Stochastic Boundaries Research Articles

Comparison of stochastic stability boundaries for parametrically forced systems with application to ship rolling motion

Numerous accidents caused by parametric rolling have been reported on container ships and pure car carriers (PCCs). Considering this dangerous phenomenon, the parametric rolling in irregular seas is examined in this paper based on the stability of the systems origin, which corresponds to the upright condition of the vessel. It provides a novel theoretical explanation of the instability mechanism for two cases: white-noise parametric excitation and colored-noise parametric excitation. Moreover, the authors confirm the usefulness of the previously provided formulae by Roberts and Dostal by means of numerical examples.

Mapping material-property space of cellular metamaterials under uncertainty

Cellular mechanical metamaterials (CMMs) are assemblies of periodic representative volume elements which can be engineered to exhibit unique mechanical properties. Recent advances in additive manufacturing (AM) have enabled us to fabricate sophisticated architected materials with high precision. To increase and diversify applications in both science and engineering practice, the rapid development of fabrication technologies necessities a novel and effective algorithm to outline a theoretical property space for material design problems, allowing designers to acknowledge the limitations of material performance, and to make informed trade-off between target objectives and competing industrial requirements. Inspired by biological evolution, the proposed work addresses a methodology for intuitively mapping material-property spaces of CMMs by using the genetic algorithm as a sampling algorithm, consisting selection of objective properties and stochastic search of property points. It approximates the property space more faithfully and comprehensively than traditional mapping approaches. Considering the manufacturing defects of the AM, uncertainties in properties are quantified via the deep learning method. Variations of properties induced by the defects are illustrated as stochastic boundaries of the property space.

Design of stochastic trigger boundaries for rural communities evacuating from a wildfire

Well-planned evacuation is an effective and often necessary tool to protect life in communities threatened by wildfire. In previous incidents, community evacuations have been called too late resulting in entrapment, such as in Mati, Greece in 2018. The most reliable method to determine the safe time to call an evacuation is through trigger boundaries, perimeters on the landscape surrounding a community where the approaching wildfire is an amount of time away from the community equal to the evacuation time. This paper presents a new tool, k-PERIL, that calculates stochastic trigger boundaries, based on the variability of wildfire behaviour around a community due to the influence of historic wind, weather and vegetation variations on the wildfire. k-PERIL is applied to the rural community of Roxborough Park, Colorado, USA, producing probabilistic trigger boundaries and showing the model's ability to find and quantify areas of elevated uncertainty of evacuation. The concept of uncertainty rosettes is introduced, which show the areas where incoming wildfires cause larger variation to the boundary location because of higher sensitivity to changes in fuel, wind or evacuation. The k-PERIL tool can be used to inform effective evacuation preparation and enhance long term planning, improving community safety and wildfire resilience.

Bounding pandemic spread by heat spread

The beginning of a pandemic is a crucial stage for policymakers. Proper management at this stage can reduce overall health and economical damage. However, knowledge about the pandemic is insufficient. Thus, the use of complex and sophisticated models is challenging. In this study, we propose analytical and stochastic heat spread-based boundaries for the pandemic spread as indicated by the Susceptible-Infected-Recovered (SIR) model. We study the spread of a pandemic on an interaction (social) graph as a diffusion and compared it with the stochastic SIR model. The proposed boundaries are not requiring accurate biological knowledge such as the SIR model does.

Discounted optimal stopping problems in first-passage time models with random thresholds

AbstractWe derive closed-form solutions to some discounted optimal stopping problems related to the perpetual American cancellable dividend-paying put and call option pricing problems in an extension of the Black–Merton–Scholes model. The cancellation times are assumed to occur when the underlying risky asset price process hits some unobservable random thresholds. The optimal stopping times are shown to be the first times at which the asset price reaches stochastic boundaries depending on the current values of its running maximum and minimum processes. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and modified normal-reflection conditions. We show that the optimal stopping boundaries are characterised as the maximal and minimal solutions of certain first-order nonlinear ordinary differential equations.

Perpetual American Double Lookback Options on Drawdowns and Drawups with Floating Strikes

We present closed-form solutions to the problems of pricing of the perpetual American double lookback put and call options on the maximum drawdown and the maximum drawup with floating strikes in the Black-Merton-Scholes model. It is shown that the optimal exercise times are the first times at which the underlying risky asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum as well as the maximum drawdown or maximum drawup. The proof is based on the reduction of the original double optimal stopping problems to the appropriate sequences of single optimal stopping problems for the three-dimensional continuous Markov processes. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state spaces. We show that the optimal exercise boundaries are determined as either the unique solutions of the associated systems of arithmetic equations or the minimal and maximal solutions of the appropriate first-order nonlinear ordinary differential equations.

