Nonlinear Oscillator Stochastic Response and Survival Probability Determination via the Wiener Path Integral

2015 ◽  
Vol 1 (2) ◽  
Author(s):  
Yuanjin Zhang ◽  
Ioannis A. Kougioumtzoglou
Keyword(s):  
Closed Form ◽  
Path Integral ◽  
Survival Probability ◽  
Nonlinear Oscillator ◽  
Transition Probability ◽  
Computational Cost ◽  
Discrete Version ◽  
Stochastic Response ◽  
Transition Probability Density ◽  
Approximate Analytical Expression

A Wiener path integral (WPI) technique based on a variational formulation is developed for nonlinear oscillator stochastic response determination and reliability assessment. This is done in conjunction with a stochastic averaging/linearization treatment of the problem. Specifically, first, the nonlinear oscillator is cast into an equivalent linear one with time-varying stiffness and damping elements. Next, relying on the concept of the most probable path, a closed-form approximate analytical expression for the oscillator joint transition probability density function (PDF) is derived for small time intervals. Finally, the transition PDF in conjunction with a discrete version of the Chapman–Kolmogorov (C–K) equation is utilized for advancing the solution in short-time steps. In this manner, not only the nonstationary response PDF but also the oscillator survival probability and first-passage PDF are determined. In comparison with existing numerical path integral schemes, a significant advantage of the proposed WPI technique is that closed-form analytical expressions are derived for the involved multidimensional integrals; thus, the computational cost is kept at a minimum level. The hardening Duffing and the bilinear hysteretic oscillators are considered as numerical examples. Comparisons with pertinent Monte Carlo simulation (MCS) data demonstrate the reliability of the developed technique.

An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems

2015 ◽  
Vol 82 (10) ◽  
Author(s):  
Ioannis A. Kougioumtzoglou ◽  
Alberto Di Matteo ◽  
Pol D. Spanos ◽  
Antonina Pirrotta ◽  
Mario Di Paola
Keyword(s):  
Path Integral ◽  
Computational Cost ◽  
High Dimensional ◽  
System Response ◽  
Stochastic Response ◽  
Mdof Systems ◽  
Low Dimensional ◽  
Response Determination ◽  
Efficiency And Reliability

The recently developed approximate Wiener path integral (WPI) technique for determining the stochastic response of nonlinear/hysteretic multi-degree-of-freedom (MDOF) systems has proven to be reliable and significantly more efficient than a Monte Carlo simulation (MCS) treatment of the problem for low-dimensional systems. Nevertheless, the standard implementation of the WPI technique can be computationally cumbersome for relatively high-dimensional MDOF systems. In this paper, a novel WPI technique formulation/implementation is developed by combining the “localization” capabilities of the WPI solution framework with an appropriately chosen expansion for approximating the system response PDF. It is shown that, for the case of relatively high-dimensional systems, the herein proposed implementation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard WPI technique and an MCS approach. Several numerical examples are included, whereas comparisons with pertinent MCS data demonstrate the efficiency and reliability of the technique.

scholarly journals PRICING STEP OPTIONS UNDER THE CEV AND OTHER SOLVABLE DIFFUSION MODELS

2013 ◽  
Vol 16 (05) ◽  
pp. 1350027 ◽  
Author(s):  
G. CAMPOLIETI ◽  
R. MAKAROV ◽  
K. WOUTERLOOT
Keyword(s):  
Closed Form ◽  
Diffusion Processes ◽  
Transition Probability ◽  
Spectral Expansion ◽  
Cev Model ◽  
Transition Probability Density ◽  
Constant Elasticity ◽  
Resolvent Kernel ◽  
Spectral Expansions ◽  
Expansion Formulae

We consider a special family of occupation-time derivatives, namely proportional step options introduced by [18]. We develop new closed-form spectral expansions for pricing such options under a class of nonlinear volatility diffusion processes which includes the constant-elasticity-of-variance (CEV) model as an example. In particular, we derive a general analytically exact expression for the resolvent kernel (i.e. Green's function) of such processes with killing at an exponential stopping time (independent of the process) of occupation above or below a fixed level. Moreover, we succeed in Laplace inverting the resolvent kernel and thereby derive newly closed-form spectral expansion formulae for the transition probability density of such processes with killing. The spectral expansion formulae are rapidly convergent and easy-to-implement as they are based simply on knowledge of a pair of fundamental solutions for an underlying solvable diffusion process. We apply the spectral expansion formulae to the pricing of proportional step options for four specific families of solvable nonlinear diffusion asset price models that include the CEV diffusion model and three other multi-parameter state-dependent local volatility confluent hypergeometric diffusion processes.

scholarly journals A path integral formulation for particle detectors: the Unruh-DeWitt model as a line defect

