Independent Increments Research Articles

Option Surface Econometrics with Applications

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.

Resistance behavior of Constant-Resistance-Large-Deformation bolt considering surrounding rock pressure

This paper proposes a new analytical solution of the constant resistance of Constant-Resistance-Large-Deformation (CRLD) bolt. The solution incorporates two novel developments: (i) a new solution of the constant resistance based on the thin-walled circle ring theory with the consideration of surrounding rock pressure; and (ii) a new analysis of the interaction between the sleeve and the cone considering both the friction force and compression force. The proposed solution is then validated via the available experimental data and numerical simulations. The results demonstrate that the rock pressure p leads to an independent increment term of the constant resistance, and this increment is proportional to the value of rock pressure; the consideration of compression between the sleeve and the cone results in an amplification of the classical theoretical solution. Moreover, the amplification factor k increases with the increase of the half cone angle. When p = 0 and k = 1, the proposed solution becomes back to the classical theoretical solution.

Continuity and variation analysis of fractional uncertain processes

Abstract The canonical Liu process is a stationary and independent increment uncertain process with normal increments. As one of the most important types of uncertain processes, the fractional Liu process is presented as a variant of canonical Liu process, which has a potential application in modeling an irregular movement with long memory in a system associated with human uncertainty rather than stochastic factors. This paper is devoted to studying the mathematical properties of fractional Liu process such as the increments, variation and sample continuity, especially the Holder continuity of its sample paths. The obtained results lay the theoretical foundation and promote the applications of fractional Liu processes.

Burkholder\u2013Davis\u2013Gundy Inequalities in UMD Banach Spaces

In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X. Assuming that M_0=0, we show that the following two-sided inequality holds for all 1le p<infty : Here gamma ([![M]!]_t) is the L^2-norm of the unique Gaussian measure on X having [![M]!]_t(x^*,y^*):= [langle M,x^*rangle , langle M,y^*rangle ]_t as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of (star ) was proved for UMD Banach functions spaces X. We show that for continuous martingales, (star ) holds for all 0<p<infty , and that for purely discontinuous martingales the right-hand side of (star ) can be expressed more explicitly in terms of the jumps of M. For martingales with independent increments, (star ) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of (star ) for arbitrary martingales implies the UMD property for X. As an application we prove various Itô isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide Itô isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures.

Global sensitivity analysis for stochastic processes with independent increments

This paper is a first attempt to develop a numerical technique to analyze the sensitivity and the propagation of uncertainty through a system with stochastic processes having independent increments as input. Similar to Sobol’ indices for random variables, a meta-model based on Chaos expansions is used and it is shown to be well suited to address such problems. New global sensitivity indices are also introduced to tackle the specificity of stochastic processes. The accuracy and the efficiency of the proposed method is demonstrated on an analytical example with three different input stochastic processes: a Wiener process; an Ornstein–Uhlenbeck process and a Brownian bridge process. The considered output, which is function of these three processes, is a non-Gaussian process. Then, we apply the same ideas on an example without known analytical solution.

Numerical solutions for asymmetric Lévy flights

Levy flights are generalised random walk processes where the independent stationary increments are drawn from a long-tailed α-stable jump length distribution. We consider the formulation of Levy flights, for 0 < α < 1, in terms of a space-fractional diffusion equation which fundamental solutions are the probability density functions. First, we present how to obtain the governing equation of Levy motion from the Fourier transform of the jump distribution. Then, we derive a family of implicit numerical methods to determine the numerical solutions and we study their consistency and stability. Although numerical algorithms for the case 1 < α < 2 have been widely discussed, very few works paid attention to the case we discuss here. We present numerical experiments to show the performance of the numerical methods and to highlight the advantages and disadvantages of the different approaches. In the end we determine the numerical solutions of an initial value problem, that considers an approximation of the Dirac delta function as the initial condition, in order to obtain approximations of the probability density functions.

