Stochastic Bridge Research Articles

Unified perspective on exponential tilt and bridge algorithms for rare trajectories of discrete Markov processes.

This article analyzes and compares two general techniques of rare event simulation for generating paths of Markov processes over fixed time horizons: exponential tilting and stochastic bridge. These two methods allow us to accurately compute the probability that a Markov process ends within a rare region which is unlikely to be attained. Exponential tilting is a general technique for obtaining an alternative or tilted sampling probability measure, under which the Markov process becomes likely to hit the rare region at terminal time. The stochastic bridge technique involves conditioning paths towards two endpoints: the terminal point and the initial one. The terminal point is generated from some appropriately chosen probability distribution that covers well the rare region. We show that both methods belong to the class of importance sampling procedures by providing a common mathematical framework of these two conceptually different methods of sampling rare trajectories. We also conduct a numerical comparison of these two methods, revealing distinct areas of application for each Monte Carlo method, where they exhibit superior efficiency. Detailed simulation algorithms are provided.

Missing data interpolation and multi‐sensors integration and its application in accelerated degradation data

AbstractMultiple sensors are commonly used for accelerated degradation monitoring. Since different sensors may be sensitive at different stages of the accelerated degradation process and each sensor dataset may contain only partial information of the unit degradation, then integration approaches of the accelerated degradation data from multiple sensors can effectively improve degradation modeling and life prediction accuracy. We present a non‐parametric approach that assigns weights to each sensor based on the dynamic clustering of the sensors' observations. Missing data are common in degradation data acquisition, especially when multiple sensors are used. We provide two approaches for data interpolation: the nonlinear Brownian bridge and the inverse Gaussian bridge when the underlying degradation paths follow the Brownian motion process and inverse Gaussian process, respectively. The stochastic bridges capture the nonlinearity and uncertainties of the degradation processes. The data integration model and stochastic bridge models are validated with real accelerated fatigue crack growth data monitored with multiple NDT sensors. The proposed models provide an accurate accelerated degradation path and reliability prediction.

Fast sampling from time-integrated bridges using deep learning

We propose a methodology for sampling from time-integrated stochastic bridges, i.e., random variables defined as ∫t1t2f(Y(t))dt conditional on Y(t1)=a and Y(t2)=b, with a,b∈R. The techniques developed in Grzelak et al. (2019) – the Stochastic Collocation Monte Carlo sampler – and in Liu et al. (2020) – the Seven-League scheme – are applied for this purpose. Notably, the time-integrated bridge distribution is approximated using a polynomial chaos expansion constructed over an appropriate set of stochastic collocation points. In addition, artificial neural networks are employed to learn the collocation points. The result is a robust, data-driven procedure for Monte Carlo sampling from time-integrated conditional processes, which guarantees high accuracy and generates thousands of samples in milliseconds. Applications are also presented, with a focus on finance.

Sampling rare trajectories using stochastic bridges.

The numerical quantification of the statistics of rare events in stochastic processes is a challenging computational problem. We present a sampling method that constructs an ensemble of stochastic trajectories that are constrained to have fixed start and end points (so-called stochastic bridges). We then show that by carefully choosing a set of such bridges and assigning an appropriate statistical weight to each bridge, one can focus more processing power on the rare events of a target stochastic process while faithfully preserving the statistics of these rare trajectories. Further, we also compare the stochastic bridges we produce to the Wentzel-Kramers-Brillouin (WKB) optimal paths of the target process, derived in the limit of low noise. We see that the generated paths, encoding the full statistics of the process, collapse onto the WKB optimal path as the level of noise is reduced. We propose that the method can also be used to judge the accuracy of the WKB approximation at finite levels of noise.

Option pricing with polynomial chaos expansion stochastic bridge interpolators and signed path dependence

Recent technological advances have made possible the obtention of vast amounts of market data and strong computing power for advanced models which would not have been practicable for use in real market settings before. In this manuscript we devise a model-free empirical risk-neutral distribution based on Polynomial Chaos Expansions coupled with stochastic bridge interpolators that includes information from the entire set of observable European call option prices under all available strikes and maturities for a given underlying asset in a way that is guaranteed by construction to produce a valid state price distribution function at all times. We also obtain a non parametric model for the risk premium behaviour via an optimisation problem that joins the risk-neutral Polynomial Chaos Expansion result with any general model for the real-world distribution. Finally, we show an empirical application on SP500 Options on Futures using a real-world distribution that assumes the presence of signed path dependence in the returns of the underlying asset.

