Stochastic Discontinuities Research Articles

Term structure modeling with overnight rates beyond stochastic continuity

AbstractOvernight rates, such as the Secured Overnight Financing Rate (SOFR) in the United States, are central to the current reform of interest rate benchmarks. A striking feature of overnight rates is the presence of jumps and spikes occurring at predetermined dates due to monetary policy interventions and liquidity constraints. This corresponds to stochastic discontinuities (i.e., discontinuities occurring at ex ante known points in time) in their dynamics. In this work, we propose a term structure modeling framework based on overnight rates and characterize absence of arbitrage in a generalized Heath–Jarrow–Morton (HJM) setting. We extend the classical short‐rate approach to accommodate stochastic discontinuities, developing a tractable setup driven by affine semimartingales. In this context, we show that simple specifications allow to capture stylized facts of the jump behavior of overnight rates. In a Gaussian setting, we provide explicit valuation formulas for bonds and caplets. Furthermore, we investigate hedging in the sense of local risk‐minimization when the underlying term structures feature stochastic discontinuities.

Term structure modelling for multiple curves with stochastic discontinuities

We develop a general term structure framework taking stochastic discontinuities explicitly into account. Stochastic discontinuities are a key feature in interest rate markets, as for example the jumps of the term structures in correspondence to monetary policy meetings of the ECB show. We provide a general analysis of multiple curve markets under minimal assumptions in an extended HJM framework and provide a fundamental theorem of asset pricing based on NAFLVR. The approach with stochastic discontinuities permits to embed market models directly, unifying seemingly different modelling philosophies. We also develop a tractable class of models, based on affine semimartingales, going beyond the requirement of stochastic continuity.

Nonparametric Regression with Stochastic Boundary and Regression Discontinuity with Endogenous Cutoff

We augment the usual regression discontinuity design model by considering an endogenously chosen cutoff, perhaps chosen to maximize certain criterion that the treatment provider has. This regime faces the challenge that, conditional on realization of the cutoff, observations are no longer i.i.d. We develop conditions under which an asymptotic expansion of the locally linear estimator contains a bias term caused by the endogeneity of order op(h2 +1/√nh). The lower order bias justifies the usual optimal bandwidth selection and bias correction procedures in this setting, though it places constraints on the maximal degree of undersmoothing.

Itô’s formula for Gaussian processes with stochastic discontinuities

We introduce a Skorokhod type integral and prove an Itô formula for a wide class of Gaussian processes which may exhibit stochastic discontinuities. Our Itô formula unifies and extends the classical one for general (i.e., possibly discontinuous) Gaussian martingales in the sense of Itô integration and the one for stochastically continuous Gaussian non-martingales in the Skorokhod sense, which was first derived in Alòs et al. (Ann. Probab. 29 (2001) 766–801).

Meticulous three-dimensional demarcation and separate presentation of statistically homogeneous structural domains within a large-scale jointed rock mass

Traditional methods used to demarcate homogeneous structural domains are mainly concerned with small-scale discontinuities; this approach can be regarded as a process for evaluating the degree of homogeneity among stochastic discontinuities from one site to another. This process forms the basis for determining the distribution of stochastic discontinuities within a homogeneous structural domain; however, it neglects large-scale discontinuities. In this study, the statistically homogeneous structural domains within a large-scale jointed rock mass were meticulously demarcated in three dimensions, which was performed in four steps. 1) A three-dimensional discontinuity network model was constructed containing both deterministic and stochastic discontinuities. 2) The model was subsequently discretized into a large number of cubic grids, and the MBi values of all the grids were calculated. 3) Using the K-means algorithm, the structural domain boundaries within the tested rock mass were meticulously demarcated, and a composited model consisting of homogeneous structural domains was constructed. 4) Each homogeneous structural domain was separately visualized. This demarcation method can consider large-scale discontinuities and improve the identification sensitivity in a large-scale jointed rock mass with a high accuracy. Moreover, the separate presentation of structural domains directly reflects the morphological properties unique to each structural domain.

Level set methods for stochastic discontinuity detection in nonlinear problems

Stochastic problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zeros of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods.