Reliability assessment and lifetime prediction of degradation processes considering recoverable shock damages

Many products degrade over time and their degradation processes could be affected by instantaneous shocks during field usage. Instantaneous shocks can cause incremental increases to the degradation signals through shock damages, and can also increase the degradation rates of products. In practice, some kinds of products can recover fully or partially from shock damages in a certain period of time. In this article, a degradation model for soft failure is proposed considering continuous degradation processes with recoverable shock damages for reliability assessment and lifetime prediction of products. The random component of the degradation processes is characterized by a Wiener process and the effect of instantaneous shocks on the degradation process is expressed by an exponential function with a residual effect for either partial or full recovery. The impact of shocks affecting the degradation rate is established to be proportional to the shock size. The resulting model includes some existing models as its special cases and can easily be extended to cases where the degradation process follows other random processes with independent increments. Numerical examples are presented to illustrate the applications of the proposed model. Sensitivity analysis and validation of the two-stage parameter estimation approach are conducted based on the simulation.

Independent increment processes: a multilinearity preserving property

ABSTRACT We observe a multilinearity preserving property of conditional expectation for infinite-dimensional independent increment processes defined on some abstract Banach space B. It is similar in nature to the polynomial preserving property analysed greatly for finite-dimensional stochastic processes and thus offers an infinite-dimensional generalization. However, while polynomials are defined using the multiplication operator and as such require a Banach algebra structure, the multilinearity preserving property we prove here holds even for processes defined on a Banach space which is not necessarily a Banach algebra. In the special case of B being a commutative Banach algebra, we show that independent increment processes are polynomial processes in a sense that coincides with a canonical extension of polynomial processes from the finite-dimensional case. The assumption of commutativity is shown to be crucial and in a non-commutative Banach algebra the multilinearity concept arises naturally. Some of our results hold beyond independent increment processes and thus shed light on infinite-dimensional polynomial processes in general.

Independent particle and lot selection of increment samples in the theory of sampling

Introduction. The theory of sampling developed by Pierre Gy does not prove the compatibility and consistency of discrete and continuous selection models. Discrete (independent particle) and continuous (lot) selection models are determined by incompatible properties of increment samples, which prevented from creating the consistent theory of sampling. Research methodology. The inconsistencies are removed at assuming the idea that there are differences in both separate ore lumps or mineral-dressing products and any locally selected parts of the rock mass under test called increment samples simultaneously available in any rock mass under test. The said differences are described by independent particle dispersion and increment samples dispersion. Composite sample permissible error formed in both discrete and continuous selection is attained by selection of the number of particles collected into the increment sample or their parts and the number of increment samples. Particle dispersion, increment sample dispersion, the number of particles in an increment sample and the number of increment samples combined in one formula make up the complete formula of the fundamental sampling error. The development of the theory of sampling. Based on the complete formula, the possibility of obtaining minimum masses of various sizes has been shown. Thus, for one and the same preset permissible error and from one and the same massif under test, it is possible to collect minimum mass of 17.55 kg (individual particle selection), minimum mass of 170.3 kg (collecting with the bucket sampler), and minimum mass of 10 g under the individual particle selection of particle parts (which is fulfilled under X-ray fluorescent on-stream analysis of material). Discrete selection of increment samples without particles destruction is an especial case of continuous selection method when it is possible to accept the condition that increment samples dispersion is equal to zero. It is possible under ideal mixing of the massif under test. Discussion. Minimum mass of a sample is not a constant. It is a function of increment sample mass. Minimum mass in individual particle selection can be accepted as a reference value of the minimum mass. In lot selection it can be significantly higher than the reference one, while in case of reducing particle size it can be significantly lower than the reference one.

