Showing papers on "Subordinator published in 2010"

Time-changed markov processes in unified credit-equity modeling

Abstract: This paper develops a novel class of hybrid credit-equity models with state-dependent jumps, local-stochastic volatility and default intensity based on time changes of Markov pro- cesses with killing. We model the defaultable stock price process as a time changed Markov difiusion process with state-dependent local volatility and killing rate (default intensity). When the time change is a Levy subordinator, the stock price process exhibits jumps with state-dependent Levy measure. When the time change is a time integral of an activity rate process, the stock price process has local-stochastic volatility and default intensity. When the time change process is a Levy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state-dependent jumps, local-stochastic volatility and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace trans- form of the time change are both available in closed form, the expectation operator of the time changed process is expressed in closed form as a single integral in the complex plane. If the payofi is square-integrable, the complex integral is further reduced to a spectral ex- pansion. To illustrate our general framework, we time change the jump-to-default extended CEV model (JDCEV) of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local-stochastic volatility and default intensity. These models can be used to jointly price equity and credit derivatives.

Fokker-Planck-Kolmogorov equations associated with SDEs driven by time-changed fractional Brownian motion

Abstract: In this paper Fokker-Planck-Kolmogorov type equations associated with stochastic differential equations driven by a time-changed fractional Brownian motion are derived. Two equivalent forms are suggested. The time-change process considered is either the first hitting time process for a stable subordinator or a mixture of stable subordinators. A family of operators arising in the representation of the Fokker-Plank-Kolmogorov equations is shown to have the semigroup property.

A subordinated kinematic wave equation for heavy‐tailed flow responses from heterogeneous hillslopes

Abstract: [1] Analytical expressions of hillslope-scale subsurface stormflow discharge are currently restricted to hillslopes with homogeneous or mildly heterogeneous conductivity fields In steep, straight hillslopes with uniform recharge these exhibit a classical piston flow response, which arises from an assemblage of impulses all moving at a constant velocity but with different starting locations Heterogeneity within a hillslope soil creates variations in the downslope velocity of these impulses, which may lead to nonpiston flow responses with either exponential or heavy (power law) tails The presence of heavy tails suggests that heterogeneity imparts a temporal memory on the motion of the impulses Using this assumption, a subordinated kinematic wave equation is proposed for moderately to highly heterogeneous hillslopes This equation convolves the piston response from a homogenous hillslope with a stable subordinator The stable subordinator randomizes the time that impulses spend in motion and produces nonpiston solutions with heavy tails Through comparisons of synthetic data generated from numerical hillslope simulations with physically realistic parameters, this equation faithfully reproduces both early and late time characteristics of heavy-tailed flow responses from moderate to highly heterogenous hillslopes A systematic evaluation of hillslope responses under different degrees of heterogeneity revealed a quantitative link between the statistical properties of the heterogeneous random fields and the parameters of the subordination framework This suggests that the subordinator can be parameterized with the physical measurement of hillslope properties

Particle tracking for fractional diffusion with two time scales

Abstract: Previous work [Y. Zhang, M.M. Meerschaert, B. Baeumer, Particle tracking for time-fractional diffusion, Phys. Rev. E 78 (2008) 036705] showed how to solve time-fractional diffusion equations by particle tracking. This paper extends the method to the case where the order of the fractional time derivative is greater than one. A subordination approach treats the fractional time derivative as a random time change of the corresponding Cauchy problem, with a first derivative in time. One novel feature of the time-fractional case of order greater than one is the appearance of clustering in the operational time subordinator, which is non-Markovian. Solutions to the time-fractional equation are probability densities of the underlying stochastic process. The process models movement of individual particles. The evolution of an individual particle in both space and time is captured in a pair of stochastic differential equations, or Langevin equations. Monte Carlo simulation yields particle location, and the ensemble density approximates the solution to the variable coefficient time-fractional diffusion equation in one or several spatial dimensions. The particle tracking code is validated against inverse transform solutions in the simplest cases. Further applications solve model equations for fracture flow, and upscaling flow in complex heterogeneous porous media. These variable coefficient time-fractional partial differential equations in several dimensions are not amenable to solution by any alternative method, so that the grid-free particle tracking approach presented here is uniquely appropriate.

