Two-sided Jumps Research Articles

Dividend problem of an investment risk model under random observation

<p>We mainly studied the dividend payout with a two-sided jumps risk model under random observation. The two-sided jumps in the model represent random claims and random returns. First, we obtained the integral differential equation of the expected dividend under the boundary conditions. Because the equations cannot be solved directly under normal circumstances, we chose the sinc numerical method here to approximate the solution of the equations. Then the error analysis of the approximate solution was carried out to illustrate the rationality of the numerical method. Finally, some concrete numerical examples were given.</p>

Penalization method for reflected BDSDEs with two-sided jumps and driven by Lévy process

In this paper, we prove the existence and uniqueness of the solution to reflected backward doubly stochastic differential equations driven by Teugels martingales associated with a Lévy process where the barrier process is not necessarily right continuous by approximating such equations by a new version of penalization method.

Estimating the Gerber–Shiu Function in the Two-Sided Jumps Risk Model by Laguerre Series Expansion

In this paper, we consider an insurance risk model with two-sided jumps, where downward and upward jumps typically represent claim amounts and random gains, respectively. We use the Laguerre series to expand the Gerber–Shiu function and estimate it based on observed information. Moreover, we show that the estimator is easily computed and has a fast convergence rate. Numerical examples are also provided to show the efficiency of our method when the sample size is finite.

Numerical Method for a Risk Model with Two-Sided Jumps and Proportional Investment

In this paper, we consider a risk model with two-sided jumps and proportional investment. The upward jumps and downward jumps represent gains and claims, respectively. Suppose the company invests all of its surplus in a certain proportion in two types of investments, one is risk-free (such as bank accounts) and the other is risky (such as stocks). Our aim is to find the optimal admissible strategy (including the optimal dividend rate and the optimal ratio of investment in risky assets), to maximize the dividend value function, and discuss the effects of a number of parameters on dividend payments. Firstly, the HJB equation of the dividend value function is obtained by the stochastic analysis theory and the dynamic programming method, and the optimal admissible strategy is obtained. Since the integro-differential equation satisfied by the dividend value function is difficult to solve, we turn to the sinc numerical method to approximate solve it. Then, the error between the exact solution (ES) and the sinc approximate solution (SA) is analyzed. Finally, the relative error of a special numerical solution and an ES is given, and some examples of sensitivity analysis are discussed. This study provides a theoretical basis for insurance companies to prevent risks better.

The two-barrier escape problem for compound renewal processes with two-sided jumps

We consider the problem of determining two-sided exit probabilities for a compound renewal process with drift and two-sided jumps. In certain cases the problem can be reduced to determining the distribution of a random sum of i.i.d. random variables. In a general situation this problem is reduced to solving a certain integral equation. We obtain explicit expressions for the escape probability for several classes of jump times and sizes.

Two-sided jumps risk model with proportional investment and random observation periods

<abstract><p>In this paper, we consider a two-sided jumps risk model with proportional investments and random observation periods. The downward jumps represent the claim while the upward jumps represent the random returns. Suppose an insurance company invests all of their surplus in risk-free and risky investments in proportion. In real life, corporate boards regularly review their accounts rather than continuously monitoring them. Therefore, we assume that insurers regularly observe surplus levels to determine whether they will ruin and that the random observation periods are exponentially distributed. Our goal is to study the Gerber-Shiu function (i.e., the expected discounted penalty function) of the two-sided jumps risk model under random observation. First, we derive the integral differential equations (IDEs) satisfied by the Gerber-Shiu function. Due to the difficulty in obtaining explicit solutions for the IDEs, we utilize the sinc approximation method to obtain the approximate solution. Second, we analyze the error between the approximate and explicit solutions and find the upper bound of the error. Finally, we discuss examples of sensitivity analysis.</p></abstract>

Gerber-Shiu Function for a Class of Markov-Modulated Lévy Risk Processes with Two-Sided Jumps

Gerber-Shiu Function for a Class of Markov-Modulated Lévy Risk Processes with Two-Sided Jumps

Ruin-related problems in the dual risk model under two different randomized observations

In this article, the dual risk model with two-sided jumps and two different randomized observations is considered. The dividend observation and ruin observation are supervised by two departments respectively. While in practice, the financial position of a company is usually monitored frequently, dividend decisions are only made periodically along with the publication of its books. So there are two situations. First, dividend observation and ruin observation are independent of each other, under this circumstance, we researched the integral-differential equation of the expected discounted dividend function until ruin. Second, dividend decision time is a multiple of ruin observation time. We deduced the expected discounted dividend function until ruin. Moreover, numerical analyses are provided to illustrate our results.

