Levy Process Research Articles

On Fractional Lévy Processes: Tempering, Sample Path Properties and Stochastic Integration

We define two new classes of stochastic processes, called tempered fractional Levy process of the first and second kinds (TFLP and TFLP II, respectively). TFLP and TFLP II make up very broad finite-variance, generally non-Gaussian families of transient anomalous diffusion models that are constructed by exponentially tempering the power law kernel in the moving average representation of a fractional Levy process. Accordingly, the increment processes of TFLP and TFLP II display semi-long range dependence. We establish the sample path properties of TFLP and TFLP II. We further use a flexible framework of tempered fractional derivatives and integrals to develop the theory of stochastic integration with respect to TFLP and TFLP II, which may not be semimartingales depending on the value of the memory parameter and choice of marginal distribution.

Empirical likelihood method for longitudinal data generated from unequally-spaced Lèvy processes

By introducing the notion of “empirical likelihood function of observing sums”, unequally-spaced time series data and longitudinal data generated from Levy processes can be analyzed. Characteristic function is further incorporated to handle the situations where the density function of the increments is difficult to obtain. In the situations where both characteristic function and the density function are available, it is shown through the simulation examples that the proposed empirical maximum likelihood method does not suffer from significant information loss comparing to the maximum likelihood estimation method. The performances of the proposed method is tested for both equally-spaced and unequally-spaced observations.

Methods of simulation mathematical modeling of the Russian derivatives market in modern times

Introduction. The paper is devoted to simulation modeling. Basic methods of the simulation mathematical modeling in the derivatives market are described. A group of realistic nonGaussian Levy processes that generalize the classical BlackScholes model is considered. The work objective is to study the most efficient methods of market forecasting, as well as the software implementation of the simulation mathematical modeling technique of the Russian derivatives market based on the Levy model. This research is relevant due to the demand for applications that simulate the dynamics of financial assets and evaluate options in realistic models of the derivatives market, allowing for jumps.Materials and Methods. Basic methods for forecasting the derivatives market, methods for determining the volatility rate at a known option price, are considered. The most effective types of Levy processes for the simulation mathematical modeling of the Russian derivatives market at the present stage are highlighted. The possibilities of the Java language for the implementation of mathematical methods are considered.Research Results. A program is developed in the Java programming language that implements the Levy mathematical model, which includes Gaussian and generalized Poisson processes. The program for calculating the mathematical method is created in the free integrated application development environment NetBeans IDE to work with any operating system.Discussion and Conclusions. The result of the simulation mathematical modeling analysis has shown that the most efficient methods in the derivatives market are those based on realistic non-Gaussian Levy processes. The software implementation of such mathematical methods can be used for educational purposes. The developed application has demonstrated high quality and speed of calculations using software resources.

Darling–Erdős theorem for Lévy processes at zero

Darling–Erdős theorem for Lévy processes at zero

On the Ruin Problem with Investment When the Risky Asset Is a Semimartingale

In this paper, we study the ruin problem with investment in a general framework where the business part X is a Levy process and the return on investment R is a semimartingale. We obtain upper bounds on the finite and infinite time ruin probabilities that decrease as a power function when the initial capital increases. When R is a Levy process, we retrieve the well-known results. Then, we show that these bounds are asymptotically optimal in the finite time case, under some simple conditions on the characteristics of X. Finally, we obtain a condition for ruin with probability one when X is a Brownian motion with negative drift and express it explicitly using the characteristics of R.

Evaluation of energy forward curves with jumps under the general Levy process

In this study, we evaluate the relationship between the forward rates and the future delivery period with the consideration of the Levy process for a time-inhomogeneous exponential jump-diffusion process and model the forward curve. This is a large variety of stylized features observed in the Samuelson effect of increasing volatilities close to maturity. However, a new method based on characteristic functions is used to estimate the jump component in a finite-activity Levy process, which includes the jump frequency and the jump size distribution which enables ´ the further investigation of the properties of estimators without the presence of high frequency data ∆. Numerical implementation of the approach was applied on sample electricity data of about 10,000 observations between the period of 2 years and then a seasonalized forecast for an extra year was implemented to normalize the volatility in forwards contracts.