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Ivan M. Burbano ◽  
T. Rick Perche ◽  
Bruno de S. L. Torres
Keyword(s):  
Path Integral ◽  
Degrees Of Freedom ◽  
Wilson Line ◽  
Transition Probability ◽  
Relativistic Quantum ◽  
Quantum Fields ◽  
Line Defect ◽  
Particle Detectors ◽  
Nonlocal Operators ◽  
Detector Model

Abstract Particle detectors are an ubiquitous tool for probing quantum fields in the context of relativistic quantum information (RQI). We formulate the Unruh-DeWitt (UDW) particle detector model in terms of the path integral formalism. The formulation is able to recover the results of the model in general globally hyperbolic spacetimes and for arbitrary detector trajectories. Integrating out the detector’s degrees of freedom yields a line defect that allows one to express the transition probability in terms of Feynman diagrams. Inspired by the light-matter interaction, we propose a gauge invariant detector model whose associated line defect is related to the derivative of a Wilson line. This is another instance where nonlocal operators in gauge theories can be interpreted as physical probes for quantum fields.

scholarly journals Parametrix construction for certain Lévy-type processes

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Victoria Knopova ◽  
Alexei Kulik
Keyword(s):  
Markov Process ◽  
Probability Density ◽  
Lower Bounds ◽  
Transition Probability ◽  
Local Operator ◽  
Upper And Lower Bounds ◽  
Transition Probability Density ◽  
Strong Markov Process ◽  
Non Local ◽  
Strong Markov

AbstractIn this paper, we show that a non-local operator of certain type extends to the generator of a strong Markov process, admitting the transition probability density. For this transition probability density we construct the intrinsic upper and lower bounds, and prove some smoothness properties. Some examples are provided.

Fourth Moment Structure of Markov Switching Multivariate GARCH Models

2019 ◽  
Author(s):  
Maddalena Cavicchioli
Keyword(s):  
Closed Form ◽  
Moving Average ◽  
Markov Switching ◽  
Computational Cost ◽  
Sufficient Conditions ◽  
Impulse Response Function ◽  
Real Data ◽  
Multivariate Garch ◽  
Fourth Moment ◽  
Spectral Density Matrix

Abstract We derive sufficient conditions for the existence of second and fourth moments of Markov switching multivariate generalized autoregressive conditional heteroscedastic processes in the general vector specification. We provide matrix expressions in closed form for such moments, which are obtained by using a Markov switching vector autoregressive moving-average representation of the initial process. These expressions are shown to be readily programmable in addition of greatly reducing the computational cost. As theoretical applications of the results, we derive the spectral density matrix of the squares and cross products, propose a new definition of multivariate kurtosis measure to recognize heavy-tailed features in financial real data, and provide a matrix expression in closed form of the impulse-response function for the volatility. An empirical example illustrates the results.

Transition probability density of a certain diffusion process concentrated on a finite spatial interval

1992 ◽  
Vol 29 (2) ◽  
pp. 334-342
Author(s):  
A. Milian
Keyword(s):  
Diffusion Process ◽  
Probability Density ◽  
Transition Probability ◽  
Transition Probability Density ◽  
One Dimensional ◽  
Spatial Interval

We show that under some assumptions a diffusion process satisfying a one-dimensional Itô's equation has a transition probability density concentrated on a finite spatial interval. We give a formula for this density.

scholarly journals Numerical simulation of the two-locus Wright-Fisher stochastic differential equation with application to approximating transition probability densities

2020 ◽  
Author(s):  
Zhangyi He ◽  
Mark Beaumont ◽  
Feng Yu
Keyword(s):  
Differential Equation ◽  
Natural Selection ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Genetic Recombination ◽  
Transition Probability ◽  
Superior Performance ◽  
Transition Probability Density

AbstractOver the past decade there has been an increasing focus on the application of the Wright-Fisher diffusion to the inference of natural selection from genetic time series. A key ingredient for modelling the trajectory of gene frequencies through the Wright-Fisher diffusion is its transition probability density function. Recent advances in DNA sequencing techniques have made it possible to monitor genomes in great detail over time, which presents opportunities for investigating natural selection while accounting for genetic recombination and local linkage. However, most existing methods for computing the transition probability density function of the Wright-Fisher diffusion are only applicable to one-locus problems. To address two-locus problems, in this work we propose a novel numerical scheme for the Wright-Fisher stochastic differential equation of population dynamics under natural selection at two linked loci. Our key innovation is that we reformulate the stochastic differential equation in a closed form that is amenable to simulation, which enables us to avoid boundary issues and reduce computational costs. We also propose an adaptive importance sampling approach based on the proposal introduced by Fearnhead (2008) for computing the transition probability density of the Wright-Fisher diffusion between any two observed states. We show through extensive simulation studies that our approach can achieve comparable performance to the method of Fearnhead (2008) but can avoid manually tuning the parameter ρ to deliver superior performance for different observed states.

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