Option Pricing with Polynomial Chaos Expansion Stochastic Bridge Interpolators and Signed Path Dependence

Recent empirical studies in Equity markets show evidence that, while asset log-returns are largely uncorrelated, it is possible to predict with some accuracy their future sign. Such prediction is made over a given forecast horizon based solely on the observed sign of the cumulative log-return over a lookback horizon. This manuscript proposes a methodology to study the impact of such findings on option pricing by embedding into the risk premium the effects of signed path dependence. This is achieved by devising a model-free empirical risk-neutral distribution based on Polynomial Chaos Expansions coupled with stochastic bridge interpolators that includes information from the entire set of observable European call option prices under all available strikes and maturities for a given underlying asset. Under the real-world measure we propose a price dynamics model that is compatible with an asset price process that is largely uncorrelated but still exhibits signed path dependence. The risk premium behaviour is subsequently inferred non-parametrically via a stochastic bridge interpolation that couples the risk neutral Polynomial Chaos Expansion result with the signed path dependence mixture binomial tree dynamic model to obtain a dynamic stochastic model for the implied risk premium process.

Theoretical study on protected unit fluctuation of chemically amplified resists used for photomask production by electron beam lithography

Stochasticity in the sensitization processes of resist materials is a significant concern in fine patterning. Such stochasticity results in protected unit fluctuation in the case of a chemically amplified resist, which is a standard resist for photomask production. After development, protected unit fluctuation comes to the surface as line edge roughness, stochastic pinching, and stochastic bridging. In this study, protected unit fluctuation after postexposure baking was investigated, assuming the fabrication of line-and-space patterns. The standard deviation of the number of protected units connected to a polymer molecule was calculated using a Monte Carlo method on the basis of the sensitization mechanism of chemically amplified electron beam resists. The dependences of the standard deviation on half-pitch, sensitivity, sensitizer concentration, and exposure pattern width were clarified.

Multiscale stochastic optimization: modeling aspects and scenario generation

Real-world multistage stochastic optimization problems are often characterized by the fact that the decision maker may take actions only at specific points in time, even if relevant data can be observed much more frequently. In such a case there are not only multiple decision stages present but also several observation periods between consecutive decisions, where profits/costs occur contingent on the stochastic evolution of some uncertainty factors. We refer to such multistage decision problems with encapsulated multiperiod random costs, as multiscale stochastic optimization problems. In this article, we present a tailor-made modeling framework for such problems, which allows for a computational solution. We first establish new results related to the generation of scenario lattices and then incorporate the multiscale feature by leveraging the theory of stochastic bridge processes. All necessary ingredients to our proposed modeling framework are elaborated explicitly for various popular examples, including both diffusion and jump models.

Multiscale stochastic computational homogenization of the thermomechanical properties of woven Cf/SiCm composites

This paper demonstrates a set of two-scale asymptotic computational homogenization formalisms of Cf/SiCm thermomechanical properties and proposes a 3D micro-to-mesoscale stochastic bridging methodology considering the uncertainties of the constituents’ properties and geometries. For multiscale thermomechanical analyses, the thermal conductivity (TC), the coefficient of thermal expansion (CTE) and the elastic stiffness (ES) properties are homogenized at the microscale within the fiber tow. The proposed consistent multiscale bridging method models spatial random fields using the three-dimensional Karhunen-Loève (KL) expansion method with integration with a 3D FE-based computational homogenization model. We compare effects of two different uncertainties (i.e., geometric and spatial material randomness) to statistical variations of the effective thermomechanical properties at the microscale. By comparing with threshold COV values, we determine the thermomechanical properties to be modeled as random fields. At the mesoscale RVEs, we randomize the key twelve thermomechanical properties, perform two-scale computational homogenization and draw statistical variations at the mesoscale.

Bridgeout: Stochastic Bridge Regularization for Deep Neural Networks

A major challenge in training deep neural networks is overfitting , i.e. inferior performance on unseen test examples compared to performance on training examples. To reduce overfitting, stochastic regularization methods have shown superior performance compared to deterministic weight penalties on a number of image recognition tasks. Stochastic methods, such as Dropout and Shakeout, in expectation, are equivalent to imposing a ridge and elastic-net penalty on the model parameters, respectively. However, the choice of the norm of the weight penalty is problem dependent and is not restricted to $\{L_{1},L_{2}\}$ . Therefore, in this paper, we propose the Bridgeout stochastic regularization technique and prove that it is equivalent to an $L_{q}$ penalty on the weights, where the norm $q$ can be learned as a hyperparameter from data. Experimental results show that Bridgeout results in sparse model weights, improved gradients, and superior classification performance compared with Dropout and Shakeout on synthetic and real data sets.