Modelling Covid-19 Deaths in Ghana as a Discrete State Process in Continuous Time

We propose a stochastic process modelling of covid-19 deaths in Ghana. The objective is to accurately capture the death processes resulting from the pandemic and to predict future deaths resulting from Covid-19 infections in Ghana. The mathematical derivation is based strictly on the compound Poisson process, a class of a Levy process. The model is verified by using empirical data of deaths resulting from Covid-19 from the onset of the pandemic up to the time of writing this report. That is, Covid-19 deaths in Ghana from March to August 2020. The method departs slightly from the usual differential equations used in modeling pandemics due to the unique occurrence of deaths from the disease in Ghana. As the methods are basically compound Poisson process, we delve into Levy processes as it allows us to effectively simulate the future behaviour of the death process. To test the effectiveness of the model, we compared the simulated results to the actual reported number of deaths from Covid-19 cases in Ghana from March to August 2020. The results show that at a 95% confidence interval there is no significant difference between the actual deaths and the simulated results. The results of the simulation, when extended to February 2021 (one year after the advent of the pandemic) shows that if the current conditions remain same, that is, if there is no immediate intervention by the discovery of an effective drug or a vaccine, then the number of deaths could reach four hundred and forty six (446) by February 28, 2020.We propose a stochastic process modelling of covid-19 deaths in Ghana. The objective is to accurately capture the death processes resulting from the pandemic and to predict future deaths resulting from Covid-19 infections in Ghana. The mathematical derivation is based strictly on the compound Poisson process, a class of a Levy process. The model is verified by using empirical data of deaths resulting from Covid-19 from the onset of the pandemic up to the time of writing this report. That is, Covid-19 deaths in Ghana from March to August 2020. The method departs slightly from the usual differential equations used in modeling pandemics due to the unique occurrence of deaths from the disease in Ghana. As the methods are basically compound Poisson process, we delve into Levy processes as it allows us to effectively simulate the future behaviour of the death process. To test the effectiveness of the model, we compared the simulated results to the actual reported number of deaths from Covid-19 cases in Ghana from March to August 2020. The results show that at a 95% confidence interval there is no significant difference between the actual deaths and the simulated results. The results of the simulation, when extended to February 2021 (one year after the advent of the pandemic) shows that if the current conditions remain same, that is, if there is no immediate intervention by the discovery of an effective drug or a vaccine, then the number of deaths could reach four hundred and forty six (446) by February 28, 2020.

An Analytical Valuation Framework for Financial Assets with Trading Suspensions

In this paper we propose a derivative valuation framework based on Levy processes which takes into account the possibility that the underlying asset is subject to information-related trading halts/...

Weak symmetries of stochastic differential equations driven by semimartingales with jumps

Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general cadlag semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by Levy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of affine iterated random maps as well as symmetries for numerical schemes (of Euler and Milstein type) for Brownian motion driven SDEs.

Levy Processes Simulation for Modeling in Bioinformatics

This article discusses the capabilities of the R language for modeling Levy processes, processes that currently most closely correspond to the nature of the evolution of stock price movements. The efficient algorithm of the CGMY process simulation as a difference of the tempered stable independent Levy is processed and programmed with the R language. The efficient algorithm of variance gamma process simulation using variance gamma random variables is processed and programmed with the R language. The article is focused on an entirely new area relevant to the scope of the International Journal of Applied Research in Bioinformatics (IJARB).

Cut-off phenomenon for Ornstein-Uhlenbeck processes driven by Lévy processes

In this paper, we study the cut-off phenomenon under the total variation distance of $d$-dimensional Ornstein-Uhlenbeck processes which are driven by Levy processes. That is to say, under the total variation distance, there is an abrupt convergence of the aforementioned process to its equilibrium, i.e. limiting distribution. Despite that the limiting distribution is not explicit, its distributional properties allow us to deduce that a profile function always exists in the reversible cases and it may exist in the non-reversible cases under suitable conditions on the limiting distribution. The cut-off phenomena for the average and superposition processes are also determined.

Ожидаемые обратные дважды стохастические дифференциальные уравнения, основанные на мартингалах Тюгеля с отражающим барьером или без него

We are interested in this paper on reflected anticipated backward doubly stochastic differential equations (RABDSDEs) driven by teugels martingales associated with Levy process. We obtain the existence and uniqueness of solutions to these equations by means of the fixed-point theorem where the coefficients of these BDSDEs depend on the future and present value of the solution $\left( Y,Z\right)$. We also show the comparison theorem for a special class of RABDSDEs under some slight stronger conditions. The novelty of our result lies in the fact that we allow the time interval to be infinite.

The stochastic Cauchy problem driven by a cylindrical Lévy process

In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Levy process. Our approach requires to establish a stochastic Fubini result for stochastic integrals with respect to cylindrical Levy processes. This approach enables us to conclude that the solution process has almost surely scalarly square integrable paths. Further properties of the solution such as the Markov property and stochastic continuity are derived.

