Levy-type Processes Research Articles

Poisson-stopped sum Lévy-type processes with application to stochastic modeling of hospital arrivals

Patient arrivals at a hospital typically occur in clusters and the arrivals count data exhibits over-dispersion. To model these characteristics, the Lévy-type processes based on Poisson-stopped sum distributions are proposed as alternatives to the non-homogeneous Poisson process (NHPP), a popular model for hospital arrivals. The arrival rate for the NHPP is a deterministic function of time that does not account for arrivals in clusters while the genesis of Poisson-stopped sum distributions arises from modeling event clusters. Data on daily scheduled patient arrivals over 15-minute intervals collected from a Malaysian public hospital were used to illustrate the application of the Levy-type processes in healthcare management. It is shown that the proposed Lévy-type negative binomial and Lévy-type Thomas processes fit the data better than the NHPP. In addition, the Levy-type processes are computationally simpler and hence, it is envisaged that they will be potentially useful for implementation in staff scheduling and resource allocations.

Fractal nature of arterial blood oxygen saturation data

The subject matter of this study was the processing of arterial blood oxygen saturation data (SaO2). The aim was to investigate the downsampling procedure of the SaO2 records on a broad range of scales. The object of study was a small data set (20 subjects, about 164 seconds duration, sampling rate 300 Hz) borrowed from the well-known portal of medical databases Physionet. The tasks to be solved are a test of the dataset heterogeneity, downsampling of the SaO2 series and its increments in a broad range of possible, checking the randomness of SaO2 series increments, argumentation in favor of applying the theory of Levy-type processes to the SaO2 increments and proving of their self-similarity, the definition of the geometrical fractal and its Hausdorff dimension. The methods used are the Levy-type processes theory, statistical methods, boxes-covering method for fractal structures, the autocorrelation function, and programming within MAPLE 2020. The authors obtained the following results: the dataset comprises three subsets with different variability; the records and their increments remain scale-invariant if the switching frequencies remain lower than the reduced sample rate; the increments of SaO2 records are a Levy-type and self-similar random process; the fractal is the set of positions of the non-zero increments (switch-overs) from a geometrical viewpoint. Conclusions. The scientific novelty of the results obtained is as follows: 1) the fractal nature and the self-similarity of SaO2 records and their increments were proved for the first time; 2) authors found the fractal Hausdorff dimensions for the subsets in the range (0.48…0.73) in dependence on variability; 3) authors found the principal possibility of the SaO2 data sizes essential reducing without losses of vital information.

Geometric approximations to transition densities of Jump-type Markov processes

Abstract This paper is concerned with the transition functions of symmetric Levy-type processes generated by a pseudo-differential operator with variable coefficients. We first give the general estimates of heat kernels of jump diffusion semigroups, which leads to diagonal estimates of transition function and subordination in the context of two-dimensional Cauchy semigroup. Then off-diagonal estimates of special classes of Levy-type processes where transition function can be expressed using the diagonal estimation results and related metrics are derived. Furthermore, we show geometric approximation of the general two-dimensional Levy processes, and graphical experiments have been made by freezing the coefficients of the generators.

Multifractal properties of sample paths of ground state-transformed jump processes

We consider a class of L\'evy-type processes with unbounded coefficients, arising as Doob $h$-transforms of Feynman-Kac type representations of non-local Schr\"odinger operators, where the function $h$ is chosen to be the ground state of such an operator. First, we show the existence of a c\`adl\`ag version of the so-obtained ground state-transformed processes. Next, we prove that they satisfy a related stochastic differential equation with jumps. Making use of this SDE, we then derive and prove the multifractal spectrum of local H\"older exponents of sample paths of ground state-transformed processes.

Approximation in law of locally $\alpha $-stable Lévy-type processes by non-linear regressions

We study a real-valued Levy-type process $X$, which is locally $\alpha $-stable in the sense that its jump kernel is a combination of a ‘principal’ (state dependent) $\alpha $-stable part with a ‘residual’ lower order part. We show that under mild conditions on the local characteristics of a process (the jump kernel and the velocity field) the process is uniquely defined, is Markov, and has the strong Feller property. We approximate $X$ in law by a non-linear regression $\widetilde{X} ^{x}_{t}=\mathfrak{f} _{t}(x)+t^{1/\alpha }U^{x}_{t} $ with a deterministic regressor term $\mathfrak{f} _{t}(x)$ and $\alpha $-stable innovation term $U^{x}_{t}$, and provide error estimates for such an approximation. A case study is performed, revealing different types of assumptions which lead to various choices of regressor/innovation terms and various types of the estimates. The assumptions are quite general, cover the super-critical case $\alpha <1$, and allow non-symmetry of the Levy kernel and unboundedness of the drift coefficient.