Simulating Strength Parameters and Size Effect of Stochastic Jointed Rock Mass using DEM Method

The strength parameters and the size effect of stochastic jointed limestone rock mass is investigated in this paper. Based on extensive statistics of joint parameters of rock mass in the research region, the probable distribution of geometric characteristic parameters of discontinuities are obtained by the probability graph method. Then the Monte-Carlo method is used for discontinuities network modeling. In addition, 3DEC software and its built-in FISH programming language are used to establish the stochastic jointed rock mass network model based on discrete element method. Triaxial numerical simulation tests under variable confining pressure are conducted with different model sizes and dip angles of bedding planes. The numerical simulation results indicate that the jointed rock mass exhibits weak anisotropy property and significant size effect when it is cut by stochastic discontinuities; the mechanical strength parameters of rock mass begins to fluctuate distinctly as the model size increases, and tend to be stable once the model size reaches or exceeds 4 m × 4 m × 8 m. Besides, the comprehensive mechanical parameters of rock mass in the research region are determined and failure modes of rock mass are analyzed as well based on the numerical simulation results.

Study on strength parameters of rock mass with stochastic combination discontinuities and their application in a tunnel in China

The rock mass has special properties, such as inhomogeneity, anisotropy, discontinuity, and nonelastic, due to various internal or external stress in the process of its formation. In this study, similar materials are considered to simulate the rock mass and analyze the failure law using laboratory tri-axial tests based on the similarity principle. The rock mass and discontinuity are constituted in the process of model construction by considering the influence of the orientation, spacing, and number of discontinuity, respectively. Then, the influences of the orientation, spacing, number of discontinuity, and the combination of different discontinuous strength on rupture mechanism of the rock mass are analyzed by considering lots of numerical test schemes using the discontinuous deformation analysis method. Finally, considering water conveyance tunnel in Jurong Pumped Storage Power Station in China as a test case, the tunnel stability under different discontinuities combination is analyzed using discontinuous deformation analysis method. The influence of the optimal tunnel axis and tunnel shape on the stability of surrounding rock is studied by comparative analysis of surrounding rock stability under different tunnel axis and tunnel shapes. The stress concentration in case of the circular tunnel with corner is somehow less than that of the rectangular one.

Analysing reward measures of LARES performability models by discontinuous Markov chains

This paper presents a new method for specifying and analysing Markovian performability models. An extension of the LARES modelling language is considered which offers both delayed and immediate transitions, as well as rate and impulse rewards on whose basis different types of reward measures can be defined. The paper describes the evaluation path, starting from the modular and hierarchical LARES description and leading via a flat labelled transition system to the underlying stochastic model. The latter is a continuous-time Markov chain with fast transitions which, by taking the limit of the fast transition rates, can be interpreted as a CTMC with stochastic discontinuities. Finally, by continuisation of impulse rewards and an aggregation process, a standard Markov reward model is obtained.

Generation and verification of three-dimensional network of fractured rock masses stochastic discontinuities based on digitalization

Accuracy of seepage simulation in rock masses is directly influenced by related geometrical parameters of DFN (discrete fracture network) models. In this paper a well-planned investigation of rock mass discontinuities was conducted on well-exposed outcrops in Beishan (Gansu province), one of the main candidate sites for the Chinese high-level radioactive waste repository using digital techniques, e.g. global positioning system, close range digital photogrammetry and geographical information system. This allows for very efficient collection of large volumes of data, over 10,000 discontinuities were obtained at the study site (Jijicao block), based on statistical parameters two homogeneities were identified, and the dominant sets were also delineated, which suggested that two sets exist in homogeneity I, while three sets in homogeneity II. In homogeneity I, The optimal fracture diameter probability distributions of both sets were lognormal. Three-dimensional stochastic DFN models were established for both homogeneities based on stochastic process as well as statistical parameters. The reliability of the model was validated by both graphical and numerical methods. Results showed that: digital techniques used in this study can overcome the problem of sample size and precision restriction posed by traditional methods, which leads to a more effective application of DFN model, and are thus suitable for tasks such as detailed mapping of rock masses in large area. It is believed that this verified DFN model is more appropriate for subsequent calculations of mechanical properties and seepage of fractured rock masses.

