Jump Process Research Articles

Jumps in High-Frequency Data on the Chinese Stock Market

This study adopts two nonparametric methods, the activity signature function (ASF) and ratio analysis of cojumps, to test jumps in China’s stock market. Jumps in the stock price, stock-index futures, and volatility of China Securities Index (CSI) 300 index are analyzed using data on the continuous main-contract price of the index. The findings are as follows. First, in the long run, the CSI 300 index process is a continuous process exhibiting jumps at all sampling intervals. In the short run, the index becomes a pure-jump process in times of recession while exhibiting the characteristics of a continuous or even semimartingale process in certain intervals. Second, the stock-index futures process is a continuous process with jumps at all sampling intervals and, in the short run, exhibits the characteristics of a pure-jump process every 6 months. Moreover, the volatility process generally exhibits the characteristics of pure-jump processes. Third, the CSI 300 index price process and the continuous main-contract price process of the CSI 300 stock-index futures are significantly and positively related, with jumps occurring with a time lag of less than 5 minutes; by contrast, the volatility and price processes of the index are nonsignificantly related.

Mobility-Aware Coded Probabilistic Caching Scheme for MEC-Enabled Small Cell Networks

Caching on the edge has been recognized as an effective solution to tackle the backhaul constraint of network densification. However, most related works ignored user mobility in wireless networks, which is unreasonable under the background of network densification. For a more flexible and context-aware caching decision, the concept of caching on the edge can be extended to mobile edge computing (MEC) that enables computation and storage resources at mobile edge networks. With MEC servers deployed on base stations, a huge amount of collected radio access network context data can be analyzed and utilized to render a caching scheme adaptive to user’s context-aware information. In this regard, a novel mobility-aware coded probabilistic caching scheme is proposed for MEC-enabled small cell networks (SCNs). Different from previous mobility-aware caching schemes, user mobility and distributed storage are incorporated into a conventional probabilistic caching scheme, with the aim of throughput maximization. Based on stochastic geometry theory and a modified mobility model of discrete random jumps, the explicit expression of throughput is derived. Due to the complexity of the expression, two light-weight heuristic algorithms are provided to numerically obtain the optimal solutions. Moreover, a significant trade-off among the gains of mobility diversity, content diversity, and channel selection diversity is discussed, and we further numerically analyze how such a trade-off is influenced by user mobility, content popularity, and backhaul capacity, with some fundamental insights into the application of the proposed scheme in MEC-enabled SCNs. The superiority of our proposed scheme is demonstrated by the comparisons with the classical M most popular caching scheme and the conventional probabilistic caching scheme. Numerical results show that our proposed caching scheme achieves higher throughput than those of the other two, especially when users of intense mobility request contents, of which the popularity profile is not skewed, in MEC-enabled SCNs with poor backhaul capacity, indicating that the proposed caching scheme is a promising solution for network densification.

Numerical investigation of a simple model of human jumping on an oscillating structure

Cantilever grandstands are susceptible to vertical vibrations due to jumping or bobbing (vertical motion without loss of contact) of human occupants. Several major sports stadia have had vibration problems as a result. For bobbing, a simple mass-spring-damper-actuator model of the human (or group of humans) has been proposed and has been incorporated in the Institution of Structural Engineers guidelines on the dynamic performance of grandstands. For jumping, laboratory experiments have shown that in certain conditions regular periodic jumping close to the natural frequency of the supporting structure is not possible. However, there has previously been no model of the human-structure interaction in this situation. A fundamental difference from the bobbing case is that there are discontinuities between contact and non-contact phases of the motion. The behaviour is therefore highly nonlinear. This paper proposes a model similar to the human bobbing model but allowing for the loss of contact during jumping and the condition for landing, which is clearly influenced by the motion of the structure during the no-contact phase. The behaviour of the model is explored numerically. It is shown that in certain parameter ranges there are stable periodic solutions, but in others the periodic motion is unstable and instead the system exhibits chaotic motion. This could possibly explain the physical difficulty of regular periodic jumping close to the natural frequency of the supporting structure in the previous laboratory experiments.

