Levy Process Research Articles

Self-Standardized Central Limit Theorems for Trimmed Lévy Processes

We prove under general conditions that a trimmed subordinator satisfies a self-standardized central limit theorem (SSCLT). Our basic tool is a powerful distributional approximation result of Zaitsev (Probab Theory Relat Fields 74:535–566, 1987). Among other results, we obtain as special cases of our subordinator result the recent SSCLTs of Ipsen et al. (Stoch Process Appl 130:2228–2249, 2020) for trimmed subordinators and a trimmed subordinator analog of a central limit theorem of Csorgő et al. (Probab Theory Relat Fields 72:1–16, 1986) for intermediate trimmed sums in the domain of attraction of a stable law. We then use our methods to prove a similar theorem for general Levy processes.

Variational solutions of stochastic partial differential equations with cylindrical Lévy noise

<p style='text-indent:20px;'>In this article, the existence of a unique solution in the variational approach of the stochastic evolution equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \, \mathrm{d}X(t) = F(X(t)) \, \mathrm{d}t + G(X(t)) \, \mathrm{d}L(t) $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>driven by a cylindrical Lévy process <inline-formula><tex-math id="M1">\begin{document}$ L $\end{document}</tex-math></inline-formula> is established. The coefficients <inline-formula><tex-math id="M2">\begin{document}$ F $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula> are assumed to satisfy the usual monotonicity and coercivity conditions. The noise is modelled by a cylindrical Lévy processes which is assumed to belong to a certain subclass of cylindrical Lévy processes and may not have finite moments.

Sharing and Avatar-Based Innovation Tools on Digital Economy Perspectives Using Levy Processes Simulation

This article is the Enhancement of the Mkrttchian and Vertakova article “Digital Sharing Economy” published in the International Journal of Innovation in Digital Economy (IJIDE, Volume 10, issue 2) and the chapter “Avatar-Based Innovation Tools for Managerial Perspectives on Digital Sharing Economy” in the book “Avatar-Based Models, Tools, and Innovation in the Digital Economy,” focused on an entirely new area relevant to the scope of IJIDE. The article discusses the capabilities of the R language for modeling Levy processes - processes that currently 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 at R language. The efficient algorithm of variance gamma process simulation using variance gamma random variables is processed and programmed at R language, as Modelling in the Digital Globalization Era.

PERSAMAAN DIFERENSIAL ORNSTEIN-UHLENBECK DALAM PERAMALAN HARGA SAHAM

Geometric Brownian motion is one of the most widely used stock price model. One of the assumptions that is filled with stock return volatility is constant. Gamma Ornstein-Uhlenbeck process a model to describe volatility in finance. Additionally, Gamma Ornstein-Uhlenbeck process driven by Background Driving Levy Process (BDLP) compound Poisson process and the marginal law of volatility follows a Gamma distribution. Barndorff-Nielsen and Shepard (BNS) Gamma Ornstein-Uhlenbeck model can to sample the process for the stock price with volatility follows Gamma Ornstein-Uhlenbeck process. Based on these, the simulation result are compared BNS Gamma Ornstein-Uhlenbeck model with geometric Brown motion for Standard and Poor (SP) 500 stock data. Simulation result give BNS Gamma Ornstein-Uhlenbeck model and Geometric Brownian motion a Root Mean Square Error (RMSE) are 0,13 and 0,24 respectively. These result indicate that the BNS Gamma Ornstein-Uhlenbeck model gives a more accurate than Geometric Brownian motion

Maximum Drawdown and Drawdown Duration of Spectrally Negative Lévy Processes Decomposed at Extremes

Path decomposition is performed to characterize the law of the pre/post-supremum, post-infimum and the intermediate processes of a spectrally negative Levy process taken up to an independent exponential time T: As a result, mainly the distributions of the supremum of the post-infimum process and the maximum drawdown of the pre/postsupremum, post-infimum processes and the intermediate processes are obtained together with the law of drawdown durations.

