Optimal dividend policy when risk reserves follow a jump–diffusion process with a completely monotone jump density under Markov-regime switching

The paper studies optimal dividend distribution for an insurance company whose risk reserves in the absence of dividends follow a Markov-modulated jump–diffusion process with a completely monotone jump density where jump densities and parameters including discount rate are modulated by a finite-state irreducible Markov chain. The major goal is to maximize the expected cumulative discounted dividend payments until ruin time when risk reserve is less than or equal to zero for the first time. I extend the results of Jiang (2015) for a Markov-modulated jump–diffusion process from exponential jump densities to completely monotone jump densities by proving that it is also optimal to take a modulated barrier strategy at some positive regime-dependent levels and that value function as the fixed point of a contraction is explicitly characterized.     

Change point estimators by local polynomial fits under a dependence assumption

We study a random design regression model generated by dependent observations, when the regression function itself (or its ν-th derivative) may have a change or discontinuity point. A method based on the local polynomial fits with one-sided kernels to estimate the location and the jump size of the change point is applied in this paper. When the jump location is known, a central limit theorem for the estimator of the jump size is established; when the jump location is unknown, we first obtain a functional limit theorem for a local dilated-rescaled version estimator of the jump size and then give the asymptotic distributions for the estimators of the location and the jump size of the change point. The asymptotic results obtained in this paper can be viewed as extensions of corresponding results for independent observations. Furthermore, a simulated example is given to show that our theory and method perform well in practice.     

Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-Homogeneous Jump Conditions

<正>In this paper,a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions.Simple non-body-fitted meshes are used.For homogeneous jump conditions,both non-conforming and conforming basis functions are constructed in such a way that they satisfy the natural jump conditions. For non-homogeneous jump conditions,a pair of functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface.With such a pair of functions,the discontinuities across the interface in the solution and flux are removed;and an equivalent elasticity interface problem with homogeneous jump conditions is formulated.Numerical examples are presented to demonstrate that such methods have second order convergence.     

沪深300仿真股指期货价格不对称跳跃波动的实证分析

本研究利用2006年10月30日至2009年3月13日期间的仿真的沪深300指数期货每日结算价,探讨了期货价格的不对称跳跃波动行为。在实证研究方法上,本文以Chan和Maheu的GARCH(1,1)-ARJI模型为基础并进行了扩展,以EGARCH(1,1)-CJI和EGARCH(1,1)-ARJI两种模型来刻画股指期货价格的不对称和跳跃波动行为。实证结果显示:(1)沪深300仿真股指期货价格存在不对称跳跃波动,而且跳跃强度不为一固定常数,异常信息所产生的跳跃强度是随着时间变动的。(2)经过似然比检验,结果显示EGARCH(1,1)-ARJI模型比EGARCH(1,1)-CJI模型具有更好的拟合能力。     

The jump operation for structure degrees

One of the main problems in effective model theory is to find an appropriate information complexity measure of the algebraic structures in the sense of computability. Unlike the commonly used degrees of structures, the structure degree measure is total. We introduce and study the jump operation for structure degrees. We prove that it has all natural jump properties (including jump inversion theorem, theorem of Ash), which show that our definition is relevant. We study the relation between the structure degree jump (in the sense of Soskov) and the jump degrees of a structure (in the sense of Jockusch) and give necessary and sufficient conditions for their existence in the terms of structure degrees. We show some properties, distinguishing the structure degrees from the enumeration degrees.     

Dam processes with state dependent batch sizes and intermittent production processes with state dependent rates

We consider a dam process with a general (state dependent) release rule and a pure jump input process, where the jump sizes are state dependent. We give sufficient conditions under which the process has a stationary version in the case where the jump times and sizes are governed by a marked point process which is point (Palm) stationary and ergodic. We give special attention to the Markov and Markov regenerative cases for which the main stability condition is weakened. We then study an intermittent production process with state dependent rates. We provide sufficient conditions for stability for this process and show that if these conditions are satisfied, then an interesting new relationship exists between the stationary distribution of this process and a dam process of the type we explore here.Supported in part by The Israel Science Foundation, grant no. 372/93-1.     

Rare Shock,Two-Factor Stochastic Volatility and Currency Option Pricing

Abstract

In this paper, we develop an option valuation model where the dynamics of the spot foreign exchange rate is governed by a two-factor Markov-modulated jump-diffusion process. The short-term fluctuation of stochastic volatility is driven by a Cox–Ingersoll–Ross (CIR) process and the long-term variation of stochastic volatility is driven by a continuous-time Markov chain which can be interpreted as economy states. Rare events are governed by a compound Poisson process with log-normal jump amplitude and stochastic jump intensity is modulated by a common continuous-time Markov chain. Since the market is incomplete under regime-switching assumptions, we determine a risk-neutral martingale measure via the Esscher transform and then give a pricing formula of currency options. Numerical results are presented for investigating the impact of the long-term volatility and the annual jump intensity on option prices.     