A Survey of Bioinspired Jumping Robot: Takeoff, Air Posture Adjustment, and Landing Buffer.

A bioinspired jumping robot has a strong ability to overcome obstacles. It can be applied to the occasion with complex and changeable environment, such as detection of planet surface, postdisaster relief, and military reconnaissance. So the bioinspired jumping robot has broad application prospect. The jumping process of the robot can be divided into three stages: takeoff, air posture adjustment, and landing buffer. The motivation of this review is to investigate the research results of the most published bioinspired jumping robots for these three stages. Then, the movement performance of the bioinspired jumping robots is analyzed and compared quantitatively. Then, the limitation of the research on bioinspired jumping robots is discussed, such as the research on the mechanism of biological motion is not thorough enough, the research method about structural design, material applications, and control are still traditional, and energy utilization is low, which make the robots far from practical applications. Finally, the development trend is summarized. This review provides a reference for further research of bioinspired jumping robots.

High-Order Compact Finite Difference Scheme for Option Pricing in Stochastic Volatility Jump Models

We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility models. The scheme is fourth order accurate in space and second order accurate in time. Under some restrictions, theoretical results like unconditional stability in the sense of von Neumann are presented. Where the analysis becomes too involved we validate our findings by a numerical study. Numerical experiments for the European option pricing problem are presented. We observe fourth order convergence for non-smooth payoff.

Research on the Price Features of Oil Stochastic Model Based on the Continuous Jump Model

Aiming at calculating the price changes under the price features of oil stochastic model, the continuous jump model is proposed in this paper for data processing. The procedure is flexible, may be used with market prices of any oil contingent claim with closed form pricing solution, and easily deals with missing data problems. The results show that the accuracy can thus be improved overall the proposed system substantially.

Mathematical Formalism of Nonequilibrium Thermodynamics for Nonlinear Chemical Reaction Systems with General Rate Law

This paper studies a mathematical formalism of nonequilibrium thermodynamics for chemical reaction models with $N$ species, $M$ reactions, and general rate law. We establish a mathematical basis for J. W. Gibbs' macroscopic chemical thermodynamics under G. N. Lewis' kinetic law of entire equilibrium (detailed balance in nonlinear chemistry kinetics). In doing so, the equilibrium thermodynamics is then naturally generalized to nonequilibrium settings without detailed balance. The kinetic models are represented by a Markovian jumping process. A generalized macroscopic chemical free energy function and its associated balance equation with nonnegative source and sink are the major discoveries. The proof is based on the large deviation principle of this type of Markov processes. A general fluctuation dissipation theorem for stochastic reaction kinetics is also proved. The mathematical theory illustrates how a novel macroscopic dynamic law can emerges from the mesoscopic kinetics in a multi-scale system.

Speed of Vertex-Reinforced Jump Process on Galton–Watson Trees

We give an alternative proof of the fact that the vertex-reinforced jump process on Galton–Watson tree has a phase transition between recurrence and transience as a function of $$c$$ , the initial local time, see Basdevant et al. (Ann Appl Probab 22(4):1728–1743, 2012). Further, applying techniques in Aidekon (Probab Theory Relat Fields 142(3–4):525–559, 2008), we show a phase transition between positive speed and null speed for the associated discrete-time process in the transient regime.