Optimal Double Stopping Problems for Maxima and Minima of Geometric Brownian Motions

We present closed-form solutions to some double optimal stopping problems with payoffs representing linear functions of the running maxima and minima of a geometric Brownian motion. It is shown that the optimal stopping times are th first times at which the underlying process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original double optimal stopping problems to sequences of single optimal stopping problems for the resulting three-dimensional continuous Markov process. The latter problems are solved as the equivalent free-boundary problems by means of the smooth-fit and normal-reflection conditions for the value functions at the optimal stopping boundaries and the edges of the three-dimensional state space. We show that the optimal stopping boundaries are determined as the extremal solutions of the associated first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of perpetual real double lookback options with floating sunk costs in the Black-Merton-Scholes model.

Exact solution of the “Rule 150” reversible cellular automaton

We study the dynamics and statistics of the Rule 150 reversible cellular automaton (RCA). This is a one-dimensional lattice system of binary variables with synchronous (Floquet) dynamics that corresponds to a bulk deterministic and reversible discretized version of the kinetically constrained "exclusive one-spin facilitated" (XOR) Fredrickson-Andersen (FA) model, where the local dynamics is restricted: A site flips if and only if its adjacent sites are in different states from each other. Similar to other RCA that have been recently studied, such as Rule 54 and Rule 201, the Rule 150 RCA is integrable, however, in contrast is noninteracting: The emergent quasiparticles, which are identified by the domain walls, behave as free fermions. This property allows us to solve the model by means of matrix product ansatz. In particular, we find the exact equilibrium and nonequilibrium stationary states for systems with closed (periodic) and open (stochastic) boundaries, respectively, resolve the full spectrum of the time evolution operator and, therefore, gain access to the relaxation dynamics, and obtain the exact large deviation statistics of dynamical observables in the long-time limit.

First passage times over stochastic boundaries for subdiffusive processes

Let X = ( X t ) t ≥ 0 \mathbb {X}=(\mathbb {X}_t)_{t\geq 0} be the subdiffusive process defined, for any t ≥ 0 t\geq 0 , by X t = X ℓ t \mathbb {X}_t = X_{\ell _t} where X = ( X t ) t ≥ 0 X=(X_t)_{t\geq 0} is a Lévy process and ℓ t = inf { s > 0 ; K s > t } \ell _t=\inf \{s>0; \mathcal {K}_s>t \} with K = ( K t ) t ≥ 0 \mathcal {K}=(\mathcal {K}_t)_{t\geq 0} a subordinator independent of X X . We start by developing a composite Wiener-Hopf factorization to characterize the law of the pair ( T a ( b ) , ( X − b ) T a ( b ) ) (\mathbb {T}_a^{({b})}, (\mathbb {X} - {b})_{\mathbb {T}_a^{({b})}}) where T a ( b ) = inf { t > 0 ; X t > a + b t } \begin{equation*}\mathbb {T}_a^{({b})} = \inf \{t>0; \mathbb {X}_t > a+ {b}_t \} \end{equation*} with a ∈ R a \in \mathbb {R} and b = ( b t ) t ≥ 0 {b}=({b}_t)_{t\geq 0} a (possibly degenerate) subordinator independent of X X and K \mathcal {K} . We proceed by providing a detailed analysis of the cases where either X \mathbb {X} is a self-similar or is spectrally negative. For the later, we show the fact that the process ( T a ( b ) ) a ≥ 0 (\mathbb {T}_a^{({b})})_{a\geq 0} is a subordinator. Our proofs hinge on a variety of techniques including excursion theory, change of measure, asymptotic analysis and on establishing a link between subdiffusive processes and a subclass of semi-regenerative processes. In particular, we show that the variable T a ( b ) \mathbb {T}_a^{({b})} has the same law as the first passage time of a semi-regenerative process of Lévy type, a terminology that we introduce to mean that this process satisfies the Markov property of Lévy processes for stopping times whose graph is included in the associated regeneration set.

Optimal stopping problems for maxima and minima in models with asymmetric information

We derive closed-form solutions to optimal stopping problems related to the pricing of perpetual American withdrawable standard and lookback put and call options in an extension of the Black-Merton-Scholes model with asymmetric information. It is assumed that the contracts are withdrawn by their writers at the last hitting times for the underlying risky asset price of its running maximum or minimum over the infinite time interval which are not stopping times with respect to the observable filtration. We show that the optimal exercise times are the first times at which the asset price process reaches some lower or upper stochastic boundaries depending on the current values of its running maximum or minimum. The proof is based on the reduction of the original necessarily two-dimensional optimal stopping problems to the associated free-boundary problems and their solutions by means of the smooth-fit and normal-reflection conditions. We prove that the optimal exercise boundaries are the maximal and minimal solutions of some first-order nonlinear ordinary differential equations.