Forward, backward, and weighted stochastic bridges.

We define stochastic bridges as conditional distributions of stochastic paths that leave a specified point in phase-space in the past and arrive at another one in the future. These can be defined relative to either forward or backward stochastic differential equations and with the inclusion of arbitrary path-dependent weights. The underlying stochastic equations are not the same except in linear cases. Accordingly, we generalize the theory of stochastic bridges to include time-reversed and weighted stochastic processes. We show that the resulting stochastic bridges are identical, whether derived from a forward or a backward time stochastic process. A numerical algorithm is obtained to sample these distributions. This technique, which uses partial stochastic equations, is robust and easily implemented. Examples are given, and comparisons are made to previous work. In stochastic equations without a gradient drift, our results confirm an earlier conjecture, while generalizing this to cases with path-dependent weights. An example of a two-dimensional stochastic equation with no potential solution is analyzed and numerically solved. We show how this method can treat unexpectedly large excursions occurring during a tunneling or escape event, in which a system escapes from one quasistable point to arrive at another one at a later time.

Theoretical study on effects of exposure pattern width on line edge roughness and stochastic defect generation in fabrication of 16-nm-half-pitch line-and-space patterns by electron beam lithography

The resolution of electron beam (EB) mask writers has reached a 16 nm half-pitch. The suppression of stochastic phenomena such as line edge roughness (LER) and stochastic defect generation is an important issue in the high-resolution fabrication of photomasks and nanoimprint molds. In this study, the effects of exposure pattern width on LER and stochastic defect generation were theoretically investigated on the basis of the sensitization and reaction mechanisms of chemically amplified EB resists. The exposure pattern width affected stochastic bridging more than stochastic pinching and LER. The optimum pattern widths for the suppression of stochastic bridging in the fabrication of 16-nm-half-pitch line-and-space patterns were 8, 8, 10, and 12 nm for the sensitivities of 300, 240, 180, and 120 µC cm−2, respectively.

Yet more on a stochastic economic model: Part 3C: stochastic bridging for share yields and dividends and interest rates

AbstractThis is the third and last subpart of a long paper in which we consider stochastic interpolation for the Wilkie asset model, considering both Brownian bridges and Ornstein–Uhlenbeck (OU) bridges. In Part 3A, we developed certain properties for both these types of stochastic bridge, and in Part 3B we investigated retail prices and wages. In this paper, we investigate the remainder of many of our data series, relating to shares and interest rates. We conclude that, regardless of the form of the annual model, the monthly data within each year can be modelled by Brownian bridges, usually on the logarithm of the principal variable. But in no case is a simple Brownian bridge enough, and all series have their own peculiarities. Overall, however, our modelling produces simulations that are realistic in comparison with the known data. Many of our findings would apply to any similar model used for simulation over time. Our results have considerable importance for financial economics. We reconcile the conflict between the long-term mean-reverting modelling of Schiller and the short-term random walk modelling of Fama. This conclusion therefore has very wide significance.

Yet more on a stochastic economic model: Part 3B: stochastic bridging for retail prices and wages

AbstractThis is the second subpart of three in a long paper in which we consider stochastic interpolation for the Wilkie asset model, considering both Brownian bridges and Ornstein–Uhlenbeck (OU) bridges. In Part 3A, we developed certain properties for both these types of stochastic bridge, and we investigate the properties of many of our data series on the same lines. We have several economic or investment series, which all have their own peculiarities. In this paper, we cover only retail prices and wages. The other series are dealt with in Part 3C. We find that, although the annual series for the rate of inflation is generated by an AR(1) model, which is the discrete time equivalent of an OU process, an OU bridge is not suitable. We need to use a Brownian bridge on the logarithm of the Price Index. Further, the standard deviation of the monthly increments in any year is, as we find empirically from the data, a function of the change in the annual value, and further there is correlation between the monthly increments in successive years.

Analysis of stochastic effects in chemically amplified poly(4-hydroxystyrene-co-t-butyl methacrylate) resist

Understanding of stochastic phenomena is essential to the development of a highly sensitive resist for nanofabrication. In this study, we investigated the stochastic effects in a chemically amplified resist consisting of poly(4-hydroxystyrene-co-t-butyl methacrylate), triphenylsulfonium nonafluorobutanesulfonate (acid generator), and tri-n-octylamine (quencher). Scanning electron microscopy (SEM) images of resist patterns were analyzed by Monte Carlo simulation on the basis of the sensitization and reaction mechanisms of chemically amplified extreme ultraviolet resists. It was estimated that a ±0.82σ fluctuation of the number of protected units per polymer molecule led to line edge roughness formation. Here, σ is the standard deviation of the number of protected units per polymer molecule after postexposure baking (PEB). The threshold for the elimination of stochastic bridge generation was 4.38σ (the difference between the average number of protected units after PEB and the dissolution point). The threshold for the elimination of stochastic pinching was 2.16σ.