Speed and Biases of Fourier-Based Pricing Choices: A Numerical Analysis

This paper compares the CPU effort and numerical biases of six Fourier-based implementations. Our analyses focus on two jump-related models that can consistently price options with different strikes and maturities: (i) the Bates jump-diffusion model, which combines jumps with stochastic volatility and (ii) the Asymmetric Variance Gamma (AVG) model, a pure-jump process where an infinite number of jumps can occur in any interval of time. We show that both truncation and discretization errors significantly increase as we move away from the diffusive Black-Scholes-Merton dynamics. While most pricing choices converge to the Bates reference values, Attari’s formula is the only Fourier-based method that does not completely blow up in any AVG problematic region. In terms of CPU speed, the strike vector computations proposed by Zhu (2010) significantly improve the computational burden, rendering the use of fast Fourier transforms and plain delta-probability decompositions inefficient.

Modeling of heat capacity peaks and enthalpy jumps of phase-change materials used for thermal energy storage

We consider three phase-change materials (PCMs) in which a change between two phases may be used to store/release thermal energy. Their enthalpy and heat capacity were measured in a quasistatic regime by adiabatic scanning calorimetry and show, within a certain temperature range, a single distinct jump and peak, respectively. We present a microscopic development from which the jumps and peaks can be accurately fitted and that could be analogously applied even to other PCMs. In addition, we determine the baseline and excess part of the heat capacity and thus the latent heat associated with the phase change. The development is based on the observation that PCMs often have polycrystalline structure, being composed of many single-crystalline grains. The enthalpy and heat capacity measured in experiments are therefore interpreted as superpositions of many contributions that come from the individual grains.

Stability and structural properties of gene regulation networks with coregulation rules

Coregulation of the expression of groups of genes has been extensively demonstrated empirically in bacterial and eukaryotic systems. Such coregulation can arise through the use of shared regulatory motifs, which allow the coordinated expression of modules (and module groups) of functionally related genes across the genome. Coregulation can also arise through the physical association of multi-gene complexes through chromosomal looping, which are then transcribed together. We present a general formalism for modeling coregulation rules in the framework of Random Boolean Networks (RBN), and develop specific models for transcription factor networks with modular structure (including module groups, and multi-input modules (MIM) with autoregulation) and multi-gene complexes (including hierarchical differentiation between multi-gene complex members). We develop a mean-field approach to analyse the dynamical stability of large networks incorporating coregulation, and show that autoregulated MIM and hierarchical gene-complex models can achieve greater stability than networks without coregulation whose rules have matching activation frequency. We provide further analysis of the stability of small networks of both kinds through simulations. We also characterize several general properties of the transients and attractors in the hierarchical coregulation model, and show using simulations that the steady-state distribution factorizes hierarchically as a Bayesian network in a Markov Jump Process analogue of the RBN model.

Distributional representations and dominance of a Lévy process over its maximal jump processes

Distributional identities for a Lévy process $X_{t}$, its quadratic variation process $V_{t}$ and its maximal jump processes, are derived, and used to make “small time” (as $t\downarrow0$) asymptotic comparisons between them. The representations are constructed using properties of the underlying Poisson point process of the jumps of $X$. Apart from providing insight into the connections between $X$, $V$, and their maximal jump processes, they enable investigation of a great variety of limiting behaviours. As an application, we study “self-normalised” versions of $X_{t}$, that is, $X_{t}$ after division by $\sup_{0<s\le t}\Delta X_{s}$, or by $\sup_{0<s\le t}\vert \Delta X_{s}\vert $. Thus, we obtain necessary and sufficient conditions for $X_{t}/\sup_{0<s\le t}\Delta X_{s}$ and $X_{t}/\sup_{0<s\le t}\vert \Delta X_{s}\vert $ to converge in probability to 1, or to $\infty$, as $t\downarrow0$, so that $X$ is either comparable to, or dominates, its largest jump. The former situation tends to occur when the singularity at 0 of the Lévy measure of $X$ is fairly mild (its tail is slowly varying at 0), while the latter situation is related to the relative stability or attraction to normality of $X$ at 0 (a steeper singularity at 0). An important component in the analyses is the way the largest positive and negative jumps interact with each other. Analogous “large time” (as $t\to\infty$) versions of the results can also be obtained.

