A new regularized stochastic approximation framework for stochastic inverse problems

We study the nonlinear inverse problem of estimating stochastic parameters in the fourth-order partial differential equation with random data. The primary focus is on developing a novel stochastic approximation framework for inverse problems consisting of three key components. As a first step, we reformulate the inverse problem into a stochastic convex optimization problem. The second step includes developing a new regularized stochastic extragradient framework for a nonlinear variational inequality, which subsumes the optimality conditions for the optimization formulation of the inverse problem. The third step involves modeling random variables by a Karhunen–Loève type finite-dimensional noise representation, allowing the direct and the inverse problems to be conveniently discretized. We show that the regularized extragradient methods are strongly convergent in a Hilbert space setting, and we also provide several auxiliary results for the inverse problem, including Lipschitz continuity and a derivative characterization of the solution map. We provide the outcome of computational experiments to estimate stochastic and deterministic parameters. The numerical results demonstrate the feasibility and effectiveness of the developed framework and validate stochastic approximation as an effective method for stochastic inverse problems.     

Stochastic processes adapted by neural networks with application to climate, energy, and finance

Local climate parameters may naturally effect the price of many commodities and their derivatives. Therefore we propose a joint framework for stochastic modeling of climate and commodity prices. In our setting, a stable Levy process is drift augmented to a generalized SDE. The related nonlinear function on the state space typically exhibits deterministic chaos. Additionally, a neural network adapts the parameters of the stable process such that the latter produces increasingly optimal differences between simulated output and observed data. Thus we propose a novel method of “intelligent” calibration of the stochastic process, using learning neural networks in order to dynamically adapt the parameters of the stochastic model.     

A probabilistic model associated with the pressureless gas dynamics

Using a method of stochastic perturbation of a Langevin system associated with the non-viscous Burgers equation we introduce a system of PDE that can be considered as a regularization of the pressureless gas dynamics describing sticky particles. By means of this regularization we describe how starting from smooth data a δ-singularity arises in the component of density. Namely, we find the asymptotics of solution at the point of the singularity formation as the parameter of stochastic perturbation tends to zero. Then we introduce a generalized solution in the sense of free particles (FP-solution) as a special limit of the solution to the regularized system. This solution corresponds to a medium consisting of non-interacting particles. The FP-solution is a bridging step to constructing solutions to the Riemann problem for the pressureless gas dynamics describing sticky particles. We analyze the difference in the behavior of discontinuous solutions for these two models and the relations between them. In our framework we obtain a unique entropy solution to the Riemann problem in 1D case.     

On stochastic management modeling with dependent variables

This paper presents an integrated framework for handling dependent random variables in a large class of stochastic management models, a class that includes stochastic break-even analysis and stochastic present-value analysis. We first demonstrate that the common approach of modeling dependent random variables is usually surprisingly inadequate, and a general “functional approach” is presented as a practical modeling alternative. Adopting this modeling approach, we then present a procedure for deriving the stochastic characteristics of the model's objective variable. In the context of stochastic breakeven analysis, this means determining the probabilities of achieving various profit levels and the expected utility of the stochastic profit. The procedure allows the model's random variables to assume diverse distribution and dependency forms, and the simplicity and reliability of the procedure is demonstrated by a numerical example.     

Modeling Dependencies in Operational Risk with Hybrid Bayesian Networks

This paper addresses the problem of quantifying and modeling financial institutions’ operational risk in accordance with the Advanced Measurement Approach put forth in the Basel II Accord. We argue that standard approaches focusing on modeling stochastic dependencies are not sufficient to adequately assess operational risk. In addition to stochastic dependencies, causal topological dependencies between the risk classes are typically encountered. These dependencies arise when risk units have common information- and/or work-flows and when failure of upstream processes imply risk for downstream processes. In this paper, we present a modeling strategy that explicitly captures both topological and stochastic dependencies between risk classes. We represent the operational-risk taxonomy in the framework of a hybrid Bayesian network (BN) and provide an intuitively compelling approach for handling causal relationships and external influences. We demonstrate the use of hybrid BNs as a tool for mapping causal dependencies between frequencies and severities of risk events and for modeling common shocks. Monte-Carlo simulations illustrate that the impact of topological dependencies on triggering overall system breakdowns can be substantial.     