Asymptotic Expansion for the Transition Densities of Stochastic Differential Equations Driven by the Gamma Processes

In this paper, enlightened by the asymptotic expansion methodology developed by Li (2013b) and Li and Chen (2016), we propose a Taylor-type approximation for the transition densities of the stochastic differential equations (SDEs) driven by the gamma processes, a special type of Levy processes. After representing the transition density as a conditional expectation of Dirac delta function acting on the solution of the related SDE, the key technical method for calculating the expectation of multiple stochastic integrals conditional on the gamma process is presented. To numerically test the efficiency of our method, we examine the pure jump Ornstein-Uhlenbeck (OU) model and its extensions to two jump-diffusion models. For each model, the maximum relative error between our approximated transition density and the benchmark density obtained by the inverse Fourier transform of the characteristic function is suffciently small, which shows the efficiency of our approximated method.

Generalized scale functions of standard processes with no positive jumps

As a generalization of scale functions of spectrally negative Levy processes, we define generalized scale functions of general standard processes with no positive jumps. For this purpose, we utilize the excursion theory. Using the generalized scale functions, we study Laplace transforms of hitting times, potential measures and duality.

On distributions of exponential functionals of the processes with independent increments

The aim of this paper is to study the laws of the exponential functionals of the processes X with independent increments , namely I t = t 0 exp(−X s)ds, t ≥ 0, and also I ∞ = ∞ 0 exp(−X s)ds. Under suitable conditions we derive the integro-differential equations for the density of I t and I ∞. We give sufficient conditions for the existence of smooth density of the laws of these function-als. In the particular case of Levy processes these equations can be simplified and, in a number of cases, solved explicitly.

Expectation Anchoring and Brownian Motion in Financial Markets

I use coarse Bayesian updating to explain three stylized facts: trading anomalies around major stock market milestones, excess price volatility and trade volume, and heavy-tailed prices. When expectations are coarse and traders are heterogeneous, traders make heterogeneous expectation errors. This disagreement generates substantial trade volume and I show that the ensuing price discovery process converges in distribution to an empirically relevant class of Levy processes. Notably, I obtain this result without any exogenous shocks or processes; the only exogenous random variable is a heterogeneity parameter drawn at time zero. I then establish a perfectly non-revealing equilibrium: because traders are different, their aggregated trades generate volatile prices, yet this volatility impedes traders from learning their differences.

Optimisation-based representations for branching processes

It is shown that a certain functional of a branching process has representations in terms of both a maximisation problem and a minimisation problem. A consequence of these representations is that upper and lower bounds on the functional can be found easily, yielding a non-asymptotic Trotter product formula. As an application, the speed of the right-most particle of a branching Levy process is calculated.

Fat Tails Arise Endogenously in Asset Prices From Supply/Demand, With or Without Jump Processes

We show that the quotient of Levy processes of jump-diffusion type has a fat-tailed distribution. An application is to price theory in economics, with the result that fat tails arise endogenously from modeling of price change based on an excess demand analysis resulting in a quotient of arbitrarily correlated demand and supply whether or not jump discontinuities are present. The assumption is that supply and demand are described by drift terms, Brownian (i.e., Gaussian) and compound Poisson jump processes. If P^(−1)dP/dt (the relative price change in an interval dt) is given by a suitable function of relative excess demand, (D − S) /S (where D and S are demand and supply), then the distribution has tail behavior F (x) ∼ x(−ζ) for a power ζ that depends on the function G in P^(−1)dP/dt = G (D/S). For G (x) ∼ |x|^(1/q) one has ζ = q. The empirical data for assets typically yields a value, ζ=˜ 3, or ζ ∈ [3, 5] for many financial markets. Many theoretical explanations have been offered for the disparity between the tail behavior of the standard asset price equation and empirical data. This issue never arises if one models price dynamics using basic economics methodology, i.e., generalized Walrasian adjustment, rather than the usual starting point for classical finance which assumes a normal distribution of price changes. The function G is deterministic, and can be calibrated with a smaller data set. The results establish a simple link between the decay exponent of the density function and the price adjustment function, a feature that can improve methodology for risk assessment. The mathematical results can be applied to other problems involving the relative difference or quotient of Levy processes of jump-diffusion type.

Game theoretic valuation of deposit insurance under jump risk: from too small to survive to too big to fail

This study examines the valuation problem in deposit insurance as a game option between the deposit insurer and the insured bank with asymmetric bankruptcy costs. The asset-to-deposit ratio of the insured bank is modeled as an exponential Levy process with a spectrally negative jump. The study examines a wide range of scenarios in which the optimal closure policies of both parties are fully characterized. Explicit solutions are derived under the exponential jump diffusion case. This model captures several important issues in banking supervision, including the too big to fail and too small to survive phenomena, bank reorganization, and regulatory forbearance.