Process Xt Research Articles

Modeling and reliability analysis of systems subject to multiple sources of degradation based on Lévy processes

In this paper we present a framework to model the effect of multiple sources of degradation on deteriorating systems, and to find easy-to-evaluate expressions for important reliability quantities. By considering the system deterioration as a general increasing Lévy process (i.e., a subordinator), which is a process with independent, stationary and non-negative increments, the proposed methodology allows the modeling of shock-based degradation (in the form of compound Poisson processes) with different distributions of shock sizes, progressive degradation as deterministic linear drift plus a Gamma process, and multiple sources of degradation by the superposition of any of these models into the same mathematical formalism. In addition, we obtain expressions for the reliability quantities (reliability function, and probability density and moments of the lifetime L), and the mean and central moments of the deterioration process Xt. Moreover, we present efficient and accurate numerical methods to compute the reliability quantities and to simulate sample paths. Several deterioration models are compared in terms of their reliability quantities, the simulated sample paths and the feasible moments of the deterioration Xt. Furthermore, we propose a method to mix different Lévy models that extend the available moments with possible applications to data fitting. The results demonstrate the generality, versatility, efficiency and accuracy of the proposed Lévy degradation framework, which can open a new productive research field in the area of probabilistic degradation models.

Intrinsic ultracontractivity of Feynman–Kac semigroups for symmetric jump processes

Consider a symmetric non-local Dirichlet form (D,D(D)) given byD(f,f)=∫Rd∫Rd(f(x)−f(y))2J(x,y)dxdy with D(D) the closure of the set of C1 functions on Rd with compact support under the norm D1(f,f), where D1(f,f):=D(f,f)+∫f2(x)dx and J(x,y) is a nonnegative symmetric measurable function on Rd×Rd. Suppose that there is a Hunt process (Xt)t⩾0 on Rd corresponding to (D,D(D)), and that (L,D(L)) is its infinitesimal generator. We study the intrinsic ultracontractivity for the Feynman–Kac semigroup (TtV)t⩾0 generated by LV:=L−V, where V⩾0 is a non-negative locally bounded measurable function such that the Lebesgue measure of the set {x∈Rd:V(x)⩽r} is finite for every r>0. By using intrinsic super Poincaré inequalities and establishing an explicit lower bound estimate for the ground state, we present general criteria for the intrinsic ultracontractivity of (TtV)t⩾0. In particular, ifJ(x,y)≍|x−y|−d−α1{|x−y|⩽1}+e−|x−y|γ1{|x−y|>1} for some α∈(0,2) and γ∈(1,∞], and the potential function V(x)=|x|θ for some θ>0, then (TtV)t⩾0 is intrinsically ultracontractive if and only if θ>1. When θ>1, we have the following explicit estimates for the ground state ϕ1c1exp⁡(−c2θγ−1γ|x|logγ−1γ⁡(1+|x|))⩽ϕ1(x)⩽c3exp⁡(−c4θγ−1γ|x|logγ−1γ⁡(1+|x|)), where ci>0(i=1,2,3,4) are constants. We stress that our method efficiently applies to the Hunt process (Xt)t⩾0 with finite range jumps, and some irregular potential function V such that lim|x|→∞⁡V(x)≠∞.

The multifractal nature of Boltzmann processes

We consider the spatially homogeneous Boltzmann equation for (true) hard and moderately soft potentials. We study the pathwise properties of the stochastic process (Vt)t≥0, which describes the time evolution of the velocity of a typical particle. We show that this process is almost surely multifractal and compute its spectrum of singularities. For hard potentials, we also compute the multifractal spectrum of the position process (Xt)t≥0.

Algebraic convergence of diffusion processes on ℝ n with radial diffusion and drift coefficients

We consider the diffusion process Xt on R n with radial diffusion and drift coefficients. We prove that once the one-dimensional diffusion |Xt| has algebraic L 2 -convergence, so does Xt. And some classical examples are discussed.

Convergence of trimmed Lévy processes to trimmed stable random variables at [formula omitted

Let (r,s)Xt be the Lévy process Xt with r largest jumps and s smallest jumps up till time t deleted and let (r)X˜t be Xt with r largest jumps in modulus up till time t deleted. We show that ((r,s)Xt−at)/bt or ((r)X˜t−at)/bt converges to a proper nondegenerate nonnormal limit distribution as t↓0 if and only if (Xt−at)/bt converges as t↓0 to an α-stable random variable, with 0<α<2, where at and bt>0 are nonstochastic functions in t. Together with the asymptotic normality case treated in Fan (2015) [7], this completes the domain of attraction problem for trimmed Lévy processes at 0.

