Showing papers on "Subordinator published in 2021"

Some Properties of the Kilbas-Saigo Function

Abstract: We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at −∞, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy function. The main ingredient for the proof is a probabilistic representation of these functions in terms of the stable subordinator.

Exact simulation of Ornstein-Uhlenbeck tempered stable processes

Abstract: There are two types of tempered stable (TS) based Ornstein–Uhlenbeck (OU) processes: (i) the OU-TS process, the OU process driven by a TS subordinator, and (ii) the TS-OU process, the OU process with TS marginal law. They have various applications in financial engineering and econometrics. In the literature, only the second type under the stationary assumption has an exact simulation algorithm. In this paper we develop a unified approach to exactly simulate both types without the stationary assumption. It is mainly based on the distributional decomposition of stochastic processes with the aid of an acceptance–rejection scheme. As the inverse Gaussian distribution is an important special case of TS distribution, we also provide tailored algorithms for the corresponding OU processes. Numerical experiments and tests are reported to demonstrate the accuracy and effectiveness of our algorithms, and some further extensions are also discussed.

A Fourier-cosine method for finite-time ruin probabilities

Abstract: In this paper, we study the finite-time ruin probability in the risk model driven by a Levy subordinator, by incorporating the popular Fourier-cosine method. Our interest is to propose a general approximation for any specified precision provided that the characteristic function of the Levy Process is known. To achieve this, we derive an explicit integral expression for the finite-time ruin probability, which is expressed in terms of the density function and the survival function of L t . Moreover, we apply the rearrangement inequality to further improve our approximations. In addition, with only mild and practically relevant assumptions, we prove that the approximation error can be made arbitrarily small (actually an algebraic convergence rate up to 3, which is the fastest possible approximant known upon all in the literature), and has a linear computation complexity in a number of terms of the Fourier-cosine expansion. The effectiveness of our results is demonstrated in various numerical studies; through these examples, the supreme power of the Fourier-cosine method is once demonstrated.

Two Theorems on Hunt’s Hypothesis (H) for Markov Processes

Abstract: Hunt’s hypothesis (H) and the related Getoor’s conjecture is one of the most important problems in the basic theory of Markov processes. In this paper, we investigate the invariance of Hunt’s hypothesis (H) for Markov processes under two classes of transformations, which are change of measure and subordination. Our first theorem shows that for two standard processes (Xt) and (Yt), if (Xt) satisfies (H) and (Yt) is locally absolutely continuous with respect to (Xt), then (Yt) satisfies (H). Our second theorem shows that a standard process (Xt) satisfies (H) if and only if $(X_{\tau _{t}})$ satisfies (H) for some (and hence any) subordinator (τt) which is independent of (Xt) and has a positive drift coefficient. Applications of the two theorems are given.

Fractional Cauchy problem on random snowflakes

Abstract: We consider time-changed Brownian motions on random Koch (pre-fractal and fractal) domains where the time change is given by the inverse to a subordinator. In particular, we study the fractional Cauchy problem with Robin condition on the pre-fractal boundary obtaining asymptotic results for the corresponding fractional diffusions with Robin, Neumann and Dirichlet boundary conditions on the fractal domain.

Estimation of model parameters of dependent processes constructed using Lévy Copulas

Abstract: The degradations of engineering components can be dependent. This study considers that a d-dimensional subordinator constructed based on Levy copula can be used to model the dependency. Three proce...

Lévy Processes Linked to the Lower-Incomplete Gamma Function

Abstract: We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Levy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior.

Reaction-Subdiffusion Equations with Species-Dependent Movement

Abstract: Reaction-diffusion equations are one of the most common mathematical models in the natural sciences and are used to model systems that combine reactions with diffusive motion. However, rather than ...