The detection and estimation of the change point in a disccrete-time stochastic system

A piecewise-constant process containing a single jump is observed under noise in the context of discrete time. The conditional density and maximum a posteriori (MAP) estimator of the jump time as well as the Bayes detector of the jump itself are determined using the powerful measure transformation approach. The Bayes detector provides a convenient sequential detection rule for practical on-line implementation. An asymptotic result for the distribution of the MAP estimator's estimation error and the corresponding convergence rate are derived. This result provides a reference measure of optimal performance for jump-time estimators in discrete-time stochastic systems that does not depend on the jump time's prior distribution     

On the notion of weak stability and related issues of hybrid diffusion systems

This work is concerned with the weak stability of hybrid diffusion systems, which consist of a number of diffusions modulated by a jump process. First, we show that when the jump component is nearly completely decomposable into a number of ergodic jump processes, the overall system is still weakly stable (or positive recurrent). Next we show that even if the process contains transient states, the positive recurrence can still be preserved. Then, we examine the asymptotic distribution when only one ergodic group of states is involved in the jump process and the state space of the continuous state belongs to a compact set. Our attention is devoted to asymptotic distribution in this case. The distribution is obtained by utilizing a spectrum gap property of the underlying process.     

Asymptotic Distribution of the Jump Change-Point Estimator

The asymptotic distribution of the change-point estimator in a jump changepoint model is considered.For the jump change-point model Xi =a + θI{[nTo] ＜ i ≤n} + εi,where εi (i =1,…,n) are independent ide...     

Mean field games with controlled jump–diffusion dynamics: Existence results and an illiquid interbank market model

We study a family of mean field games with a state variable evolving as a multivariate jump–diffusion process. The jump component is driven by a Poisson process with a time-dependent intensity function. All coefficients, i.e. drift, volatility and jump size, are controlled. Under fairly general conditions, we establish existence of a solution in a relaxed version of the mean field game and give conditions under which the optimal strategies are in fact Markovian, hence extending to a jump–diffusion setting previous results established in Lacker (2015). The proofs rely upon the notions of relaxed controls and martingale problems. Finally, to complement the abstract existence results, we study a simple illiquid inter-bank market model, where the banks can change their reserves only at the jump times of some exogenous Poisson processes with a common constant intensity, and provide some numerical results.     

Detecting derivative discontinuity locations in piecewise continuous functions from Fourier spectral data

We present a method that uses Fourier spectral data to locate jump discontinuities in the first derivatives of functions that are continuous with piecewise smooth derivatives. Since Fourier spectral methods yield strong oscillations near jump discontinuities, it is often difficult to distinguish true discontinuities from artificial oscillations. In this paper we show that by incorporating a local difference method into the global derivative jump function approximation, we can reduce oscillations near the derivative jump discontinuities without losing the ability to locate them. We also present an algorithm that successfully locates both simple and derivative jump discontinuities. This work was partially supported by NSF grants CNS 0324957 and DMS 0510813, and NIH grant EB 02553301 (AG).     

On some functionals of the first passage times in jump models of stochastic volatility

Abstract

We compute some functionals related to the generalized joint Laplace transforms of the first times at which two-dimensional jump processes exit half strips. It is assumed that the state space components are driven by Cox processes with both independent and common (positive) exponential jump components. The method of proof is based on the solutions of the equivalent partial integro-differential boundary-value problems for the associated value functions. The results are illustrated on several two-dimensional jump models of stochastic volatility which are based on non-affine analogs of certain mean-reverting or diverting diffusion processes representing closed-form solutions of the appropriate stochastic differential equations.     

A 3/2-approximation algorithm for the jump number of interval orders

The jump number of a partial order P is the minimum number of incomparable adjacent pairs in some linear extension of P. The jump number problem is known to be NP-hard in general. However some particular classes of posets admit easy calculation of the jump number.The complexity status for interval orders still remains unknown. Here we present a heuristic that, given an interval order P, generates a linear extension , whose jump number is less than 3/2 times the jump number of P.This work was supported by the Deutsche Forschungsgemeinschaft (DFG).     

The Immersed Interface Method for Simulating Two-Fluid Flows

We develop the immersed interface method (IIM) to simulate a two-fluid flow of two immiscible fluids with different density and viscosity. Due to the surface tension and the discontinuous fluid properties, the two-fluid flow has nonsmooth velocity and discontinuous pressure across the moving sharp interface separating the two fluids. The IIM computes the flow on a fixed Cartesian grid by incorporating into numerical schemes the necessary jump conditions induced by the interface. We present how to compute these necessary jump conditions from the analytical principal jump conditions derived in [Xu, DCDS, Supplement 2009, pp. 838-845]. We test our method on some canonical two-fluid flows. The results demonstrate that the method can handle large density and viscosity ratios, is second-order accurate in the infinity norm, and conserves mass inside a closed interface.     