Numerical Computation of First-Passage Times of Increasing Lévy Processes

Abstract: Let {D(s), s ≥ 0} be a non-decreasing Levy process. The first-hitting time process {E(t), t ≥ 0} (which is sometimes referred to as an inverse subordinator) defined by $E(t) = \inf \{s: D(s) > t \}$ is a process which has arisen in many applications. Of particular interest is the mean first-hitting time $U(t)=\mathbb{E}E(t)$ . This function characterizes all finite-dimensional distributions of the process E. The function U can be calculated by inverting the Laplace transform of the function $\widetilde{U}(\lambda) = (\lambda \phi(\lambda))^{-1}$ , where ϕ is the Levy exponent of the subordinator D. In this paper, we give two methods for computing numerically the inverse of this Laplace transform. The first is based on the Bromwich integral and the second is based on the Post-Widder inversion formula. The software written to support this work is available from the authors and we illustrate its use at the end of the paper.

Mittag-Leffler Waiting Time, Power Laws,Rarefaction, Continuous Time Random Walk, Diffusion Limit

Abstract: We discuss some applications of the Mittag-Leffler function and related probability distributions in the theory of renewal processes and continuous time random walks. In particular we show the asymptotic (long time) equivalence of a generic power law waiting time to the Mittag-Leffler waiting time distribution via rescaling and respeeding the clock of time. By a second respeeding (by rescaling the spatial variable) we obtain the diffusion limit of the continuous time random walk under power law regimes in time and in space. Finally, we exhibit the time-fractional drift process as a diffusion limit of the fractional Poisson process and as a subordinator for space-time fractional diffusion.

Information-based models for finance and insurance

Abstract: In financial markets, the information that traders have about an asset is reflected in its price. The arrival of new information then leads to price changes. The `information-based framework' of Brody, Hughston and Macrina (BHM) isolates the emergence of information, and examines its role as a driver of price dynamics. This approach has led to the development of new models that capture a broad range of price behaviour. This thesis extends the work of BHM by introducing a wider class of processes for the generation of the market filtration. In the BHM framework, each asset is associated with a collection of random cash flows. The asset price is the sum of the discounted expectations of the cash flows. Expectations are taken with respect (i) an appropriate measure, and (ii) the filtration generated by a set of so-called information processes that carry noisy or imperfect market information about the cash flows. To model the flow of information, we introduce a class of processes termed L\'evy random bridges (LRBs), generalising the Brownian and gamma information processes of BHM. Conditioned on its terminal value, an LRB is identical in law to a L\'evy bridge. We consider in detail the case where the asset generates a single cash flow $X_T$ at a fixed date $T$. The flow of information about $X_T$ is modelled by an LRB with random terminal value $X_T$. An explicit expression for the price process is found by working out the discounted conditional expectation of $X_T$ with respect to the natural filtration of the LRB. New models are constructed using information processes related to the Poisson process, the Cauchy process, the stable-1/2 subordinator, the variance-gamma process, and the normal inverse-Gaussian process. These are applied to the valuation of credit-risky bonds, vanilla and exotic options, and non-life insurance liabilities.

On the potential theory of one-dimensional subordinate Brownian motions with continuous components

Abstract: Suppose that S is a subordinator with a nonzero drift and W is an independent 1-dimensional Brownian motion. We study the subordinate Brownian motion X defined by X t = W(S t ). We give sharp bounds for the Green function of the process X killed upon exiting a bounded open interval and prove a boundary Harnack principle. In the case when S is a stable subordinator with a positive drift, we prove sharp bounds for the Green function of X in (0, ∞ ), and sharp bounds for the Poisson kernel of X in a bounded open interval.

On time-changed Gaussian processes and their associated Fokker-Planck-Kolmogorov equations

Abstract: This paper establishes Fokker-Planck-Kolmogorov type equations for time-changed Gaussian processes. Examples include those equations for a time-changed fractional Brownian motion with time-dependent Hurst parameter and for a time-changed Ornstein-Uhlenbeck process. The time-change process considered is the inverse of either a stable subordinator or a mixture of independent stable subordinators.