First passage times for some classes of fractional time-changed diffusions

Abstract: We consider some time-changed diffusion processes obtained by applying the Doob transformation rule to a time-changed Brownian motion. The time-change is obtained via the inverse of an α-stable subordinator. These processes are specified in terms of time-changed Gauss-Markov processes and fractional time-changed diffusions. A fractional pseudo-Fokker-Planck equation for such processes is given. We investigate their first passage time densities providing a generalized integral equation they satisfy and some transformation rules. First passage time densities for time-changed Brownian motion and Ornstein-Uhlenbeck processes are provided in several forms. Connections with closed form results and numerical evaluations through the level zero are given.

Lévy-walk-like Langevin dynamics affected by a time-dependent force.

Abstract: The L\'evy walk is a popular and more `physical' model to describe the phenomena of superdiffusion, because of its finite velocity. The movements of particles are under the influence of external potentials at almost any time and anywhere. In this paper, we establish a Langevin system coupled with a subordinator to describe the L\'evy walk in a time-dependent periodic force field. The effects of external force are detected and carefully analyzed, including the nonzero first moment (even though the force is periodic), adding an additional dispersion on the particle position, a consistent influence on the ensemble- and time-averaged mean-squared displacement, etc. Besides, the generalized Klein-Kramers equation is obtained, not only for the time-dependent force but also for the space-dependent one.

An insurance risk process with a generalized income process: A solvency analysis

Abstract: In ruin theory, an insurer’s income process is usually assumed to grow at a deterministic rate of c > 0 over time. For instance, both the well-known Cramer–Lundberg risk process and the Sparre Andersen risk model have this assumption built in the construction of their respective surplus processes. This assumption is mainly considered for purposes of mathematical tractability, but generally fails to accurately model an insurer’s income dynamics. To better characterize the variability and uncertainty of an insurer’s income process, several papers have studied insurance risk models with random incomes where the main emphasis is placed on carrying the related Gerber–Shiu analysis. However, a systematic and quantitative understanding of how the more volatile income processes impact an insurer’s solvency risk is still lacking. This paper aims to fill this gap in the literature by quantitatively assessing the impact of the choice of income process on some finite-time and infinite-time ruin quantities. To carry this analysis, we consider a generalized Sparre Andersen risk model with a random income process which renews at claim instants. For exponentially distributed claim sizes, we derive explicit expressions for some joint distributions involving the time to ruin and the number of claims until ruin. As special cases of the proposed insurance risk process, we consider income processes modelled by a subordinator or a particular varying premium rate model. Numerical examples are then carried to draw important risk management implications of a solvency nature for the insurer.

Spectral projections correlation structure for short-to-long range dependent processes

Abstract: Let X = (Xt)t≥0 be a stochastic process issued from that admits a marginal stationary measure v, i.e. vPt f = vf for all t ≥ 0, where . In this paper, we introduce the (resp. biorthogonal) spectral projections correlation functions which are expressed in terms of projections.” Also, update first published online date, if available. into the eigenspaces of Pt (resp. and of its adjoint in the weighted Hilbert space L2 (v)). We obtain closed-form expressions involving eigenvalues, the condition number and/or the angle between the projections in the following different situations: when X = X with X = (Xt)t ≥ 0 being a Markov process, X is the subordination of X in the sense of Bochner, and X is a non-Markovian process which is obtained by time-changing X with an inverse of a subordinator. It turns out that these spectral projections correlation functions have different expressions with respect to these classes of processes which enables to identify substantial and deep properties about their dynamics. This interesting fact can be used to design original statistical tests to make inferences, for example, about the path properties of the process (presence of jumps), distance from symmetry (self-adjoint or non-self-adjoint) and short-to-long-range dependence. To reveal the usefulness of our results, we apply them to a class of non-self-adjoint Markov semigroups studied in Patie and Savov (to appear, Mem. Amer. Math. Soc., 179p), and then time-change by subordinators and their inverses.

