a Lévy process \(\xi \), we find necessary and sufficient conditions for almost sure finiteness of the perpetual integral \(\int _0^\infty f(\xi _s)\hbox {d}s\), where \(f\) is a positive locally integrable function. If \(\mu =\mathbb {E}[\xi _1]\in (0,\infty )\) and \(\xi \) has local times we prove the 0–1 law with the exact characterization The proof uses spatially stationary Lévy processes, local time calculations, Jeulin’s lemma and the Hewitt–Savage 0–1 law.
相似文献
$$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )\in \{0,1\} \end{aligned}$$
$$\begin{aligned} \mathbb {P}\Big (\int _0^\infty f(\xi _s)\,\hbox {d}s<\infty \Big )=0\qquad \Longleftrightarrow \qquad \int ^\infty f(x)\,\hbox {d}x=\infty . \end{aligned}$$