有向跳图的生成欧拉子有向图

一个有向多重图D的跳图$J(D)$是一个顶点集为$D$的弧集,其中$(a,b)$是$J(D)$的一条弧当且仅当存在有向多重图$D$中的顶点$u_1$, $v_1$, $u_2$, $v_2$,使得$a=(u_1,v_1)$, $b=(u_2,v_2)$ 并且$v_1\neq u_2$.本文刻画了有向多重图类$\mathcal{H}_1$和$\mathcal{H}_2$,并证明了一个有向多重图$D$的跳图$J(D)$是强连通的当且仅当$D\not\in \mathcal{H}_1$.特别地, $J(D)$是弱连通的当且仅当$D\not\in \mathcal{H}_2$.进一步, 得到以下结果: (i) 存在有向多重图类$\mathcal{D}$使得有向多重图$D$的强连通跳图$J(D)$是强迹连通的当且仅当$D\not\in\mathcal{D}$. (ii) 每一个有向多重图$D$的强连通跳图$J(D)$是弱迹连通的,因此是超欧拉的. (iii) 每一个有向多重图D的弱连通跳图$J(D)$含有生成迹.     

Constrained MPC design of nonlinear Markov jump system with nonhomogeneous process

In this paper, a differential-inclusion-based MPC scheme is developed for the controller design for a discrete time nonlinear Markov jump system with nonhomogeneous transition probability. By adopting a differential-inclusion-based convex model predictive control mechanism, the nonlinear Markov jump system with nonhomogeneous transition probability is enclosed by a set of linear Markov jump systems. In this way, the controller design for the nonlinear Markov jump system can be solved via solving a set of linear Markov jump systems. Two numerical examples with different weighting parameters R are presented to illustrate the applicability of the results obtained.     

Linear correlation between fractal dimension of surface EMG signal from Rectus Femoris and height of vertical jump

Fractal dimension was demonstrated to be able to characterize the complexity of biological signals. The EMG time series are well known to have a complex behavior and some other studies already tried to characterize these signals by their fractal dimension.This paper is aimed at studying the correlation between the fractal dimension of surface EMG signal recorded over Rectus Femoris muscles during a vertical jump and the height reached in that jump.Healthy subjects performed vertical jumps at different heights. Surface EMG from Rectus Femoris was recorded and the height of each jump was measured by an optoelectronic motion capture system.Fractal dimension of sEMG was computed and the correlation between fractal dimension and eight of the jump was studied.Linear regression analysis showed a very high correlation coefficient between the fractal dimension and the height of the jump for all the subjects.The results of this study show that the fractal dimension is able to characterize the EMG signal and it can be related to the performance of the jump. Fractal dimension is therefore an useful tool for EMG interpretation.     

Subgraph distances in graphs defined by edge transfers

For two edge-induced subgraphs F and H of the same size in a graph G, the subgraph H can be obtained from F by an edge jump if there exist four distinct vertices u, v, w, and x in G such that uv ε E(F), wx ε E(G) - E(F), and H = F - uv + wx. The subgraph F is j-transformed into H if H can be obtained from F by a sequence of edge jumps. Necessary and sufficient conditions are presented for a graph G to have the property that every edge-induced subgraph of a fixed size in G can be j-transformed into every other edge-induced subgraph of that size. The minimum number of edge jumps required to transform one subgraph into another is called the jump distance. This distance is a metric and can be modeled by a graph. The jump graph J(G) of a graph G is defined as that graph whose vertices are the edges of G and where two vertices of J(G) are adjacent if and only if the corresponding edges of G are independent. For a given graph G, we consider the sequence {{J^k(G)}} of iterated jump graphs and classify each graph as having a convergent, divergent, or terminating sequence.     

Constrained LQ Problem with a Random Jump and Application to Portfolio Selection

This paper deals with a constrained stochastic linear-quadratic(LQ for short)optimal control problem where the control is constrained in a closed cone. The state process is governed by a controlled SDE with random coefficients. Moreover, there is a random jump of the state process. In mathematical finance, the random jump often represents the default of a counter party. Thanks to the It-Tanaka formula, optimal control and optimal value can be obtained by solutions of a system of backward stochastic differential equations(BSDEs for short). The solvability of the BSDEs is obtained by solving a recursive system of BSDEs driven by the Brownian motions. The author also applies the result to the mean variance portfolio selection problem in which the stock price can be affected by the default of a counterparty.