Rule 54: exactly solvable model of nonequilibrium statistical mechanics

We review recent results on an exactly solvable model of nonequilibrium statistical mechanics, specifically the classical rule 54 reversible cellular automaton and some of its quantum extensions. We discuss the exact microscopic description of nonequilibrium dynamics as well as the equilibrium and nonequilibrium stationary states. This allows us to obtain a rigorous handle on the corresponding emergent hydrodynamic description, which is treated as well. Specifically, we focus on two different paradigms of rule 54 dynamics. Firstly, we consider a finite chain driven by stochastic boundaries, where we provide exact matrix product descriptions of the nonequilibrium steady state, most relevant decay modes, as well as the eigenvector of the tilted Markov chain yielding exact large deviations for a broad class of local and extensive observables. Secondly, we treat the explicit dynamics of macro-states on an infinite lattice and discuss exact closed form results for dynamical structure factor, multi-time-correlation functions and inhomogeneous quenches. Remarkably, these results prove that the model, despite its simplicity, behaves like a regular fluid with coexistence of ballistic (sound) and diffusive (heat) transport. Finally, we briefly discuss quantum interpretation of rule 54 dynamics and explicit results on dynamical spreading of local operators and operator entanglement.

Closed form valuation of barrier options with stochastic barriers

This article deals with the computation of the probability, for a GBM (geometric Brownian motion) process, to hit sequences of one-sided stochastic boundaries defined as GBM processes, over a closed time interval. Explicit formulae are obtained, allowing the analytical valuation of all the main kinds of barrier options in a much more general setting than the usual one assuming constant or time-dependent, deterministic barriers. The numerical implementation of all stated formulae is shown to be easy, fast and accurate. The practical applications are potentially substantial, since barrier options play a major role in quantitative finance, not only as intensively traded contracts on their own, but also as the building blocks of a large variety of structured products. Barrier options are also an important tool in financial modelling, used to measure default risk in the so-called “structural” models.

Multi-objective optimal formation reconfiguration with scalarization and stochastic boundaries

Abstract A new method is proposed for multi-objective optimal control of satellite formation reconfiguration. First, necessary optimality conditions of single-objective fuel-optimal reconfiguration are studied. Problem of initial guess of the costate and control discontinuity are addressed by using a new developed stochastic based method. Terminal conditions of Two Point Boundary Value Problem (TPBVP) of the optimal control equations are relaxed using augmented Gaussian. Variances or bandwidths of the terminal conditions are set to decision variables to be optimized. Responses of varied initial costate are modeled and propagated using Riccati equation. Using quadratic convolution of relaxed terminal conditions, objective function of the fuel-optimal problem is reformulated into a quadratic equation, and a gradient based optimizer is used to search the optimal initial guess. In the second part of the paper, optimal control of format flying reconfiguration with two competitive objectives:fuel and time of flight (ToF) is studied. With Pascoletti-Serafini scalarization, the Multi-objective Optimal Control Problem (MOCP) is scalarized into a set of weighted Single-objective Optimal Control Problems (SOCP). Hamiltonian and switch function for the weighted SOCP are developed and constructed. New developed stochastic based approach is then used to search the optimal Pareto solutions. Verification and performances of proposed algorithms are demonstrated through numerical simulations of single-objective and multi-objective optimal control of low-thrust elliptical orbit formation reconfiguration. Potential applications of the algorithms to some other preliminary space mission designs are also discussed.

Two-species hardcore reversible cellular automaton: matrix ansatz for dynamics and nonequilibrium stationary state

In this paper we study the statistical properties of a reversible cellular automaton in two out-of-equilibrium settings. In the first part we consider two instances of the initial value problem, corresponding to the inhomogeneous quench and the local quench. Our main result is an exact matrix product expression of the time evolution of the probability distribution, which we use to determine the time evolution of the density profiles analytically. In the second part we study the model on a finite lattice coupled with stochastic boundaries. Once again we derive an exact matrix product expression of the stationary distribution, as well as the particle current and density profiles in the stationary state. The exact expressions reveal the existence of different phases with either ballistic or diffusive transport depending on the boundary parameters.