Simplified Stochastic Calculus: Multiplicative Compensators and Changes of Measure

The paper develops multiplicative compensation for complex-valued semimartingales and studies some of its consequences. It is shown that the stochastic exponential of any complex-valued semimartingale with independent increments becomes a true martingale after multiplicative compensation, where such compensation is meaningful. This generalization of the Levy-Khintchin formula fills an existing gap in the literature. We further report Girsanov-type results based on non-negative multiplicatively compensated semimartingales. In particular, we obtain a simplified expression for the multiplicative compensator under the new measure.

Continuous-time limits of multi-period cost-of-capital margins

Abstract We consider multi-period cost-of-capital valuation of a liability cash flow subject to repeated capital requirements that are partly financed by capital injections from capital providers with limited liability. Limited liability means that, in any given period, the capital provider is not liable for further payment in the event that the capital provided at the beginning of the period turns out to be insufficient to cover both the current-period payments and the updated value of the remaining cash flow. The liability cash flow is modeled as a continuous-time stochastic process on [ 0 , T ] {[0,T]} . The multi-period structure is given by a partition of [ 0 , T ] {[0,T]} into subintervals, and on the corresponding finite set of times, a discrete-time cost-of-capital-margin process is defined. Our main objective is the analysis of existence and properties of continuous-time limits of discrete-time cost-of-capital-margin processes corresponding to a sequence of partitions whose meshes tend to zero. Moreover, we provide explicit expressions for the limit processes when cash flows are given by Itô diffusions and processes with independent increments.

Judgment on Convergence-in-Distribution of Monte Carlo Tallies Under Autocorrelation

In the Monte Carlo method for criticality calculation, the convergence-in-distribution check of the sample mean of tallies can be approached in terms of the influence range of autocorrelation compared to the total number of generations iterated. In this context, it is necessary to evaluate the attenuation of autocorrelation coefficients (ACCs) over lags. However, in just one replica of calculation, it is difficult to accurately estimate small ACCs at large lags because of the comparability with statistical uncertainty. This paper proposes a method to overcome such an issue. Its essential component is the transformation of a standardized time series of tallies so that the resulting series asymptotically converges in distribution to Brownian motion. The convergence-in-distribution check is then constructed based on the independent increment property of Brownian motion. The judgment criterion is set by way of the spectral analysis of fractional Brownian motion. Numerical results are demonstrated for extreme and standard types of criticality calculation and different numbers of histories per generation. Excellent performance is observed for most replicas of calculation. An issue related to small numbers of generations is addressed for strongly autocorrelated tallies in the extreme type.

Operator-valued chordal Loewner chains and non-commutative probability

We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a C⁎-algebra A. We define an A-valued chordal Loewner chain as a subordination chain of analytic self-maps of the A-valued upper half-plane, such that each Ft is the reciprocal Cauchy transform of an A-valued law μt, such that the mean and variance of μt are continuous functions of t.We relate A-valued Loewner chains to processes with A-valued free or monotone independent increments just as was done in the scalar case by Bauer [8] and Schleißinger [48]. We show that the Loewner equation ∂tFt(z)=DFt(z)[Vt(z)], when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains Ft and vector fields Vt(z) of the form Vt(z)=−Gνt(z) where νt is a generalized A-valued law.Based on the Loewner equation, we derive a combinatorial expression for the moments of μt in terms of νt. We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws μt. Finally, we prove a version of the monotone central limit theorem which describes the behavior of Ft as t→+∞ when νt has uniformly bounded support.

Limit theorems for integrated trawl processes with symmetric Lévy bases

We study long time behavior of integrated trawl processes introduced by Barndorff-Nielsen. The trawl processes form a class of stationary infinitely divisible processes, described by an infinitely divisible random measure (L\'evy base) and a family of shifts of a fixed set (trawl). We assume that the L\'evy base is symmetric and homogeneous and that the trawl set is determined by the trawl function that decays slowly. Depending on the geometry of the trawl set and on the L\'evy measure corresponding to the L\'evy base we obtain various types of limits in law of the normalized integrated trawl processes for large times. The limit processes are always stable and self-similar with stationary increments. In some cases they have independent increments - they are stable L\'evy processes where the index of stability depends on the parameters of the model. We show that stable limits with stability index smaller than 2 may appear even in cases when the underlying L\'evy base has all its moments finite. In other cases, the limit process has dependent increments and it may be considered as a new extension of fractional Brownian motion to the class of stable processes.