Subcell resolution in simplex stochastic collocation for spatial discontinuities

Subcell resolution has been used in the Finite Volume Method (FVM) to obtain accurate approximations of discontinuities in the physical space. Stochastic methods are usually based on local adaptivity for resolving discontinuities in the stochastic dimensions. However, the adaptive refinement in the probability space is ineffective in the non-intrusive uncertainty quantification framework, if the stochastic discontinuity is caused by a discontinuity in the physical space with a random location. The dependence of the discontinuity location in the probability space on the spatial coordinates then results in a staircase approximation of the statistics, which leads to first-order error convergence and an underprediction of the maximum standard deviation. To avoid these problems, we introduce subcell resolution into the Simplex Stochastic Collocation (SSC) method for obtaining a truly discontinuous representation of random spatial discontinuities in the interior of the cells discretizing the probability space. The presented SSC–SR method is based on resolving the discontinuity location in the probability space explicitly as function of the spatial coordinates and extending the stochastic response surface approximations up to the predicted discontinuity location. The applications to a linear advection problem, the inviscid Burgers’ equation, a shock tube problem, and the transonic flow over the RAE 2822 airfoil show that SSC–SR resolves random spatial discontinuities with multiple stochastic and spatial dimensions accurately using a minimal number of samples.

A Simple Genetic Algorithm for Calibration of Stochastic Rock Discontinuity Networks

We present a novel approach for calibration of stochastic discontinuity network parameters based on genetic algorithms (GAs). To validate the approach, examples of application of the method to cases with known parameters of the original Poisson discontinuity network are presented. Parameters of the model are encoded as chromosomes using a binary representation, and such chromosomes evolve as successive generations of a randomly generated initial population, subjected to GA operations of selection, crossover and mutation. Such back-calculated parameters are employed to make assessments about the inference capabilities of the model using different objective functions with different probabilities of crossover and mutation. Results show that the predictive capabilities of GAs significantly depend on the type of objective function considered; and they also show that the calibration capabilities of the genetic algorithm can be acceptable for practical engineering applications, since in most cases they can be expected to provide parameter estimates with relatively small errors for those parameters of the network (such as intensity and mean size of discontinuities) that have the strongest influence on many engineering applications.

Self-organized criticality in superferromagnets

Superferromagnetic structures are studied accounting for quantum fluctuations due to the discrete level structure and disorder within the randomly jumping interacting moments model. The occurrence of self-organized criticality is found to indicate an existence of spinodal regions and critical points in magnetic state equation and phase diagram. The magnetodynamics show jerky behaviour displayed as erratic stochastic discontinuities for magnetic induction. Magnetic noise correlations are proposed as model-independent analytical tools employed in order to specify, quantify and analyse magnetic structure and origin of superferromagnetism.

Identifying rock blocks based on hierarchical rock-mass structure model

Rock-masses are divided into many closed blocks by deterministic and stochastic discontinuities and engineering interfaces in complex rock-mass engineering. Determining the sizes, shapes, and adjacent relations of blocks is important for stability analysis of fractured rock masses. Here we propose an algorithm for identifying spatial blocks based on a hierarchical 3D Rock-mass Structure Model (RSM). First, a model is built composed of deterministic discontinuities, engineering interfaces, and the earth’s surface, and the deterministic blocks surrounded by these interfaces are traced. Then, in each deterministic block, a network model of stochastic discontinuities is built and the stochastic blocks are traced. Building a unitary wire frame that connects all interfaces seamlessly is the key for our algorithm to identify the above two kinds of blocks. Using this algorithm, geometric models can be built for block theory, discrete element method, and discontinuous deformation analysis.

A Template Model for Defect Simulation for Evaluating Nondestructive Testing in X-Radiography

This paper proposes a new template model for the simulation of casting defects which are classified, according to shape, into three main types: the single defect with circular or elliptical shape, shrinkage defects with stochastic discontinuities, and the cavity or sponge shrinkage defects. For effective simulation, different nesting stencil plates are designed to reflect the characteristics of different casting defects. These include intensity, orientation, size, and shape. The proposed approach also uses geometric diffusion to demonstrate the production of simulated defects with effective shading and contrast when compared to their background. In order to evaluate the effectiveness of the proposed approach, the simulated casting defects are superposed on real radioscopic images of casting pieces and compared with real defects by extensive visual inspection. On the other hand, in order to verify the similarity of the simulated defects and the real defects, we have used our defect inspection algorithm to recognize both real and simulated defects in the same image. The experimental results show that the proposed defect simulation approach can produce a large range of simulated casting defects, which can be utilized as sample images to tune the parameters of casting inspection algorithms.