Efficient Asian option pricing under regime switching jump diffusions and stochastic volatility models

Utilizing frame duality and a FFT-based implementation of density projection we develop a novel and efficient transform method to price Asian options for very general asset dynamics, including regime switching Levy processes and other jump diffusions as well as stochastic volatility models with jumps. The method combines continuous-time Markov chain approximation, with Fourier pricing techniques. In particular, our method encompasses Heston, Hull-White, Stein-Stein, 3/2 model as well as recently proposed Jacobi, $$\alpha $$ -Hypergeometric, and 4/2 models, for virtually any type of jump amplitude distribution in the return process. This framework thus provides a ‘unified’ approach to pricing Asian options in stochastic jump diffusion models and is readily extended to alternative exotic contracts. We also derive a characteristic function recursion by generalizing the Carverhill-Clewlow factorization which enables the application of transform methods in general. Numerical results are provided to illustrate the effectiveness of the method. Various extensions of this method have since been developed, including the pricing of barrier, American, and realized variance derivatives.

A minimal contrast estimator for the linear fractional stable motion

In this paper we present an estimator for the three-dimensional parameter $$(\sigma , \alpha , H)$$ of the linear fractional stable motion, where H represents the self-similarity parameter, and $$(\sigma , \alpha )$$ are the scaling and stability parameters of the driving symmetric Levy process L. Our approach is based upon a minimal contrast method associated with the empirical characteristic function combined with a ratio type estimator for the self-similarity parameter H. The main result investigates the strong consistency and weak limit theorems for the resulting estimator. Furthermore, we propose several ideas to obtain feasible confidence regions in various parameter settings. Our work is mainly related to Ljungdahl and Podolskij (A note on parametric estimation of Levy moving average processes, p 294, 2019) and Mazur et al. (Bernoulli 26(1): 226–252, 2020) in which parameter estimation for the linear fractional stable motion and related Levy moving average processes has been studied.

Application of iterated filtering to stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck process

Abstract Barndorff-Nielsen and Shephard (2001) proposed a class of stochastic volatility models in which the volatility follows the Ornstein–Uhlenbeck process driven by a positive Levy process without the Gaussian component. The parameter estimation of these models is challenging because the likelihood function is not available in a closed-form expression. A large number of estimation techniques have been proposed, mainly based on Bayesian inference. The main aim of the paper is to present an application of iterated filtering for parameter estimation of such models. Iterated filtering is a method for maximum likelihood inference based on a series of filtering operations, which provide a sequence of parameter estimates that converges to the maximum likelihood estimate. An application to S&amp;P500 index data shows the model perform well and diagnostic plots for iterated filtering ensure convergence iterated filtering to maximum likelihood estimates. Empirical application is accompanied by a simulation study that confirms the validity of the approach in the case of Barndorff-Nielsen and Shephard’s stochastic volatility models.

Multivariate Escher Transformed Laplace Distribution and Its Generalization

This paper we introduced a new distribution namely the multivariate Esscher transformed Laplace distribution. Various properties of the distribution are studied and the applications are discussed. Further we develop an autoregressive process with multivariate ETL marginal and study its properties. A Levy process based on this multivariate infinitely divisible distribution is known as Laplace motion, and its marginal distributions are multivariate generalized Esscher transformed Laplace distribution.

Option Pricing with Mixed Lévy Subordinated Price Process and Implied Probability Weighting Function

It is essential to incorporate the impact of investor behavior when modeling the dynamics of asset returns. In this paper, we reconcile behavioral finance and rational finance by incorporating investor behavior within the framework of dynamic asset pricing theory. To include the views of investors, we employ the method of subordination which has been proposed in the literature by including business (intrinsic, market) time. We define a mixed Levy subordinated model by adding a single subordinated Levy process to the well-known log-normal model, resulting in a new log-price process. We apply the proposed models to study the behavioral finance notion of "greed and fear" disposition from the perspective of rational dynamic asset pricing theory. The greedy or fearful disposition of option traders is studied using the shape of the probability weighting function. We then derive the implied probability weighting function for the fear and greed deposition of option traders in comparison to spot traders. Our result shows the diminishing sensitivity of option traders. Diminishing sensitivity results in option traders overweighting the probability of big losses in comparison to spot traders.