On distributions of exponential functionals of the processes with independent increments

The aim of this paper is to study the laws of the exponential functionals of the processes X with independent increments , namely I t = t 0 exp(−X s)ds, t ≥ 0, and also I ∞ = ∞ 0 exp(−X s)ds. Under suitable conditions we derive the integro-differential equations for the density of I t and I ∞. We give sufficient conditions for the existence of smooth density of the laws of these function-als. In the particular case of Levy processes these equations can be simplified and, in a number of cases, solved explicitly.

Monotone increment processes, classical Markov processes, and Loewner chains

We prove one-to-one correspondences between certain decreasing Loewner chains in the upper half-plane, a special class of real-valued Markov processes, and quantum stochastic processes with monotonically independent additive increments. This leads us to a detailed investigation of probability measures on $\mathbb{R}$ with univalent Cauchy transform. We discuss several subclasses of such measures and obtain characterizations in terms of analytic and geometric properties of the corresponding Cauchy transforms. Furthermore, we obtain analogous results for the setting of decreasing Loewner chains in the unit disk, which correspond to quantum stochastic processes of unitary operators with monotonically independent multiplicative increments.

Asymptotic Properties of the Impulse Perturbation Process under the Poisson Approximation Conditions with the Equilibrium Point of Quality Criterion

For a system of stochastic differential equations with Markov switchings and impulse perturbation under the Poisson approximation scheme and the conditions of the existence of a single equilibrium point of the quality criterion, the limit generators for the impulse process and the dynamic system are constructed. The complexity of the proposed evolutionary model lies in its three properties. First, the system is under conditions of an external random effect, which is modeled using the Markov switching process. Processes with independent increments, which also depend on the Markov switching process, have certain characteristics between the moments of its restoration, and at the moments of restoration, these characteristics change. Therefore, the so-called "gluing" of trajectories of processes with independent increments occurs. Second, the model contains a Poisson approximation scheme, which is a generalization of the classical averaging scheme and is determined by normalization depending on a small parameter. In the classical approximation scheme in the limit process, we do not see large jumps in the system. The maximum that we get is the shift of the deterministic trajectory. However in the Poisson approximation scheme, which was invented by Korolyuk and Limnios in the monograph of 2005, this problem is eliminated, therefore, both a deterministic shift and large jumps will take place in the limit. Third, the system has a control function, which is determined using the Robins-Monroe stochastic approximation procedure. This procedure solves the problem of determination of the equilibrium point of the regression function and consists in finding the unique solution to the equation relative to control. Assuming the existence of a unique control on every interval, we solve a two-level problem. The question, how the behavior of the limit process depends on the prelimit normalization of the stochastic system in an ergodic Markov environment is studied in the article.