Insider Trading, Stochastic Liquidity, and Equilibrium Prices

We extend Kyle’s (1985) model of insider trading to the case where liquidity provided by noise traders follows a general stochastic process. Even though the level of noise trading volatility is observable, in equilibrium, measured price impact is stochastic. If noise trading volatility is mean-reverting, then the equilibrium price follows a multivariate ‘stochastic bridge’ process, which displays stochastic volatility. This is because insiders choose to optimally wait to trade more aggressively when noise trading activity is higher. In equilibrium, market makers anticipate this, and adjust prices accordingly. More private information is revealed when volatility is higher. In time series, insiders trade more aggressively, when measured price impact is lower. Therefore, execution costs to uninformed traders can be higher when price impact is lower.

Inequivalence of nonequilibrium path ensembles: the example of stochastic bridges

.We study stochastic processes in which the trajectories are constrained so that the process realises a large deviation of the unconstrained process. In particular we consider stochastic bridges and the question of inequivalence of path ensembles between the microcanonical ensemble, in which the end points of the trajectory are constrained, and the canonical or s ensemble in which a bias or tilt is introduced into the process. We show how ensemble inequivalence can be manifested by the phenomenon of temporal condensation in which the large deviation is realised in a vanishing fraction of the duration (for long durations). For diffusion processes we find that condensation happens whenever the process is subject to a confining potential, such as for the Ornstein–Uhlenbeck process, but not in the borderline case of dry friction in which there is partial ensemble equivalence. We also discuss continuous-space, discrete-time random walks for which in the case of a heavy tailed step-size distribution it is known that the large deviation may be achieved in a single step of the walk. Finally we consider possible effects of several constraints on the process and in particular give an alternative explanation of the interaction-driven condensation in terms of constrained Brownian excursions.

Application of random eigenvalue analysis to assess bridge flutter probability

This study examines the well-known issue of evaluating the flutter probability for a long-span bridge, necessary for assessing structural integrity and reliability. The motivation stems from the need for critically reviewing and investigating a number of existing numerical approaches (e.g., random perturbation analysis, collocation methods, Galerkin approach) for the modeling and the solution of stochastic dynamic problems, by adapting them to the specific problem. The study proposes a generalized formulation for stochastic bridge flutter in terms of random eigenvalue analysis. A 1200m suspension bridge model is used in the numerical simulations to compare the various methods and to provide indications on advantages and limitations of each method. Moreover, the link between the proposed formulation and existing studies on the propagation of uncertainty in aeroelastic systems, for example associated with measurement errors in wind tunnel, is examined.

Stochastic Bridges of Linear Systems

We consider particles obeying Langevin dynamics while being at known positions and having known velocities at the two end-points of a given interval. Their motion in phase space can be modeled as an Ornstein-Uhlenbeck process conditioned at the two end-points-a generalization of the Brownian bridge. Using standard ideas from stochastic optimal control we construct a stochastic differential equation (SDE) that generates such a bridge that agrees with the statistics of the conditioned process, as a degenerate diffusion. Higher order linear diffusions are also considered. In general, a time-varying drift is sufficient to modify the prior SDE and meet the end-point conditions. When the drift is obtained by solving a suitable differential Lyapunov equation, the SDE models correctly the statistics of the bridge. These types of models are relevant in controlling and modeling distribution of particles and the interpolation of density functions.

Typical deterministic and stochastic bridge deterioration modelling incorporating backward prediction model

A backward prediction model (BPM) has been developed to generate the missing bridge condition ratings in past years, thereby ensuring adequate condition data as required in long-term performance modelling. The BPM establishes a correlation between the known condition ratings and the non-bridge factors, including climate condition, traffic volume and population growth. The aim of this study is to confirm the ability of BPM in improving the prediction accuracy using the existing bridge deterioration models. The prediction accuracies of typical deterministic and stochastic bridge deterioration models are compared when different sets of BPM-generated historical condition ratings are used as input. Comparisons indicate that the prediction error decreases as more historical condition ratings are made available. Notwithstanding the above findings, several limitations of the current deterministic and stochastic bridge deterioration models are also worth noting and further research is essential to improve the prediction accuracy of bridge deterioration modelling.