Transportation distance between the Lévy measures and stochastic equations for Lévy-type processes

The notion of the transportation distance on the set of the Levy measures on $\mathbb{R}$ is introduced. A Levy-type process with a given symbol (state dependent analogue of the characteristic triplet) is proved to be well defined as a strong solution to a stochastic differential equation (SDE) under the assumption of Lipschitz continuity of the Levy kernel in the symbol w.r.t. the state space variable in the transportation distance. As examples, we construct Gamma-type process and $\alpha$-stable like process as strong solutions to SDEs.

Asymptotics for &lt;i&gt;d&lt;/i&gt;-Dimensional Levy-Type Processes

We consider a general d-dimensional Levy-type process with killing. Combining the classical Dyson series approach with a novel polynomial expansion of the generator A(t) of the Levy-type process, we derive a family of asymptotic approximations for transition densities and European-style options prices. Examples of stochastic volatility models with jumps are provided in order to illustrate the numerical accuracy of our approach. The methods described in this paper extend the results from Corielli et al. (2010), Pagliarani and Pascucci (2013) and Lorig et al. (2013a) for Markov diffusions to Markov processes with jumps.

Nonlocal convection-diffusion volume-constrained problems and jump processes

We introduce the Cauchy and time-dependent volume-constrained problems associated with a linear nonlocal convection-diffusion equation. These problems are shown to be well-posed and correspond to conventional convection-diffusion equations as the region of nonlocality vanishes. The problems also share a number of features such as the maximum principle, conservation and dispersion relations, all of which are consistent with their corresponding local counterparts. Moreover, these problems are the master equations for a class of finite activity Levy-type processes with nonsymmetric Levy measure. Monte Carlo simulations and finite difference schemes are applied to these nonlocal problems, to show the effects of time, kernel, nonlocality and different volume-constraints.

CEV: Constant Lévylasticity of Variance

We consider a class of assets whose risk-neutral pricing dynamics are described by an exponential Levy-type process subject to default. The class of processes we consider features locally-dependent drift, diffusion and default-intensity as well as a locally-dependent Levy measure. Using techniques from regular perturbation theory and Fourier analysis, we derive a series expansion for the price of a European-style option. We also provide precise conditions under which this series expansion converges to the exact price. Additionally, for a certain subclass of assets in our modeling framework, we derive an expansion for the implied volatility induced by our option pricing formula. The implied volatility expansion is exact within its radius of convergence. As an example of our framework, we propose a class of CEV-like Levy-type models. Within this class, approximate option prices can be computed by a single Fourier integral and approximate implied volatilities are explicit (i.e., no integration is required). Furthermore, the class of CEV-like Levy-type models is shown to provide a tight fit to the implied volatility surface of S{&}P500 index options.

Exact Scenario Simulation for Selected Multi-Dimensional Stochastic Processes

Accurate scenario simulation methods for solutions of multi-dimensional stochastic differential equations find application in stochastic analysis, the statistics of stochastic processes and many other areas, for instance, in finance. They have been playing a crucial role as standard models in various areas and dominate often the communication and thinking in a particular field of application, even that they may be too simple for more advanced tasks. Various discrete time simulation methods have been developed over the years. However, the simulation of solutions of some stochastic differential equations can be problematic due to systematic errors and numerical instabilities. Therefore, it is valuable to identify multi-dimensional stochastic differential equations with solutions that can be simulated exactly. This avoids several of the theoretical and practical problems encountered by those simulation methods that use discrete time approximations. This paper provides a survey of methods for the exact simulation of paths of some multi-dimensional solutions of stochastic differential equations including Ornstein-Uhlenbeck, square root, squared Bessel, Wishart and Levy type processes.

SLEκ: correlation functions in the coefficient problem

We apply the method of correlation functions to the coefficient problem in stochastic geometry. In particular, we give a proof for some universal patterns conjectured by M Zinsmeister for the second moments of the Taylor coefficients for special values of κ in the whole-plane Schramm–Loewner evolution (SLEκ). We propose to use multi-point correlation functions for the study of higher moments in the coefficient problem. Generalizations related to the Levy-type processes are also considered. The exact integral means β-spectrum of this version of the whole-plane SLEκ is discussed.

Stochastic Comparison for Lévy-Type Processes

The main goal of this paper is to establish necessary and sufficient conditions for stochastic comparison of two general Levy-type processes on ℝd. By refining the test functions in Wang (Acta Math. Sin. Engl. Ser. 25:741–758, 2009), mainly the test functions of diffusion coefficients, we get the necessary conditions. The sufficiency of the conditions is obtained by constructing a new sequence of finite Levy measures {νn}n≥1 different from the one in Wang (Acta Math. Sin. Engl. Ser. 25:741–758, 2009) to approach the Levy measure ν.