奖惩系统在随机优序下的极小与极大平稳分布

奖惩系统(Bonus-M a lus System)是世界各国机动车辆险中广泛采用的一种经验费率厘定机制.文[1]在最一般的框架下,给出了奖惩系统的数学建模与稳态分析.本文将进一步证明,文[1]中给出的奖惩系统两种特定的平稳分布恰分别是奖惩系统在随机优序下的极小与极大平稳分布.特别地,基于这一事实严格证明了文[1]提出的有关封闭型奖惩系统年度总保费的一个猜想.     

A general theory on spectral properties of state-homogeneous finite-state quasi-birth–death processes

In this paper a spectral theory pertaining to Quasi-Birth–Death Processes (QBDs) is presented. The QBD, which is a generalization of the birth–death process, is a powerful tool that can be utilized in modeling many stochastic phenomena. Our theory is based on the application of a matrix polynomial method to obtain the steady-state probabilities in state-homogeneous finite-state QBDs. The method is based on finding the eigenvalue–eigenvector pairs that solve a matrix polynomial equation. Since the computational effort in the solution procedure is independent of the cardinality of the counting set, it has an immediate advantage over other solution procedures. We present and prove different properties relating the quantities that arise in the solution procedure. By also compiling and formalizing the previously known properties, we present a formal unified theory on the spectral properties of QBDs, which furnishes a formal framework to embody much of the previous work. This framework carries the prospect of furthering our understanding of the behavior the modeled systems manifest.     

Pricing Variance Swaps in a Hybrid Model of Stochastic Volatility and Interest Rate with Regime-Switching

In this paper, we consider the problem of pricing discretely-sampled variance swaps based on a hybrid model of stochastic volatility and stochastic interest rate with regime-switching. Our modeling framework extends the Heston stochastic volatility model by including the Cox-Ingersoll-Ross (CIR) stochastic interest rate model. In addition, certain model parameters in our model switch according to a continuous-time observable Markov chain process. This enables our model to capture several macroeconomic issues such as alternating business cycles. A semi-closed form pricing formula for variance swaps is derived. The pricing formula is assessed through numerical implementation, where we validate our pricing formula against the Monte Carlo simulation. The impact of incorporating regime-switching for pricing variance swaps is also discussed, where variance swaps prices with and without regime-switching effects are examined in our model. We also explore the economic consequence for the prices of variance swaps by allowing the Heston-CIR model to switch across three different regimes.     

On the application of Wishart process to the pricing of equity derivatives: the multi-asset case

Given the inherent complexity of financial markets, a wide area of research in the field of mathematical finance is devoted to develop accurate models for the pricing of contingent claims. Focusing on the stochastic volatility approach (i.e. we assume to describe asset volatility as an additional stochastic process), it appears desirable to introduce reliable dynamics in order to take into account the presence of several assets involved in the definition of multi-asset payoffs. In this article we deal with the multi asset Wishart Affine Stochastic Correlation model, that makes use of Wishart process to describe the stochastic variance covariance matrix of assets return. The resulting parametrization turns out to be a genuine multi-asset extension of the Heston model: each asset is exactly described by a single instance of the Heston dynamics while the joint behaviour is enriched by cross-assets and cross-variances stochastic correlation, all wrapped in an affine modeling. In this framework, we propose a fast and accurate calibration procedure, and two Monte Carlo simulation schemes.

     

Two-stage stochastic programming under multivariate risk constraints with an application to humanitarian relief network design

In this study, we consider two classes of multicriteria two-stage stochastic programs in finite probability spaces with multivariate risk constraints. The first-stage problem features multivariate stochastic benchmarking constraints based on a vector-valued random variable representing multiple and possibly conflicting stochastic performance measures associated with the second-stage decisions. In particular, the aim is to ensure that the decision-based random outcome vector of interest is preferable to a specified benchmark with respect to the multivariate polyhedral conditional value-at-risk or a multivariate stochastic order relation. In this case, the classical decomposition methods cannot be used directly due to the complicating multivariate stochastic benchmarking constraints. We propose an exact unified decomposition framework for solving these two classes of optimization problems and show its finite convergence. We apply the proposed approach to a stochastic network design problem in the context of pre-disaster humanitarian logistics and conduct a computational study concerning the threat of hurricanes in the Southeastern part of the United States. The numerical results provide practical insights about our modeling approach and show that the proposed algorithm is computationally scalable.