Infection Spread in Random Geometric Graphs

In this paper we study the speed of infection spread and the survival of the contact process in the random geometric graph G = G(n, rn, f) of n nodes independently distributed in S = [-½, ½]2 according to a certain density f(·). In the first part of the paper we assume that infection spreads from one node to another at unit rate and that infected nodes stay in the same state forever. We provide an explicit lower bound on the speed of infection spread and prove that infection spreads in G with speed at least D1nrn2. In the second part of the paper we consider the contact process ξt on G where infection spreads at rate λ &gt; 0 from one node to another and each node independently recovers at unit rate. We prove that, for every λ &gt; 0, with high probability, the contact process on G survives for an exponentially long time; there exist positive constants c1 and c2 such that, with probability at least 1 - c1 / n4, the contact process starting with all nodes infected survives up to time tn = exp(c2n/logn) for all n.

On the properties of dynamic value functions in the problem of optimal investment in incomplete markets

Abstract We study the analytical properties of a dynamic value function and of an optimal solution to the utility maximization problem in incomplete markets for utility functions defined on the whole real line. It was shown by Kramkov and Sirbu [Ann. Appl. Probab. 16 (2006), no. 3, 1352–1384] that if the relative risk-aversion coefficient of the utility function defined on the half real line is uniformly bounded away from zero and infinity, then the value function at time t = 0 of utility maximization problem is two-times differentiable and the optimal wealth is differentiable in probability with respect to the initial capital. Similar results are true for utility functions defined on the whole real line if instead of relative risk-aversion the same condition on the risk-aversion coefficient is imposed. Besides, assuming the continuity of the filtration F we prove that the derivative of the optimal wealth is strictly positive and that the derivative exists also in the sense of L 1-convergence. This enables us to show the existence of a strictly increasing (with respect to the initial capital) and absolutely continuous modification of the optimal wealth. We also study the differentiability properties of the value function V(t,x) and of the optimal wealth process Xt (x) for any t ∈ [0,T] and give the conditions under which the second derivative of the value function and the derivative of the optimal wealth process are continuous. We need these properties to show that the value function satisfies a certain backward stochastic partial differential equation used to characterize the optimal wealth process.

Expectations of Functions of Stochastic Time with Application to Credit Risk Modeling

We develop two novel approaches to solving for the Laplace transform of a time-changed stochastic process. We discard the standard assumption that the background process (Xt) is Levy. Maintaining the assumption that the business clock (Tt) and the background process are independent, we develop two different series solutions for the Laplace transform of the time-changed process X-tildet=X(Tt). In fact, our methods apply not only to Laplace transforms, but more generically to expectations of smooth functions of random time. We apply the methods to introduce stochastic time change to the standard class of default intensity models of credit risk, and show that stochastic time-change has a very large effect on the pricing of deep out-of-the-money options on credit default swaps.

Laws of the iterated logarithm for self-normalised Lévy processes at zero

We develop tools and methodology to establish laws of the iterated logarithm (LILs) for small times (as t ↓ 0 t\downarrow 0 ) for the “self-normalised” process ( X t − a t ) / V t (X_{t}-at)/\sqrt {V_{t}} , t &gt; 0 t&gt;0 , constructed from a Lévy process ( X t ) t ≥ 0 (X_{t})_{t\geq 0} having quadratic variation process ( V t ) t ≥ 0 (V_{t})_{t\geq 0} , and an appropriate choice of the constant a a . We apply them to obtain LILs when X t X_{t} is in the domain of attraction of the normal distribution as t ↓ 0 t\downarrow 0 , when X t X_{t} is symmetric and in the Feller class at 0, and when X t X_{t} is a strictly α − \alpha - stable process. When X t X_{t} is attracted to the normal distribution, an important ingredient in the proof is a Cramér-type theorem which bounds above the distance of the distribution of the self-normalised process from the standard normal distribution.