Strongly correlated non-equilibrium steady states with currents – quantum and classical picture

In this minireview we will discuss recent progress in the analytical study of current-carrying non-equilibrium steady states (NESS) that can be constructed in terms of a matrix product ansatz. We will focus on one-dimensional exactly solvable strongly correlated cases, and will study both quantum models, and classical models which are deterministic in the bulk. The only source of classical stochasticity in the time-evolution will come from the boundaries of the system. Physically, these boundaries may be understood as Markovian baths, which drive the current through the system. The examples studied include the open XXZ Heisenberg spin chain, the open Hubbard model, and a classical integrable reversible cellular automaton, namely the Rule 54 of Bobenko {\em et al}. [Commun. Math. Phys. {\bf 158}, 127 (1993)] with stochastic boundaries. The quantum NESS can be at least partially understood through the Yang-Baxter integrability structure of the underlying integrable bulk Hamiltonian, whereas for the Rule 54 model NESS seems to come from a seemingly unrelated integrability theory. In both the quantum and the classical case, the underlying matrix product ansatz defining the NESS also allows for construction of novel conservation laws of the bulk models themselves. In the classical case, a modification of the matrix product ansatz also allows for construction of states beyond the steady state (i.e., some of the decay modes -- Liouvillian eigenvectors of the model). We hope that this article will help further the quest to unite different perspectives of integrability of NESS (of both quantum and classical models) into a single unified framework.

On the First Exit Time of Geometric Brownian Motion from Stochastic Exponential Boundaries

This article deals with the boundary crossing probability of a geometric Brownian motion (GBM) process when the boundary itself is a GBM process. An exact formula is obtained for the probability that the first exit time of $$ S\left( t \right) $$ from the stochastic interval $$ \left[ {H_{1} \left( t \right),H_{2} \left( t \right)} \right] $$ is greater than a finite time $$ T $$ using a partial differential equation approach. Applications and numerical results are provided. The possibility of an extension to higher dimension is also discussed. In particular, the steps to obtain the probability that $$ S_{1} \left( t \right) $$ , $$ S_{2} \left( t \right) $$ and $$ S_{3} \left( t \right) $$ remain above $$ S_{4} \left( t \right) $$ , $$ \forall 0 \le t \le T $$ , are outlined, while pointing out that the entailed numerical issues make the relevance of an analytical approach questionable.

Efficient Distribution Estimation and Uncertainty Quantification for Elliptic Problems on Domains with Stochastic Boundaries

We study the problem of uncertainty quantification for the numerical solution of elliptic partial differential equation boundary value problems posed on domains with stochastically varying boundari...

A stochastic control problem and related free boundaries in finance

In this paper, we investigate an optimal stopping problem (mixed with stochastic controls) for a manager whose utility is nonsmooth and nonconcave over a finite time horizon. The paper aims to develop a new methodology, which is significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature, so as to figure out the manager's best strategies. The problem is first reformulated into a free boundary problem with a fully nonlinear operator. Then, by means of a dual transformation, it is further converted into a free boundary problem with a linear operator, which can be consequently tackled by the classical method. Finally, using the inverse transformation, we obtain the properties of the optimal trading strategy and the optimal stopping time for the original problem.

Inexact Multistage Stochastic Chance Constrained Programming Model for Water Resources Management under Uncertainties

In order to formulate water allocation schemes under uncertainties in the water resources management systems, an inexact multistage stochastic chance constrained programming (IMSCCP) model is proposed. The model integrates stochastic chance constrained programming, multistage stochastic programming, and inexact stochastic programming within a general optimization framework to handle the uncertainties occurring in both constraints and objective. These uncertainties are expressed as probability distributions, interval with multiply distributed stochastic boundaries, dynamic features of the long-term water allocation plans, and so on. Compared with the existing inexact multistage stochastic programming, the IMSCCP can be used to assess more system risks and handle more complicated uncertainties in water resources management systems. The IMSCCP model is applied to a hypothetical case study of water resources management. In order to construct an approximate solution for the model, a hybrid algorithm, which incorporates stochastic simulation, back propagation neural network, and genetic algorithm, is proposed. The results show that the optimal value represents the maximal net system benefit achieved with a given confidence level under chance constraints, and the solutions provide optimal water allocation schemes to multiple users over a multiperiod planning horizon.

Quasimodes instability analysis of uncertain asymmetric rotor system based on 3D solid element model

Uncertainties are considered in the equation of motion of an asymmetric rotor system. Based on Hill's determinant method, quasimodes stability analysis with uncertain parameters is used to get stochastic boundaries of unstable regions. Firstly, A 3D finite element rotor model was built in rotating frame with four parameterized coefficients, which is assumed as random parameters representing the uncertainties existing in the rotor system. Then the influences of uncertain coefficients on the distribution of the unstable region boundaries are analyzed. The results show that uncertain parameters have various influences on the size, boundary and number of unstable regions. At last, the statistic results of the minimum and maximum spin speeds of unstable regions were got by Monte Carlo simulation. The used method is suitable for real engineering rotor system, because arbitrary configuration of rotors can be modeled by 3D finite element.