Consistent pricing of VIX and equity derivatives with the 4/2 stochastic volatility plus jumps model

In this paper, we develop a 4/2 stochastic volatility plus jumps model, namely, a new stochastic volatility model including the Heston model and 3/2 model as special cases. Our model is highly tractable by applying the Lie symmetries theory for PDEs, which means that the pricing procedure can be performed efficiently. In fact, we obtain a closed-form solution for the joint Fourier–Laplace transform so that equity and realized-variance derivatives can be priced. We also employ our model to consistently price equity and VIX derivatives. In this process, the quasi-closed-form solutions for future and option prices are derived. Furthermore, through adopting data on daily VIX future and option prices, we investigate our model along with the Heston model and 3/2 model and compare their different performance in practice. Our result illustrates that the 4/2 model with an instantaneous volatility of the form (aVt+b/Vt) for some constants a,b presents considerable advantages in pricing VIX derivatives.

A water-walking robot mimicking the jumping abilities of water striders

The highly efficient and agile water-surface locomotion of water striders has attracted substantial research attention. Compared with imitating the horizontal rowing motion, imitating the jumping capability of water striders is much more challenging because the strong interaction in the jumping process easily causes the robot to sink. This study focuses on designing a miniature robot capable of continuously jumping on the water surface. A spring-based actuating mechanism is proposed to produce a large jumping force. The center of gravity of the robot is carefully designed to allow the robot to jump on the surface continuously and smoothly. The influences of several critical factors, including the area of the supporting legs, the spring stiffness, the jumping angle, etc on jumping ability are analyzed by means of dynamic simulation and experiments. The jumping performance under different jumping angles is tested. The fabricated robot weighs approximately 10.2 g and can continuously jump on water with a maximum leap height and length of 120 and 410 mm, respectively. This study helps researchers understand the jumping mechanism of water striders and provides a reference for developing water-jumping robots that can perform various aquatic tasks in the future.

Delay-dependent H∞ filtering for singular Markovian jump systems with general uncomplete transition probabilities

This paper is devoted to the investigation of the delay-dependent H∞ filtering problem for a kind of singular Markovian jump time-delay systems with general unknown transition probabilities. In this model, the transition rates of the jumping process are assumed to be partly available, that is, some elements have been exactly known, some ones have been merely known with lower and upper bounds, others may have no information to use. Using the Lyapunov functional theory, a stochastically stable filter is designed to guarantee both the mean-square exponential admissibility and a prescribed level of H∞ performance for the singular Markovian jump time-delay systems with general unknown transition probabilities. A sufficient condition is derived for the existence of such a desired filter in terms of linear matrix inequalities (LMIs). A numerical example is provided to demonstrate the effectiveness of the proposed theory.

Stochastic Stabilization of Discrete-Time Markov Jump Systems with Generalized Delay and Deficient Transition Rates

In this paper, the problem of mode-dependent state feedback controller design is studied for discrete-time Markov jump systems with generalized delay and deficient transition rates. The time delay under consideration is subject to mode-dependent and time-varying, and the transition probabilities of the jumping process are assumed to be partially accessible. By utilizing some novel summation inequalities, and by constructing an improved Lyapunov---Krasovskii functional which comprises a mode-dependent quadratic functional and some single, double and triple summation terms, delay-dependent stabilization conditions are derived in terms of tractable linear matrix inequalities. Two numerical examples are given to illustrate the effectiveness of the proposed conditions.