Numerical Method for a Compound Poisson Risk Model with Two-Sided Jumps and Proportional Investment

Numerical Method for a Compound Poisson Risk Model with Two-Sided Jumps and Proportional Investment

On Reflection with Two-Sided Jumps

In the first part of this paper, we pose and solve a new version of Skorokhod problem with jumps, reflected in the half line $$\mathbb {R}^{+}$$ and associated to a right limited and left limited function. We show the existence and uniqueness of the solution for this problem. Then, we apply our result to prove existence and uniqueness of solutions of reflected stochastic differential equations driven by optional semimartingales on an unusual probability space.

The Expected Discounted Penalty Function in the Generalized Erlang (n) Risk Model with Two-Sided Jumps and a Constant Dividend Barrier

In this paper, the generalized Erlang(n) risk model with two-sided jumps and a constant dividend barrier is considered. We assume that the downward jump sizes follow an arbitrary distribution and the upward jump sizes follow the mixed Erlang distribution. An integro-differential equation with boundary conditions for the expected discounted penalty function is derived and the solution is provided. The defective renewal equation for the expected discounted penalty function with no barrier is derived. We also give an example to obtain the expression of the expected discounted penalty function when the claim amounts are exponentially distributed.

Analytic value function for a pairs trading strategy with a Lévy-driven Ornstein–Uhlenbeck process

This paper studies the performance of pairs trading strategy under a specific spread model. Based on the empirical evidence of mean reversion and jumps in the spread between pairs of stocks, we assume that the spread follows a Lévy-driven Ornstein–Uhlenbeck process with two-sided jumps. To evaluate the performance of a pairs trading strategy, we propose the expected return per unit time as the value function of the strategy. Significantly different from the current related works, we incorporate an excess jump component into the calculation of return and time cost. Further, we obtain the analytic expression of strategy value function, where we solve out the probabilities of crossing thresholds via the Laplace transform of first passage time of the Lévy-driven Ornstein–Uhlenbeck process in one-sided and two-sided exit problems. Through numerical illustrations, we calculate the value function and optimal thresholds for a spread model with symmetric jumps, reveal the non-negligible contribution of incorporating the excess jumps into the value function, and analyze the impact of model parameters on the strategy performance.

The Erlang(n) risk model with two-sided jumps and a constant dividend barrier

In this paper, the Erlang(n) risk model with two-sided jumps and a constant dividend barrier is considered. In the analysis of the expected discounted penalty function, the downward jumps are assumed to have an arbitrary distribution function and the upward jumps are assumed to be exponentially distributed. An integro-differential equation with boundary conditions for the expected discounted penalty function is derived and the solution is provided. The defective renewal equation for the expected discounted penalty function with no barrier is derived. In the analysis of the moments of the discounted dividend payments until ruin, we assume that the inter-jump times are generalized Erlang(n) distributed. An integro-differential equation for the mth moment function of the discounted sum of dividend payments until ruin is derived. Numerical examples are also given to obtain the expressions for the expected discounted penalty function and the expected present value of dividend payments.

Phase-type approximations perturbed by a heavy-tailed component for the Gerber-Shiu function of risk processes with two-sided jumps

We consider in this paper a risk reserve process where the claims and gains arrive according to two independent Poisson processes. While the gain sizes are phase-type distributed, we assume instead that the claim sizes are phase-type perturbed by a heavy-tailed component; that is, the claim size distribution is formally chosen to be phase-type with large probability and heavy-tailed with small probability We analyze the seminal Gerber-Shiu function coding the joint distribution of the time to ruin, the surplus immediately before ruin, and the deficit at ruin. We derive its value as an expansion with respect to powers of with known coefficients and we construct approximations from the first two terms of the aforementioned series. The main idea is based on the so-called fluid embedding that allows to put the considered risk process into the framework of spectrally negative Markov-additive processes and use its fluctuation theory developed in Ivanovs and Palmowski 2012.