     

Numerical algorithms and simulations for reflected backward stochastic differential equations with two continuous barriers

In this paper we study different algorithms for reflected backward stochastic differential equations (BSDE in short) with two continuous barriers based on the framework of using a binomial tree to simulate 1-d Brownian motion. We introduce numerical algorithms by the penalization method and the reflected method, respectively. In the end simulation results are also presented.     

State constrained reachability for stochastic hybrid systems

Many control problems can be formulated as driving a system to reach some target states while avoiding some unwanted states. We study this problem for systems with regime change operating in uncertain environments. Nowadays, it is a common practice to model such systems in the framework of stochastic hybrid system models. In this casting, the problem is formalized as a mathematical problem named state constrained stochastic reachability analysis. In the state constrained stochastic reachability analysis, this probability is computed by imposing a constraint on the system to avoid the unwanted states. The scope of this paper is twofold. First we define and investigate the state constrained reachability analysis in an abstract mathematical setting. We define the problem for a general model of stochastic hybrid systems, and we show that the reach probabilities can be computed as solutions of an elliptic integro-differential equation. Moreover, we extend the problem by considering randomized targets. We approach this extension using stochastic dynamic programming. The second scope is to define a developmental setting in which the state constrained reachability analysis becomes more tractable. This framework is based on multilayer modelling of a stochastic system using hierarchical viewpoints. Viewpoints represent a method originated from software engineering, where a system is described by multiple models created from different perspectives. Using viewpoints, the reach probabilities can be easily computed, or even symbolically calculated. The reach probabilities computed in one viewpoint can be used in another viewpoint for improving the system control. We illustrate this technique for trajectory design.     

Substructuring tools for probabilistic analysis of instrumented nonlinear moving oscillator–beam systems

The study considers the application of finite element modeling, combined with numerical solutions of governing stochastic differential equations, to analyze instrumented nonlinear moving vehicle–structure systems. The focus of the study is on achieving computational efficiency by deploying, within a single modeling framework, three substructuring schemes with different methodological moorings. The schemes considered include spatial substructuring schemes (involving free-interface coupling methods), a spatial mesh partitioning scheme for governing stochastic differential equations (involving the use of a predictor corrector method with implicit integration schemes for linear regions and explicit schemes for local nonlinear regions), and application of the Rao–Blackwellization scheme (which permits the use of Kalman's filtering for linear substructures and Monte Carlo filters for nonlinear substructures). The main effort in this work is expended on combining these schemes with provisions for interfacing of the substructures by taking into account the relative motion of the vehicle and the supporting structure. The problem is formulated with reference to an archetypal beam and multi-degrees of freedom moving oscillator with spatially localized nonlinear characteristics. The study takes into account imperfections in mathematical modeling, guide way unevenness, and measurement noise. The numerical results demonstrate notable reduction in computational effort achieved on account of introduction of the substructuring schemes.     

Numerical algorithms and simulations for reflected backward stochastic differential equations with two continuous barriers

In this paper we study different algorithms for reflected backward stochastic differential equations (BSDE in short) with two continuous barriers based on the framework of using a binomial tree to simulate 1-d Brownian motion. We introduce numerical algorithms by the penalization method and the reflected method, respectively. In the end simulation results are also presented.     

The Link between Stochastic Differential Equations with Non-Markovian Coefficients and Backward Stochastic Partial Differential Equations

In this paper, we conjecture and prove the link between stochastic differential equations with non-Markovian coefficients and nonlinear parabolic backward stochastic partial differential equations, which is an extension of such kind of link in Markovian framework to non-Markovian framework.Different from Markovian framework, where the corresponding partial differential equation is deterministic, the backward stochastic partial differential equation here has a pair of adapted solutions, and thus the link has a much different form. Moreover, two examples are given to demonstrate the applications of the derived link.     