Option Pricing with Discrete Time Jump Processes

In this paper we propose new option pricing models based on two classes of discrete Levy-type processes. By combining these Levy processes with several volatility dynamics of the GARCH type, we aim to take into account the dynamics of financial returns in a realistic way. The associated risk neutral dynamics of the time series models is obtained through two different specifications for the pricing kernel: we provide a characterization of the change in the probability measure using the Esscher transform and the Minimal Entropy Martingale Measure. We finally assess empirically the performance of this modelling approach, using a dataset of European options based on the CAC 40. Our empirical findings show that discrete time Levy based models behave well regardless of the volatility dynamics and of the pricing kernel. A different specification is however required to obtain the best results for options with a shorter or longer maturity.

Exact scenario simulation for selected multi-dimensional stochastic processes

Accurate scenario simulation methods for solutions of multi - dimensional stochastic differential equations find application in stochastic analysis, the statistics of stochastic processes and many other areas, for in- stance, in finance. Various discrete time simulation methods have been devel- oped over the years. However, the simulation of solutions of some stochastic differential equations can be problematic due to systematic errors and nu- merical instabilities. Therefore, it is valuable to identify multi-dimensional stochastic differential equations with solutions that can be simulated exactly. This avoids several of the theoretical and practical problems encountered by those simulation methods that use discrete time approximations. This paper provides a survey of methods for the exact simulation of paths of some multi- dimensional solutions of stochastic differential equations including Ornstein- Uhlenbeck, square root, squared Bessel, Wishart and Levy type processes.

Stochastic control of SDEs associated with Lévy generators and application to financial optimization

This paper is concerned with the optimal control of jump type stochastic differential equations associated with (general) Levy generators. The maximum principle is formulated for the solutions of the equations, which is inspired by N. C. Framstad, B. Oksendal and A. Sulem [J. Optim. Theory Appl., 2004, 121: 77–98] (and a continuation, J. Bennett and J. -L. Wu [Front. Math. China, 2007, 2(4): 539–558]). The result is then applied to optimization problems in financial models driven by Levy-type processes.

Stochastic comparison and preservation of positive correlations for Lévy-type processes

The stochastic comparison and preservation of positive correlations for Levy-type processes on ℝd are studied under the condition that Levy measure ν satisfies ∫{0<|z|≤1}|z||ν(x, dz) − ν(x, d(−z))| < ∞, x ∈ ℝd, while the sufficient conditions and necessary ones for them are obtained. In some cases the conditions for stochastic comparison are not only sufficient but also necessary.

On the Self-Decomposability of Euler's Gamma Function

Let Γ be Euler's Gamma function. We prove that, for all α ≠ 0, β > 0, γ > 0, δ > 0, the function (Γ(γ + iαz)/Γ(γ)βiαz)δ, z ∈ R1, is a self-decomposable characteristic function from the Thorin class \(\mathcal{T}_e \) and derive its explicit canonical form. Similarly to [1], we also describe several classes of Levy-type stochastic processes related to Γ.

The Entrance Boundary of the Multiplicative Coalescent

The multiplicative coalescent $X(t)$ is a $l^2$-valued Markov process representing coalescence of clusters of mass, where each pair of clusters merges at rate proportional to product of masses. From random graph asymptotics it is known (Aldous (1997)) that there exists a standard version of this process starting with infinitesimally small clusters at time $- \infty$. In this paper, stochastic calculus techniques are used to describe all versions $(X(t);- \infty < t < \infty)$ of the multiplicative coalescent. Roughly, an extreme version is specified by translation and scale parameters, and a vector $c \in l^3$ of relative sizes of large clusters at time $- \infty$. Such a version may be characterized in three ways: via its $t \to - \infty$ behavior, via a representation of the marginal distribution $X(t)$ in terms of excursion-lengths of a Levy-type process, or via a weak limit of processes derived from the standard version via a coloring construction.

Fractional kinetic equations: solutions and applications.

Fractional generalization of the diffusion equation includes fractional derivatives with respect to time and coordinate. It had been introduced to describe anomalous kinetics of simple dynamical systems with chaotic motion. We consider a symmetrized fractional diffusion equation with a source and find different asymptotic solutions applying a method which is similar to the method of separation of variables. The method has a clear physical interpretation presenting the solution in a form of decomposition of the process of fractal Brownian motion and Levy-type process. Fractional generalization of the Kolmogorov-Feller equation is introduced and its solutions are analyzed. (c) 1997 American Institute of Physics.