Option Surface Econometrics with Applications

Abstract: At each maturity a discrete return distribution is inferred from option prices. Option pricing models imply a comparable theoretical distribution. As both the transformed data and the option pricing model deliver points on a simplex, the data is statistically modeled by a Dirichlet distribution with expected values given by the option pricing model. The resulting setup allows for maximum likelihood estimation of option pricing model parameters with standard errors enabling the test of hypotheses. Hypothesis testing is illustrated by testing for risk neutral return distributions being consistent with Brownian motion with drift time changed by a subordinator. Models mixing processes of independent increments with processes related to solution of Ornstein Uhlenbeck (OU) equations are tested for the presence of the OU component. OU equations are a form of perpetual motion processes continuously responding to their past changes. The tests support the rejection of Brownian subordination and the presence of a perpetual motion component.

Asymptotic of densities of exponential functionals of subordinators

Abstract: In this paper we derive non-classical Tauberian asymptotic at infinity for the tail, the density and the derivatives thereof of a large class of exponential functionals of subordinators. More precisely, we consider the case when the Levy measure of the subordinator satisfies the well-known and mild condition of positive increase. This is achieved via a convoluted application of the saddle point method to the Mellin transform of these exponential functionals which is given in terms of Bernstein-gamma functions. To apply the saddle point method we improved the Stirling type of asymptotic for Bernstein-gamma functions and the latter is of interest beyond this paper as the Bernstein-gamma functions are applicable in different settings especially through their asymptotic behaviour in the complex plane. As an application we have derived the asymptotic of the density and its derivatives for all exponential functionals of non-decreasing, potentially compound Poisson processes which turns out to be precisely as that of an exponentially distributed random variable. We show further that a large class of densities are even analytic in a cone of the complex plane.

Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integrators

Abstract: The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time change given by an inverse subordinator. SDEs to be considered are unique in two different aspects: (i) they contain two drift terms, one driven by the random time change and the other driven by a regular, non-random time variable; (ii) the standard Lipschitz assumption is replaced by that with a time-varying Lipschitz bound. The difficulty imposed by the first aspect is overcome via an approach that is significantly different from a well-known method based on the so-called duality principle. On the other hand, the second aspect requires the establishment of a criterion for the existence of exponential moments of functions of the random time change.

Option pricing under time interval driven model

Abstract: A new model which we called time driven model for option pricing is proposed. In this model, the price of underlying asset is driven by different random driving source in different time interval. E...

A generalised Dickman distribution and the number of species in a negative binomial process model

Abstract: We derive the large-sample distribution of the number of species in a version of Kingman’s Poisson–Dirichlet model constructed from an -stable subordinator but with an underlying negative binomial process instead of a Poisson process. Thus it depends on parameters from the subordinator and from the negative binomial process. The large-sample distribution of the number of species is derived as sample size . An important component in the derivation is the introduction of a two-parameter version of the Dickman distribution, generalising the existing one-parameter version. Our analysis adds to the range of Poisson–Dirichlet-related distributions available for modeling purposes.

Towards Control of Dam and Reservoir Systems with Forward-Backward Stochastic Differential Equations Driven by Clustered Jumps.

Abstract: We deal with a new maximum principle-based stochastic control model for river management through operating a dam and reservoir system. The model is based on coupled forward-backward stochastic differential equations (FBSDEs) derived from jump-driven streamflow dynamics and reservoir water balance. A continuous-time branching process with immigration driven by a tempered stable subordinator efficiently describes clustered inflow streamflow dynamics. This is a completely new attempt in hydrology and control engineering. Applying a stochastic maximum principle to the dynamics based on an objective functional for designing cost-efficient control of dam and reservoir systems leads to the FBSDEs as a system of optimality equations. The FBSDEs under a linear-quadratic ansatz lead to a tractable model, while they are solved numerically in the other cases using a least-squares Monte-Carlo method. Optimal controls are found in the former, while only sub-optimal ones are computable in the latter due to a hard state constraint. Model parameters are successfully identified from a real data of a river in Japan having a dam and reservoir system. We also show that the linear-quadratic case can capture the real operation data of the system with underestimation of the outflow discharge. More complex cases with a realistic time horizon are analyzed numerically to investigate impacts of considering the environmental flows and seasonal operational purposes. Key challenges towards more sophisticated modeling and analysis with jump-driven FBSDEs are discussed as well.