A general control variate method for multi-dimensional SDEs: An application to multi-asset options under local stochastic volatility with jumps models in finance

This paper presents a new control variate method for general multi-dimensional stochastic differential equations (SDEs) including jumps in order to reduce the variance of Monte Carlo method. Our control variate method is based on an asymptotic expansion technique, and does not require an explicit characteristic function of SDEs. This is an extension of previous researches using asymptotic expansions to obtain the control variates for such general models. Moreover, in our control variate method, the regression estimators can be chosen for each number of jump times with a stratified sampling, and improve the efficiency of the variance reduction. This paper also provides the asymptotic bias and variance of our method in terms of its terminal time and a small noise parameter used in an asymptotic expansion method.For an application of our method, we evaluate multi-asset options whose payoffs are expressed as linear functions of the underlying asset price processes in general local stochastic volatility with jumps models, and show a calculation scheme of control variates for Greeks.In numerical experiments, we apply the new control variate method to pricing basket, spread, and average options and Delta of basket options under a ZABR-type local stochastic volatility model with jumps, and confirm that our method works very well.

Stochastic volatility of the futures prices of emission allowances: A Bayesian approach

Abstract Understanding the stochastic nature of the spot volatility of emission allowances is crucial for risk management in emissions markets. In this study, by adopting a stochastic volatility model with or without jumps to represent the dynamics of European Union Allowances (EUA) futures prices, we estimate the daily volatilities and model parameters by using the Markov Chain Monte Carlo method for stochastic volatility (SV), stochastic volatility with return jumps (SVJ) and stochastic volatility with correlated jumps (SVCJ) models. Our empirical results reveal three important features of emissions markets. First, the data presented herein suggest that EUA futures prices exhibit significant stochastic volatility. Second, the leverage effect is noticeable regardless of whether or not jumps are included. Third, the inclusion of jumps has a significant impact on the estimation of the volatility dynamics. Finally, the market becomes very volatile and large jumps occur at the beginning of a new phase. These findings are important for policy makers and regulators.

An Arithmetic Pure-Jump Multi-Curve Interest Rate Model

We propose an innovative multi-curve model involving interest rates and (ordered) spreads which are modeled by arithmetic martingale processes being larger than some arbitrarily chosen constant. Under our mean-reverting pure-jump approach, we derive tractable martingale representations for the OIS rate, the spread as well as the LIBOR rate and provide analytical caplet price formulae. In a second part, we introduce an extended jump-diffusion version of our model and investigate hedging and the computation of Greeks under this new specification. As a by-product, we infer the related arithmetic pure-jump single-curve model. We finally consider the modeling of future information in multi-curve interest rate markets by enlarged filtrations and deduce the related OIS and LIBOR rate representations as well as the corresponding information premium.

Focal Plane Shift Imaging for the Analysis of Dynamic Wetting Processes

Droplet-surface interactions are common to a plethora of natural and industrial processes due to their ability to rapidly exchange energy, mass, and momentum. Droplets are particularly of interest due to their large surface-to-volume ratios and hence enhanced transport properties. For example, coalescence-induced droplet jumping on superhydrophobic surfaces has recently received much attention for its potential to enhance heat transfer, anti-icing, and self-cleaning performance by passively shedding microscale water droplets. To study droplet jumping, researchers typically use a two-camera setup to observe the out-of-plane droplet motion, with limited success due to the inability to resolve the depth dimension using two orthogonal cameras. Here we develop a single-camera technique capable of providing three-dimensional (3D) information through the use of focal plane manipulation, termed "focal plane shift imaging" (FPSI). We used FPSI to study the jumping process on superhydrophobic surfaces having a wide range of structure length scales (10 nm < l < 1 μm) and droplet radii (3 μm < R < 160 μm). We benchmarked the FPSI technique and studied the effects of droplet mismatch, multidroplet coalescence, and multihop coalescence on droplet jumping speed. Furthermore, we were able to resolve the full 3D trajectory of multiple jumping events, to show that, unlike previously theorized, the departure angle during droplet jumping is not a function of droplet mismatch or number of droplets coalescing prior to jumping. Rather, angular deviation arises due to in-plane motion postcoalescence governed by droplet pinning. The outcomes of this work both elucidate key fundamental aspects governing droplet jumping and provide a powerful imaging platform for the study of dynamic droplet processes that result in out-of-plane motion such as sliding, coalescence, or impact.