Completely asymmetric stable processes conditioned to avoid an interval

AbstractIn a recent article, Döring et al. (2018) conditioned a stable process with two-sided jumps to avoid an interval. As usual, the strategy was to find an invariant function for the process killed on entering the interval and to show that the corresponding h-transformed process is indeed the process conditioned to avoid an interval in a meaningful way. In the present article we consider the case of a completely asymmetric stable process. It turns out that the invariant function found by Döring et al. does not exist or is not invariant, but nonetheless we will characterise the conditioned process as a Markov process.

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.

Extinction Time of Non-Markovian Self-Similar Processes, Persistence, Annihilation of Jumps and the Fréchet Distribution

We start by providing an explicit characterization and analytical properties, including the persistence phenomena, of the distribution of the extinction time $\mathbb{T}$ of a class of non-Markovian self-similar stochastic processes with two-sided jumps that we introduce as a stochastic time-change of Markovian self-similar processes. For a suitably chosen time-changed, we observe, for classes with two-sided jumps, the following surprising facts. On the one hand, all the $\mathbb{T}$'s within a class have the same law which we identify in a simple form for all classes and reduces, in the spectrally positive case, to the Fr\'echet distribution. On the other hand, each of its distribution corresponds to the law of an extinction time of a single Markov process without positive jumps, leaving the interpretation that the time-changed has annihilated the effect of positive jumps. The example of the non-Markovian processes associated to L\'evy stable processes is detailed.

A semi-parametric method for estimating the beta coefficients of the hidden two-sided asset return jumps

ABSTRACTWe introduce a new methodology for estimating the parameters of a two-sided jump model, which aims at decomposing the daily stock return evolution into (unobservable) positive and negative jumps as well as Brownian noise. The parameters of interest are the jump beta coefficients which measure the influence of the market jumps on the stock returns, and are latent components. For this purpose, at first we use the Variance Gamma (VG) distribution which is frequently used in modeling financial time series and leads to the revelation of the hidden market jumps' distributions. Then, our method is based on the central moments of the stock returns for estimating the parameters of the model. It is proved that the proposed method provides always a solution in terms of the jump beta coefficients. We thus achieve a semi-parametric fit to the empirical data. The methodology itself serves as a criterion to test the fit of any sets of parameters to the empirical returns. The analysis is applied to NASDAQ and Google returns during the 2006–2008 period.

Numerical Method for the Perturbed Risk Model with Randomized Observation and Two-sided Jumps

Numerical Method for the Perturbed Risk Model with Randomized Observation and Two-sided Jumps

Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps

This paper considers a renewal insurance risk model with two-sided jumps (e.g. Labbé et al., 2011), where downward and upward jumps typically represent claim amounts and random gains respectively. A generalization of the Gerber–Shiu expected discounted penalty function (Gerber and Shiu, 1998) is proposed and analyzed for sample paths leading to ruin. In particular, we shall incorporate the joint moments of the total discounted costs associated with claims and gains until ruin into the Gerber–Shiu function. Because ruin may not occur, the joint moments of the total discounted claim costs and gain costs are also studied upon ultimate survival of the process. General recursive integral equations satisfied by these functions are derived, and our analysis relies on the concept of ‘moment-based discounted densities’ introduced by Cheung (2013). Some explicit solutions are obtained in two examples under different cost functions when the distribution of each claim is exponential or a combination of exponentials (while keeping the distributions of the gains and the inter-arrival times between successive jumps arbitrary). The first example looks at the joint moments of the total discounted amounts of claims and gains whereas the second focuses on the joint moments of the numbers of downward and upward jumps until ruin. Numerical examples including the calculations of covariances between the afore-mentioned quantities are given at the end along with some interpretations.