Efficient Asian Option Pricing Under Regime Switching Jump Diffusions and Stochastic Volatility Models

Utilizing frame duality and a FFT-based implementation of density projection we develop a novel and efficient transform method to price Asian options for very general asset dynamics, including regime switching Levy processes and other jump diffusions as well as stochastic volatility models with jumps. The method combines Continuous-Time Markov Chain (CTMC) approximation, with Fourier pricing techniques. In particular, our method encompasses Heston, Hull-White, Stein-Stein, 3/2 model as well as recently proposed Jacobi, alpha-Hypergeometric, and 4/2 models, for virtually any type of jump amplitude distribution in the return process. This framework thus provides a unified approach to pricing Asian options in stochastic jump diffusion models and is readily extended to alternative exotic contracts. We also derive a characteristic function recursion by generalizing the Carverhill-Clewlow factorization which enables the application of transform methods in general. Numerical results are provided to illustrate the effectiveness of the method. Various extensions of this method have since been developed, including the pricing of barrier, American, and realized variance derivatives.

Parameter estimation of laboratory-reared Panulirus ornatus

In the fisheries study, fish growth is typically in an indeterminate fashion implying a continuous growth function occurs throughout the life. Most existing studies of fish trajectories are continuous; however, it is not true when modelling growth of the crustacean species with discontinuous growth paths. In this context, it is imperative to insure that a different type of model for describing absolute growth in crustaceans. Crustaceans must moult for them to grow. A moulting process is of periodical shedding of the exoskeleton and thus the crustacean growth is known to be a discontinuous process. The sudden growth of crustaceans through the moulting process makes the growth estimation more complex. To model the discontinuous growth, we consider stochastic approaches where the growth model only considers for a monotonically increasing function. To this end, we introduce a subordinator that is a special case of a Levy process. A subordinator is a non-decreasing Levy process, that enabling the individual variability and environmental perturbation to be included in modelling growth. A dataset in the laboratory setting (e.g. in an aquarium) is developed. The motivational dataset is from the ornate rock lobster, Panulirus ornatus, where the growth parameters can be estimated through two inter-correlated variables, namely the intermoult periods and the moult increments. We propose a joint density function, consisting of the moult increments and the intermoult periods. Both of these variables are assumed to be conditionally independent based on the Markov property. In the four-year studies from 1995 to 1999, the growth rates for females and males are estimated averagely 0.307 mm year–1 and 0.205 mm year–1, respectively. Therefore, the growth parameters of moult increments and intermoult periods can be quantified individually. The corresponding functions will then be convoluted through a simulation approach to obtain a population mean curve for crustaceans.

On the Cauchy problem for stochastic integro-differential equations with radially O-regularly varying Lévy measure

Parabolic integro-differential nondegenerate Cauchy problem is considered in the scale of L_{p} spaces of functions whose regularity is defined by a Levy measure with O-regulary varying radial profile. Existence and uniqueness of a solution is proved by deriving apriori estimates. Some probability density function estimates of the associated Levy process are used as well.

On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models

In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution in the scale of Bessel potential spaces. Our aim is to generalize known existence results for a wide class of Levy measures including with a strong singular kernel. As an application we consider a class of PIDEs arising in the financial mathematics. The classical linear Black–Scholes model relies on several restrictive assumptions such as liquidity and completeness of the market. Relaxing the complete market hypothesis and assuming a Levy stochastic process dynamics for the underlying stock price process we obtain a model for pricing options by means of a PIDE. We investigate a model for pricing call and put options on underlying assets following a Levy stochastic process with jumps. We prove existence and uniqueness of solutions to the penalized PIDE representing approximation of the linear complementarity problem arising in pricing American style of options under Levy stochastic processes. We also present numerical results and comparison of option prices for various Levy stochastic processes modelling underlying asset dynamics.

Parameter Estimation for the Discretely Observed Vasicek Model with Small Fractional Lévy Noise

The statistical inference of the Vasicek model driven by small Levy process has a long history. In this paper, we consider the problem of parameter estimation for Vasicek model dXt = (μ-θXt)dt + edLdt, t ∈ [0,1], X0 = x0, driven by small fractional Levy noise with the known parameter d less than one half, based on discrete high-frequency observations at regularly spaced time points $$\{ {t_i} = \frac{i}{n},i = 1,2,...,n\} $$. For the general case and the null recurrent case, the consistency as well as the asymptotic behavior of least squares estimation of unknown parameters μ and θ have been established as small dispersion coefficient e → 0 and large sample size n → − simultaneously.