A brief note on computing average state occupation times

AbstractOBJECTIVEThis review discusses how biometricians would probably compute or estimate expected waiting times, if they had the data.METHODSOur framework is a time-inhomogeneous Markov multistate model, where all transition hazards are allowed to be time-varying. We assume that the cumulative transition hazards are given. That is, they are either known, as in a simulation, determined by expert guesses, or obtained via some method of statistical estimation. Our basic tool is product integration, which transforms the transition hazards into the matrix of transition probabilities. Product integration enjoys a rich mathematical theory, which has successfully been used to study probabilistic and statistical aspects of multistate models. Our emphasis will be on practical implementation of product integration, which allows us to numerically approximate the transition probabilities. Average state occupation times and other quantities of interest may then be derived from the transition probabilities.(ProQuest: ... denotes formulae omitted.)1. IntroductionThis review is motivated by an interdisciplinary workshop on 'Multistate event history analysis' at the Netherlands Interdisciplinary Demographic Institute, The Hague, in April 2011. The workshop brought together demographers and biometricians, who both use multistate models, but appear to follow somewhat different methodological traditions.At the workshop, there were discussions on how to compute average state occupation times. The organizer, Frans Willekens, informed us that the most common approaches in demography use either piecewise linear approximations of the survival function or piece- wise constant transition hazards (personal communication). See also Gill and Keilman (1990) on these approaches, including a critique of the former method.We review how biometricians might compute expected waiting times, if they had the data. In biometry, the restriction is that right-censoring typically precludes evaluating waiting time distributions on the whole of their support. In other words, the maximum follow-up in most of the data sets analyzed by biometricians is considerably smaller than the assumed maximum age in the population under study. As a consequence, it is more common to consider median waiting times (e.g., Brookmeyer and Crowley 1982) or expectations restricted to a maximum follow-up (e.g., Andersen et al. 1993, Example IV.3.8). In the context of demography, expected (restricted) waiting times are arguably more relevant, for instance for policy making.The key idea is that a certain transformation, product integration, allows us to move from the transition hazards of a time-inhomogeneous Markov process to the matrix of transition probabilities. As also noted by Gill and Keilman (1990), combined with the initial distribution of the process, this allows us to derive expected waiting times in a given state and other quantities of interest.In demography, product integration appears to be rarely used. Gill and Keilman (1990) mention the relation, but then proceed to attack a different problem, namely estimation of constant transition hazards with population registry data. In a recent tutorial on multistate methods, Kuo, Suchindran, and Koo (2008) value product integration as a 'basic tool', but argue that it 'is difficult to implement.' These authors therefore proceed to work under special assumptions such as uniform right-censoring. Another reference is Schoen (2005) who mentions product integration as a tool for numerical evaluation, but then concentrates on special cases with an analytical solution.2. The relation between transition hazards and transition probabilitiesConsider a time-inhomogeneous Markov process (Xt)t≥0 with state space {0, 1,2, . . . ,J}. We assume that (Xt )t≥0 has right-continuous sample paths, which are constant between transition times. That is, if the process moves from state j to state k, j = k, at time t0 , Xt0 = k and Xt0 - = j . …

Generic Identication of Binary-Valued Hidden Markov Processes

The generic identication problem is to decide whether a stochastic process (Xt) is ahidden Markov process and if yes to infer its parameters for all but a subset of parametrizationsthat form a lower-dimensional subvariety in parameter space. Partial answers so far availabledepend on extra assumptions on the processes, which are usually centered around stationarity.Here we present a general solution for binary-valued hidden Markov processes. Our approach isrooted in algebraic statistics hence it is geometric in nature. We nd that the algebraic varietiesassociated with the probability distributions of binary-valued hidden Markov processes are zerosets of determinantal equations which draws a connection to well-studied objects from algebra. Asa consequence, our solution allows for algorithmic implementation based on elementary (linear)algebraic routines.

A prediction theory for a coloured noise-driven stochastic differential system

The standard approaches to analyse the Itô stochastic differential system are the Fokker–Planck equation and multi-dimensional Itô differential rule. In contrast to the Itô stochastic differential system, this paper develops a mathematical theory of a coloured noise process-driven stochastic differential system. More precisely, the coloured noise process-driven stochastic differential equation x˙t=f(xt)+g(xt) ξt is the subject of investigations. The statistical properties of the input noise process ξt are stationary and finite, non-zero, relatively smaller correlation time. The notion of “stochastic equivalence” coupled with stochastic differential rule plays the key role to develop the theory of this paper. The theory of the paper will be of use to analysing and control of dynamical systems embedded in the coloured noise environment.

Identification of a noncausal Itô process from the stochastic Fourier coefficients

Let Xt be a noncausal Itô process of Skorokhod type driven by the Brownian motion W., that is, a stochastic process of the form dXt=b(t,ω)dt+a(t,ω)dWt where the term a(⋅)dWt is understood as Skorokhod integral. For such an Itô process Xt we consider the Fourier coefficient Fn(dX) of the differential dXt by Fn(dX)=∫01en(t)¯dXt, en(t)=exp(2π−1nt) (n∈Z) and we are concerned with the elementary question: whether we can identify the two parameters a(⋅,ω), b(⋅,ω) from the complete set of the stochastic Fourier coefficients {Fn(dX),n∈Z}. In this note we study this problem in a framework of noncausal calculus, as we did in the previous articles (Ogawa, 2013; Ogawa and Uemura, in press), and we give an affirmative answer with a concrete scheme for the reconstruction of the parameters a(⋅,ω), b(t,ω). Our result will give another light to the theoretical background of the method of Fourier series for the volatility estimation proposed by P. Malliavin et al. (Malliavin and Mancino, 2002; Malliavin and Thalmaier, 2009).