Subordination by orthogonal martingales in $L^{p}$ and zeros of Laguerre polynomials

Abstract: In this paper we address the question of finding the best $L^p$-norm constant for martingale transforms with one-sided orthogonality. We consider two martingales on a probability space with filtration $\mathcal{B}$ generated by a two-dimensional Brownian motion $B_t$. One is differentially subordinated to the other. Here we find the sharp estimate for subordinate martingales if the subordinated martingale is orthogonal and $1 2$, but the orthogonal martingale is a subordinator. The answers are given in terms of zeros of Laguerre polynomials. As an application of our sharp constant we obtain a new estimate for the norm of theAhlfors--Beurling operator. We estimate it as $1.3922(p-1)$ asymptotically for large $p$.

Convergence of clock processes in random environments and ageing in the p-spin SK model

Abstract: We derive a general criterion for the convergence of clock processes in random dynamics in random environments that is applicable in cases when correlations are not negligible, extending recent results by Gayrard [(2010), (2011), forthcoming], based on general criterion for convergence of sums of dependent random variables due to Durrett and Resnick [Ann. Probab. 6 (1978) 829-846]. We demonstrate the power of this criterion by applying it to the case of random hopping time dynamics of the p-spin SK model. We prove that on a wide range of time scales, the clock process converges to a stable subordinator almost surely with respect to the environment. We also show that a time-time correlation function converges to the arcsine law for this subordinator, almost surely. This improves recent results of Ben Arous, Bovier and Cerny [Comm. Math. Phys. 282 (2008) 663-695] that obtained similar convergence results in law, with respect to the random environment.

Estimation of the Expected Discounted Penalty Function for Levy Insurance Risks

Abstract: We consider a generalized risk process which consists of a subordinator plus a spectrally negative Levy process. Our interest is to estimate the expected discounted penalty function (EDPF) from a set of data which is practical in the insurance framework. We construct an empirical type estimator of the Laplace transform of the EDPF, and obtain it by a regularized Laplace inversion. The asymptotic behavior of the estimator under a high frequency assumption is investigated.

The fractional Poisson process and the inverse stable subordinator

Abstract: The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also establishes an interesting connection between the fractional Poisson process and Brownian time.

Information-based models for finance and insurance

Abstract: In financial markets, the information that traders have about an asset is reflected in its price. The arrival of new information then leads to price changes. The `information-based framework' of Brody, Hughston and Macrina (BHM) isolates the emergence of information, and examines its role as a driver of price dynamics. This approach has led to the development of new models that capture a broad range of price behaviour. This thesis extends the work of BHM by introducing a wider class of processes for the generation of the market filtration. In the BHM framework, each asset is associated with a collection of random cash flows. The asset price is the sum of the discounted expectations of the cash flows. Expectations are taken with respect (i) an appropriate measure, and (ii) the filtration generated by a set of so-called information processes that carry noisy or imperfect market information about the cash flows. To model the flow of information, we introduce a class of processes termed L\'evy random bridges (LRBs), generalising the Brownian and gamma information processes of BHM. Conditioned on its terminal value, an LRB is identical in law to a L\'evy bridge. We consider in detail the case where the asset generates a single cash flow $X_T$ at a fixed date $T$. The flow of information about $X_T$ is modelled by an LRB with random terminal value $X_T$. An explicit expression for the price process is found by working out the discounted conditional expectation of $X_T$ with respect to the natural filtration of the LRB. New models are constructed using information processes related to the Poisson process, the Cauchy process, the stable-1/2 subordinator, the variance-gamma process, and the normal inverse-Gaussian process. These are applied to the valuation of credit-risky bonds, vanilla and exotic options, and non-life insurance liabilities.

AGING IN REVERSIBLE DYNAMICS OF DISORDERED SYSTEMS. I. emergence of the arcsine law in Bouchaud's asymmetric trap model on the complete graph

Abstract: In this paper the celebrated arcsine aging scheme of G. Ben Arous and J. y Cerny is taken up. Using a brand new approach based on point processes and weak convergence techniques, this scheme is implemented in a wide class of Markov processes that can best be described as Glauber dynamics of discrete disordered systems. More specifically, conditions are given for the underlying clock process (a partial sum process that measures the total time elapsed along paths of a given length) to converge to a subordinator, and this subordinator is constructed explicitly. This approach is illustrated on Bouchaud's asymmetric trap model on the complete graph for which aging is for the first time proved, and the full, optimal picture, obtained.