Stability in probability and inverse optimal control of evolution systems driven by Levy processes

This paper first develops a Lyapunov-type theorem to study global well-posedness &#x0028 existence and uniqueness of the strong variational solution &#x0029 and asymptotic stability in probability of nonlinear stochastic evolution systems &#x0028 SESs &#x0029 driven by a special class of Levy processes, which consist of Wiener and compensated Poisson processes. This theorem is then utilized to develop an approach to solve an inverse optimal stabilization problem for SESs driven by Levy processes. The inverse optimal control design achieves global well-posedness and global asymptotic stability of the closed-loop system, and minimizes a meaningful cost functional that penalizes both states and control. The approach does not require to solve a Hamilton-Jacobi-Bellman equation &#x0028 HJBE &#x0029. An optimal stabilization of the evolution of the frequency of a certain genetic character from the population is included to illustrate the theoretical developments.

The Wiener-Hopf Factorization for Pricing Options Made Easy

The paper suggest a new approach to pricing barrier options under pure non-Gaussian Levy processes with jumps of finite variation. The key idea behind the method to represent the process under consideration as a difference between subordinators (increasing Levy processes). Such splitting rule applied to the process at exponentially distributed randomized time points gives us the possibility to find the option price by analytically solving a sequence of simple Wiener-Hopf equations.

The finite-time ruin probability of time-dependent risk model with stochastic return and Brownian perturbation

This paper considers a dependent risk model with stochastic return and Brownian perturbation, where there exists a dependence structure between the claim sizes and the inter-arrival times and the price process of the investment portfolio is a geometric Levy process. When the claim sizes have heavy-tailed distributions, the asymptotic lower and upper bounds of the finite-time ruin probability have been given.

Some properties of the free stable distributions

Nous étudions certaines propriétés analytiques des densités $\alpha $-stables libres sur la droite. Nous montrons qu’elles sont classiquement infiniment divisibles pour $\alpha \le 1$ et qu’elles appartiennent à la classe de Thorin étendue pour $\alpha \le 3/4$. La mesure de Lévy est calculée explicitement pour $\alpha =1$ et ce calcul entraîne que les lois 1-stables libres n’appartiennent pas à la classe de Thorin, sauf dans le cas de la loi de Cauchy avec dérive. Dans le cas symétrique, nous montrons que les densités $\alpha $-stables libres ne sont pas infiniment divisibles quand $\alpha >1$. Dans le cas de signe constant nous montrons que les densités stables libres ont une courbe en baleine, autrement dit que leurs dérivées successives ne s’annulent qu’une seule fois sur leurs supports, ce qui constitue un raffinement de l’unimodalité. Nous établissons enfin plusieurs propriétés précises des densités stables libres spectralement de signe constant, parmi lesquelles une analyse détaillée de la variable aléatoire de Kanter, des expansions asymptotiques complètes en zéro, ainsi que plusieurs propriétés intrinsèques des courbes en baleine.

Intra-Horizon Expected Shortfall and Risk Structure in Models with Jumps

The present article deals with intra-horizon risk in models with jumps. Our general understanding of intra-horizon risk is along the lines of the approach taken in [BRSW04], [Ro08], [BMK09], [BP10], and [LV20]. In particular, we believe that quantifying market risk by strictly relying on point-in-time measures cannot be deemed a satisfactory approach in general. Instead, we argue that complementing this approach by studying measures of risk that capture the magnitude of losses potentially incurred at any time of a trading horizon is necessary when dealing with (m)any financial position(s). To address this issue, we propose an intra-horizon analogue of the expected shortfall for general profit and loss processes and discuss its key properties. Our intra-horizon expected shortfall is well-defined for (m)any popular class(es) of Levy processes encountered when modeling market dynamics and constitutes a coherent measure of risk, as introduced in [CDK04]. On the computational side, we provide a simple method to derive the intra-horizon risk inherent to popular Levy dynamics. Our general technique relies on results for maturity-randomized first-passage probabilities and allows for a derivation of diffusion and single jump risk contributions. These theoretical results are complemented with an empirical analysis, where popular Levy dynamics are calibrated to the S&P 500 index and the Brent crude oil data, and an analysis of the resulting intra-horizon risk is presented.