Recourse-based stochastic nonlinear programming: properties and Benders-SQP algorithms

In this paper, we study recourse-based stochastic nonlinear programs and make two sets of contributions. The first set assumes general probability spaces and provides a deeper understanding of feasibility and recourse in stochastic nonlinear programs. A sufficient condition, for equality between the sets of feasible first-stage decisions arising from two different interpretations of almost sure feasibility, is provided. This condition is an extension to nonlinear settings of the “W-condition,” first suggested by Walkup and Wets (SIAM J. Appl. Math. 15:1299–1314, 1967). Notions of complete and relatively-complete recourse for nonlinear stochastic programs are defined and simple sufficient conditions for these to hold are given. Implications of these results on the L-shaped method are discussed. Our second set of contributions lies in the construction of a scalable, superlinearly convergent method for solving this class of problems, under the setting of a finite sample-space. We present a novel hybrid algorithm that combines sequential quadratic programming (SQP) and Benders decomposition. In this framework, the resulting quadratic programming approximations while arbitrarily large, are observed to be two-period stochastic quadratic programs (QPs) and are solved through two variants of Benders decomposition. The first is based on an inexact-cut L-shaped method for stochastic quadratic programming while the second is a quadratic extension to a trust-region method suggested by Linderoth and Wright (Comput. Optim. Appl. 24:207–250, 2003). Obtaining Lagrange multiplier estimates in this framework poses a unique challenge and are shown to be cheaply obtainable through the solution of a single low-dimensional QP. Globalization of the method is achieved through a parallelizable linesearch procedure. Finally, the efficiency and scalability of the algorithm are demonstrated on a set of stochastic nonlinear programming test problems.     

Cylindrical fractional Brownian motion in Banach spaces

In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical fractional Brownian motion, which is analogous to the Karhunen–Loève expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical fractional Brownian motion.     

Stochastic flows and finite block frames

With the aim of understanding the mathematical structure of the fluctuation-dissipation theorem in non-equilibrium statistical physics and then constructing a mathematical principle in the modeling problem for time series analysis, we have developed the theory of KM₂O-Langevin equations for discrete time stochastic processes. In this paper, as a new method for model analysis in the theory of KM₂O-Langevin equations, we show that block frames provide a natural mathematical language for dealing with minimum norm expansions of multi-dimensional stochastic processes which do not necessarily satisfy stationarity and non-degeneracy conditions.     

Solvability of forward–backward stochastic partial differential equations

In this paper we study the solvability of a class of fully-coupled forward–backward stochastic partial differential equations (FBSPDEs). These FBSPDEs cannot be put into the framework of stochastic evolution equations in general, and the usual decoupling methods for the Markovian forward–backward SDEs are difficult to apply. We prove the well-posedness of the FBSPDEs, under various conditions on the coefficients, by using either the method of contraction mapping or the method of continuation. These conditions, especially in the higher dimensional case, are novel in the literature.     

A stochastic planning framework for the discovery of complementary,agricultural systems

One of the greatest 21st century challenges is meeting the need to feed a growing world population which is expected to increase by about 35% by 2050. To meet this challenge, it is necessary to make major improvements on current food production and distribution systems capabilities, as well as to adapt these systems to expected trends such as climate change. Changing climate patterns may present opportunities for unidentified, geographical regions with adequate climate patterns to produce high-value agricultural products in a profitable and sustainable manner.This paper focuses on the design and planning aspects of a discovery process to unearth agri-food supply chains capable of generating attractive return on investments. A stochastic optimization framework is used to develop planting and harvesting schedules for a set of identified regions with complementary weather characteristics. To address the high-level of variability in the problem context, a two-stage stochastic decomposition method is used to consider a larger number of scenarios. As part of the solution process, a modeling scheme is developed that learns past interactions between entering discretized, weather scenarios and optimal first-stage solutions. In this context, machine learning and dimensionality reduction techniques are used to iteratively estimate each region's probability of belonging to first-stage solutions based on previous solution-scenario results. The implementation of the stochastic framework is shown through a case study applied to multiple locations within the US southwest states of Arizona and New Mexico.