Limit theorems for prices of options written on semi-Markov processes

Abstract: We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator's Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

Time-changed space-time fractional Poisson process

Abstract: In this paper, we introduce and study a time-changed version of the space-time fractional Poisson process (STFPP) by time changing it by an independent Levy subordinator with finite moments of any

Gibbs partitions, Riemann-Liouville fractional operators, Mittag-Leffler functions, and fragmentations derived from stable subordinators

Abstract: Pitman (2003), and subsequently Gnedin and Pitman (2006), showed that a large class of random partitions of the integers derived from a stable subordinator of index -diversity.

Steady-state optimization of an exhaustive lévy storage process with intermittent output and random output rate

Abstract: Consider a regenerative storage process with a nondecreasing Levy input (subordinator) such that every cycle may be split into two periods. In the first (off), the output is shut off and the workload accumulates. This continues until some stopping time. In the second (on), the process evolves like a subordinator minus a positive drift (output rate) until it hits the origin. In addition, we assume that the output rate of every on period is a random variable, which is determined at the beginning of this period. For example, at each period, the output rate may depend on the workload level at the beginning of the corresponding busy period. We derive the Laplace–Stieltjes transform of the steady-state distribution of the workload process and then apply this result to solve a steady-state cost minimization problem with holding, setup and output capacity costs. It is shown that the optimal output rate is a nondecreasing deterministic function of the workload level at the beginning of the corresponding on period.

Alòs type decomposition formula for Barndorff-Nielsen and Shephard model

Abstract: The objective is to provide an Al\`os type decomposition formula of call option prices for the Barndorff-Nielsen and Shephard model: an Ornstein-Uhlenbeck type stochastic volatility model driven by a subordinator without drift. Al\`os (2012) introduced a decomposition expression for the Heston model by using Ito's formula. In this paper, we extend it to the Barndorff-Nielsen and Shephard model. As far as we know, this is the first result on the Al\`os type decomposition formula for models with infinite active jumps.

An analytical EM algorithm for sub-gaussian vectors

Abstract: The area in which a multivariate α-stable distribution could be applied is vast; however, a lack of parameter estimation methods and theoretical limitations diminish its potential. Traditionally, the maximum likelihood estimation of parameters has been considered using a representation of the multivariate stable vector through a multivariate normal vector and an α-stable subordinator. This paper introduces an analytical expectation maximization (EM) algorithm for the estimation of parameters of symmetric multivariate α-stable random variables. Our numerical results show that the convergence of the proposed algorithm is much faster than that of existing algorithms. Moreover, the likelihood ratio (goodness-of-fit) test for a multivariate α-stable distribution was implemented. Empirical examples with simulated and real world (stocks, AIS and cryptocurrencies) data showed that the likelihood ratio test can be useful for assessing goodness-of-fit.

Workload distributions in ASIP queueing networks

Abstract: The workload of a generalized n-site asymmetric simple inclusion process (ASIP) is investigated. Three models are analyzed. The first model is a serial network for which the steady-state Laplace–Stieltjes transform (LST) of the total workload in the first k sites ( $$k\le n$$ ) just after gate openings and at arbitrary epochs is derived. In a special case, the former (just after gate openings) turns out to be an LST of the sum of k independent random variables. The second model is a 2-site ASIP with leakage from the first queue. Gate openings occur at exponentially distributed intervals, and the external input processes to the stations are two independent subordinator Levy processes. The steady-state joint workload distribution right after gate openings, right before gate openings and at arbitrary epochs is derived. The third model is a shot-noise counterpart of the second model where the workload at the first queue behaves like a shot-noise process. The steady-state total amount of work just before a gate opening turns out to be a sum of two independent random variables.

On the inverse gamma subordinator

Abstract: In this paper we deal with some open problems concerned with gamma subordinators. In particular, we provide a representation for the moments of the inverse gamma subordinator. Then, we focus on $\lambda$-potentials and we study the governing equations associated with gamma subordinators and inverse processes. Such representations are given in terms of higher transcendental functions, also known as Volterra functions.