Fluctuation theory for spectrally positive additive Lévy fields

A spectrally positive additive Levy field is a multidimensional field obtained as the sum $\mathbf {X}_{\mathrm {t}}={\mathrm {X}}^{(1)}_{t_{1}}+{\mathrm {X}}^{(2)}_{t_{2}}+\dots +{\mathrm {X}}^{(d)}_{t_{d}}$, ${\mathrm {t}}=(t_{1},\dots ,t_{d})\in \mathbb {R}_{+}^{d}$, where ${\mathrm {X}}^{(j)}={}^{t} (X^{1,j},\dots ,X^{d,j})$, $j=1,\dots ,d$, are $d$ independent $\mathbb {R}^{d}$-valued Levy processes issued from $\mathbf {0}={}^{t}(0,0,\dots ,0)$, such that $X^{i,j}$ is non decreasing for $i\neq j$ and $X^{j,j}$ is spectrally positive. It can also be expressed as $\mathbf {X}_{\mathrm {t}}=\mathbb {X}_{\mathrm {t}}\cdot {\mathbf {1}}$, where ${\mathbf {1}}={}^{t}(1,1,\dots ,1)$ and $\mathbb {X}_{\mathrm {t}}=(X^{i,j}_{t_{j}})_{1\leq i,j\leq d}$. The main interest of spaLf’s lies in the Lamperti representation of multitype continuous state branching processes. In this work, we study the law of the first passage times $\mathbf {T}_{\mathrm {r}}$ of such fields at levels $-{\mathrm {r}}$, where ${\mathrm {r}}\in \mathbb {R}_{+}^{d}$. We prove that the field $\{(\mathbf {T}_{\mathrm {r}},\mathbb {X}_{\mathbf {T}_{\mathrm {r}}}),{\mathrm {r}}\in \mathbb {R}_{+}^{d}\}$ has stationary and independent increments and we describe its law in terms of this of the spaLf $\mathbf {X}$. In particular, the Laplace exponent of $(\mathbf {T}_{\mathrm {r}},\mathbb {X}_{\mathbf {T}_{\mathrm {r}}})$ solves a functional equation leaded by the Laplace exponent of $\mathbf {X}$. This equation extends in higher dimension a classical fluctuation identity satisfied by the Laplace exponents of the ladder processes. Then we give an expression of the distribution of $\{(\mathbf {T}_{\mathrm {r}},\mathbb {X}_{\mathbf {T}_{\mathrm {r}}}),{\mathrm {r}}\in \mathbb {R}_{+}^{d}\}$ in terms of the distribution of $\{\mathbb {X}_{\mathrm {t}},{\mathrm {t}}\in \mathbb {R}_{+}^{d}\}$ by the means of a Kemperman-type formula, well-known for spectrally positive Levy processes.

Extended Large Deviation Principle for Trajectories of Processes with Independent and Stationary Increments on the Half-line

Extended Large Deviation Principle for Trajectories of Processes with Independent and Stationary Increments on the Half-line

A Sequential Bayesian Updated Wiener Process Model for Remaining Useful Life Prediction

Wiener processes have been extensively used to model the degradation processes exhibiting a linear trend for predicting the remaining useful life (RUL) of degrading components. To incorporate the real-time degradation monitoring information into degradation modeling, the Bayesian method has been frequently utilized to update the model parameter, particularly for the drift parameter in Wiener process. However, due to the inherent independent increment and Markov properties of Wiener process, the Bayesian updated drift parameter only utilizes the current degradation measurement and cannot incorporate the whole degradation measurements up to now. As such, once the updated degradation model in this way is used to predict the RUL, the obtained result may be dominated by partial degradation observations or lower the prognosis accuracy. In this paper, we propose a sequential Bayesian updated Wiener process model for RUL prediction. First, a Wiener process model with random drift efficient is used to model the degradation process with the linear trend. To estimate the model parameters, the historical degradation measurements are used to determine the initial model parameters based on the maximum likelihood estimation (MLE) method. Then, for the degrading component in service, a sequential Bayesian method is proposed to update the random drift parameter in Wiener process model. Differing from existing studies using the Bayesian method, the proposed sequential method uses the Bayesian estimate for random drift parameter in the last time as the prior of the next time. As such, the Bayesian estimate for random drift parameter in the current time is dependent on the whole degradation measurements up to current time, and thus the problem of depending only on the current degradation measurement is solved. Finally, we derive the analytical expressions of the RUL distribution based on the concept of the first passage time (FPT). Two case studies associated with the gyroscope drift data and lithium-ion battery data are provided to show the effectiveness and superiority of the proposed method. The results indicate that the proposed method can improve the RUL prediction accuracy.