STOCHASTIC SOLUTIONS OF A CLASS OF HIGHER ORDER CAUCHY PROBLEMS IN ℝd

Abstract: We study solutions of a class of higher order partial differential equations in bounded domains. These partial differential equations appeared first time in the papers of Allouba and Zheng [4], Baeumer, Meerschaert and Nane [10], Meerschaert, Nane and Vellaisamy [37], and Nane [42]. We express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index 0 < β < 1, or by the absolute value of a symmetric α-stable process with 0 < α ≤ 2, independent of the Markov process. In some special cases we represent the solutions by running composition of k independent Brownian motions, called k-iterated Brownian motion for an integer k ≥ 2. We make use of a connection between fractional-time diffusions and higher order partial differential equations established first by Allouba and Zheng [4] and later extended in several directions by Baeumer, Meerschaert and Nane [10].

Quantile clocks

Abstract: Quantile clocks are defined as convolutions of subordinators $L$, with quantile functions of positive random variables We show that quantile clocks can be chosen to be strictly increasing and continuous and discuss their practical modeling advantages as business activity times in models for asset prices We show that the marginal distributions of a quantile clock, at each fixed time, equate with the marginal distribution of a single subordinator Moreover, we show that there are many quantile clocks where one can specify $L$, such that their marginal distributions have a desired law in the class of generalized $s$-self decomposable distributions, and in particular the class of self-decomposable distributions The development of these results involves elements of distribution theory for specific classes of infinitely divisible random variables and also decompositions of a gamma subordinator, that is of independent interest As applications, we construct many price models that have continuous trajectories, exhibit volatility clustering and have marginal distributions that are equivalent to those of quite general exponential L\'{e}vy price models In particular, we provide explicit details for continuous processes whose marginals equate with the popular VG, CGMY and NIG price models We also show how to perfectly sample the marginal distributions of more general classes of convoluted subordinators when $L$ is in a sub-class of generalized gamma convolutions, which is relevant for pricing of European style options

Some Remarks on Special Subordinators

Abstract: A subordinator is called special if the restriction of its potential measure to (0,∞) has a decreasing density with respect to Lebesgue measure. In this note we investigate what type of measures μ on (0,∞) can arise as Levy measures of special subordinators and what type of functions u : (0,∞)→ [0,∞) can arise as potential densities of special subordinators. As an application of the main result, we give examples of potential densities of subordinators which are constant to the right of a positive number. AMS 2000 Mathematics Subject Classification: Primary 60G51, Secondary 60J45, 60J75.

On the dynamics of trap models in Z^d

Abstract: We consider trap models on Z^d, namely continuous time Markov jump process on Z^d with embedded chain given by a generic discrete time random walk, and whose mean waiting time at x is given by tau_x, with tau = (tau_x, x in Z^d) a family of positive iid random variables in the basin of attraction of an alpha-stable law, 0

On additive time-changes of Feller processes

Abstract: In this note we generalise the Phillips theorem [1] on the subordination of Feller processes by Levy subordinators to the class of additive subordinators (i.e. subordinators with independent but possibly nonstationary increments). In the case where the original Feller process is Levy we also express the timedependent characteristics of the subordinated process in terms of the characteristics of the Levy process and the additive subordinator.

A Multivariate Pure-Jump Model with Multi-Factorial Dependence Structure

Abstract: In this work we propose a new approach to build multivariate pure jump processes. We introduce linear and nonlinear dependence, without restrictions on marginal properties, by imposing a multi-factorial structure separately on both positive and negative jumps. Such a new approach provides higher flexibility in calibrating nonlinear dependence than in other comparable Levy models in the literature. Using the notion of multivariate subordinator, this modeling approach can be applied to the class of univariate Levy processes which can be written as the difference of two subordinators. A common example in the financial literature is the variance gamma process, which we extend to the multivariate (multi-factorial) case. The model is tractable and a straightforward multivariate simulation procedure is available. An empirical analysis documents an accurate multivariate fit of stock index returns in terms of both linear and nonlinear dependence. An example of multi-asset option pricing emphasizes the importance of the proposed multivariate approach.