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to SSFEM. It approximates the solution in the stochastic space using Lagrange polynomial interpolation. The collocation method requires only repetitive calls to an existing deterministic solver, similar to the Monte Carlo method. However, both the SSFEM and current sparse grid collocation methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge very slowly or even fail to converge. In this paper, we develop an adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. Numerical examples, especially for problems related to long-term integration and stochastic discontinuity, are presented. Comparisons with Monte Carlo and multi-element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method.

Compositionality for Markov reward chains with fast and silent transitions

A parallel composition is defined for Markov reward chains with stochastic discontinuity, and with fast and silent transitions. In this setting, compositionality with respect to the relevant aggregation preorders is established. For Markov reward chains with fast transitions the preorders are τ -lumping and τ -reduction. Discontinuous Markov reward chains are ‘limits’ of Markov reward chains with fast transitions, and have related notions of lumping and reduction. Markov reward chains with silent transitions are equivalence classes of Markov reward chains with fast transitions and come equipped with the lifted preorders τ ∼ -lumping and τ ∼ -reduction. In total, six compositionality results are presented. Additionally, the parallel operators involved are related by a continuity result.

Influence of Stochastic Discontinuity Network Parameters on the Formation of Removable Blocks in Rock Slopes

We study the effects of discontinuity network parameters on the formation of removable wedges in rock slopes. Discontinuities are simulated using the Poisson disk model, and removable wedges are identified using block theory. The formation of removable wedges of different sizes is assumed to follow a Poisson process. Poisson regression and Monte Carlo simulation are then used to identify statistically relevant parameters of the model, and to study the effects that variations in their values have on formation of removable blocks. The sensitivity of the results as a function of the mean orientations of the discontinuity sets forming the blocks is also studied by means of a parametric study. The volumetric intensity of discontinuities in the rock mass is found to have a significant impact on the computed estimates of removable block formation. As predicted by theory, our results indicate that, everything else being equal, the expected rate of formation of removable wedges is proportional to the square of the intensity measure. Estimates are also sensitive to changes in discontinuity size, especially in cases in which discontinuities are smaller than one to two times the height of the slope. The interaction between the mean size of discontinuities and the coefficient of variation of discontinuity sizes is found to be significant as well. Finally, results of our sensitivity analysis suggest that the orientation of discontinuity sets significantly affects the rate of formation of removable blocks in rock slopes.

Inference of discontinuity trace length distributions using statistical graphical models

The characterization of discontinuities within rock masses is often accomplished using stochastic discontinuity network models, in which the stochastic nature of the discontinuity network is represented by means of statistical distributions. We present a flexible methodology for maximum likelihood inference of the distribution of discontinuity trace lengths based on observations at rock outcrops. The inference problem is formulated using statistical graphical models and target distributions with several Gaussian mixture components. We use the Expectation–Maximization algorithm to exploit the relations of conditional independence between variables in the maximum likelihood estimation problem. Initial results using artificially generated discontinuity traces show that the method has good inference capabilities, and inferred trace length distributions closely reproduce those used for generation. In addition, the convergence of the algorithm is shown to be fast.

Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures

We develop a multi-element generalized polynomial chaos (ME-gPC) method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs. Given a stochastic input with an arbitrary probability measure, its random space is decomposed into smaller elements. Subsequently, in each element a new random variable with respect to a conditional probability density function (PDF) is defined, and a set of orthogonal polynomials in terms of this random variable is constructed numerically. Then, the generalized polynomial chaos (gPC) method is implemented element-by-element. Numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost. Efficiency and convergence of ME-gPC are studied numerically by considering some commonly used random variables. ME-gPC provides an efficient and flexible approach to solving differential equations with random inputs, especially for problems related to long-term integration, large perturbation, and stochastic discontinuities.