Non-parametric adaptive estimation of the drift for a jump diffusion process

In this article, we consider a jump diffusion process (Xt)t≥0 observed at discrete times t=0,Δ,…,nΔ. The sampling interval Δ tends to 0 and nΔ tends to infinity. We assume that (Xt)t≥0 is ergodic, strictly stationary and exponentially β-mixing. We use a penalised least-square approach to compute two adaptive estimators of the drift function b. We provide bounds for the risks of the two estimators.

Statistical inference of spectral estimation for continuous-time MA processes with finite second moments

In this paper we investigate a continuous-time MA (moving average) process (Xt)t≥0 sampled at an equally spaced time grid {Δ,2Δ, …, nΔ}, where the grid distance Δ > 0 is fixed and n denotes the number of observations, in the frequency domain. We derive for the process (XkΔ)k∈ℕ with finite second moments the asymptotic behavior of the periodogram and of the lag-window spectral density estimator. The periodogram is not a consistent estimator for the spectral density of (XkΔ)k∈ℕ. Different periodogram frequencies are asymptotically independent and exponentially distributed like for ARMA processes in discrete time. This result is basic for frequency bootstraps. In contrast, the lag-window spectral density estimator is a consistent estimator for the spectral density of (XkΔ)k∈ℕ and moreover, it is asymptotically normally distributed.

Asymptotics for the First Passage Times of Lévy Processes and Random Walks

We study the exact asymptotics for the distribution of the first time, τx, a Lévy process Xt crosses a fixed negative level -x. We prove that ℙ{τx &gt;t} ~V(x) ℙ{Xt≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τx&gt;t} explicitly in both light- and heavy-tailed cases.

Excursions and path functionals for stochastic processes with asymptotically zero drifts

We study discrete-time stochastic processes (Xt) on [0,∞) with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at x is about c/x. Our focus is the recurrent case (when c is not too large). We give sharp asymptotics for various functionals associated with the process and its excursions, including results on maxima and return times. These results include improvements on existing results in the literature in several respects, and also include new results on excursion sums and additive functionals of the form ∑s≤tXsα, α>0. We make minimal moments assumptions on the increments of the process. Recently there has been renewed interest in Lamperti-type process in the context of random polymers and interfaces, particularly nearest-neighbour random walks on the integers; some of our results are new even in that setting. We give applications of our results to processes on the whole of R and to a class of multidimensional ‘centrally biased’ random walks on Rd; we also apply our results to the simple harmonic urn, allowing us to sharpen existing results and to verify a conjecture of Crane et al.

Expectations of Functions of Stochastic Time with Application to Credit Risk Modeling

We develop two novel approaches to solving for the Laplace transform of a time-changed stochastic process. We discard the standard assumption that the background process (Xt) is Lévy. Maintaining the assumption that the business clock (Tt) and the background process are independent, we develop two different series solutions for the Laplace transform of the time-changed process Xt=X(Tt). In fact, our methods apply not only to Laplace transforms, but more generically to expectations of smooth functions of random time. We apply the methods to introduce stochastic time change to the standard class of default intensity models of credit risk, and show that stochastic time-change has a very large effect on the pricing of deep out-of-the-money options on credit default swaps.

A Note on Malliavin Fractional Smoothness for Lévy Processes and Approximation

Assume a Levy process (Xt)t ∈ [0,1] that is an L2-martingale and let Y be either its stochastic exponential or X itself. For certain integrands φ we investigate the behavior of $$ \bigg \|\int_{(0,1]} {{\varphi}}_t dX_t - \sum_{k=1}^N v_{k-1} (Y_{t_k}-Y_{t_{k-1}}) \bigg \|_{L_2}, $$ where vk − 1 is \({\mathcal{F}}_{t_{k-1}}\)-measurable, in dependence on the fractional smoothness in the Malliavin sense of \(\int_{(0,1]} {{\varphi}}_t dX_t\). A typical situation where these techniques apply occurs if the stochastic integral is obtained by the Galtchouk–Kunita–Watanabe decomposition of some f(X1). Moreover, using the example f(X1) = 1(K, ∞ )(X1) we show how fractional smoothness depends on the distribution of the Levy process.

Nonparametric estimation of the derivatives of the stationary density for stationary processes

In this article, our aim is to estimate the successive derivatives of the stationary density f of a strictly stationary and β -mixing process (Xt )t≥0 . This process is observed at discrete times t = 0,Δ, ... ,nΔ . The sampling interval Δ can be fixed or small. We use a penalized least-square approach to compute adaptive estimators. If the derivative f (j ) belongs to the Besov space , then our estimator converges at rate (nΔ )−α /(2α +2j +1) . Then we consider a diffusion with known diffusion coefficient. We use the particular form of the stationary density to compute an adaptive estimator of its first derivative f ′. When the sampling interval Δ tends to 0, and when the diffusion coefficient is known, the convergence rate of our estimator is (nΔ )−α /(2α +1) . When the diffusion coefficient is known, we also construct a quotient estimator of the drift for low-frequency data.