Simulation of multifractal products of Ornstein–Uhlenbeck type processes

Abstract: This paper investigates and provides evidence of the multifractal properties of products of the exponential of Ornstein–Uhlenbeck processes driven by Levy motion. We demonstrate in detail the construction of a multifractal process with gamma subordinator as the background driving Levy process. Simulations are performed for the scenarios corresponding to the normal inverse Gaussian, gamma and inverse Gaussian distributions. The log periodograms and Renyi functions of the simulated processes are also computed to investigate their multifractality.

Variance‐Gamma Model

Abstract: The variance-gamma (VG) process is a stochastic process with independent stationary increments, which allows for flexible parameterization of skewness and kurtosis and is analytically tractable. It can be represented either as a Levy process with VG increments or as a subordinated Brownian motion. We briefly discuss the VG model, parameter estimation issues, applications to option pricing, and extensions allowing for dependence in increments. Keywords: variance-gamma model; normal variance-mixing; heavy tails; skew distribution; subordinator model; option pricing; strictly stationary process

Multiple stratonovich integral and Hu-Meyer formula for Lévy processes

Abstract: In the framework of vector measures and the combinatorial approach to stochastic multiple integral introduced by Rota and Wallstrom [Ann. Probab. 25 (1997) 1257–1283], we present an Ito multiple integral and a Stratonovich multiple integral with respect to a Levy process with finite moments up to a convenient order. In such a framework, the Stratonovich multiple integral is an integral with respect to a product random measure whereas the Ito multiple integral corresponds to integrate with respect to a random measure that gives zero mass to the diagonal sets. A general Hu–Meyer formula that gives the relationship between both integrals is proved. As particular cases, the classical Hu–Meyer formulas for the Brownian motion and for the Poisson process are deduced. Furthermore, a pathwise interpretation for the multiple integrals with respect to a subordinator is given.

Calibration of the subdiffusive arithmetic Brownian motion with tempered stable waiting-times

Abstract: In the classical analysis many models used to real data description are based on the standard Brownian diffusion-type processes. However, some real data exhibit characteristic periods of constant values. In such cases the popular systems seem not to be applicable. Therefore we propose an alternative approach, based on the combination of the popular arithmetic Brownian motion and tempered stable subordinator. The probability density function of the proposed model can be described by a Fokker-Planck type equation and therefore it has many similar properties as the popular arithmetic Brownian motion. In this paper we propose the estimation procedure for the considered tempered stable subdiffusive arithmetic Brownian motion and calibrate the analyzed process to the real financial data.

Absolute ruin in the ornstein-uhlenbeck type risk model

Abstract: We start by showing that the finite-time absolute ruin probability in the classical risk model with constant interest force can be expressed in terms of the transition probability of a positive Ornstein-Uhlenbeck type process, say ˆ X. Our methodology applies to the case when the dynamics of the aggregate claims process is a subordinator. From this expression, we easily deduce necessary and sufficient conditions for the infinite-time absolute ruin to occur. We proceed by showing that, under some technical conditions, the transition density of ˆ X admits a spectral type representation involving merely the limiting distribution of the process. As a by product, we obtain a series expansions for the finite-time absolute ruin probability. On the way, we also derive, for the aforementioned risk process, the Laplace transform of the first-exit time from an interval from above. Finally, we illustrate our results by detailing some examples.

Coag-Frag duality for a class of stable Poisson-Kingman mixtures

Abstract: Exchangeable sequences of random probability measures (partitions of mass) and their corresponding exchangeable bridges play an important role in a variety of areas in probability, statistics and related areas, including Bayesian statistics, physics, finance and machine learning. An area of theoretical as well as practical interest, is the study of coagulation and fragmentation operators on partitions of mass. In this regard, an interesting but formidable question is the identification of operators and distributional families on mass partitions that exhibit interesting duality relations. In this paper we identify duality relations for a large sub-class of mixed Poisson-Kingman models generated by a stable subordinator. Our results are natural generalizations of the duality relations developed in Pitman, Bertoin and Goldschmidt, and Dong, Goldschmidt and Martin for the two-parameter Poisson Dirichlet family. These results are deduced from results for corresponding bridges.