Corpus ID: 88515840

# Density Tracking by Quadrature for Stochastic Differential Equations

@article{Bhat2016DensityTB,
title={Density Tracking by Quadrature for Stochastic Differential Equations},
journal={arXiv: Computation},
year={2016}
}
• Published 2016
• Mathematics
• arXiv: Computation
We develop and analyze a method, density tracking by quadrature (DTQ), to compute the probability density function of the solution of a stochastic differential equation. The derivation of the method begins with the discretization in time of the stochastic differential equation, resulting in a discrete-time Markov chain with continuous state space. At each time step, the DTQ method applies quadrature to solve the Chapman-Kolmogorov equation for this Markov chain. In this paper, we focus on a… Expand

#### Figures and Tables from this paper

On numerical density approximations of solutions of SDEs with unbounded coefficients
• Mathematics, Computer Science
• 2018
A rigorous analysis of the numerical method to compute probability density functions of solutions of stochastic differential equations that covers systems of equations with unbounded coefficients is provided. Expand
• Computer Science, Mathematics
• ArXiv
• 2021
This paper proposes and describes a flexible quadrature rule that allows for unstructured, adaptive meshes, and demonstrates that the resulting adaptive procedure is significantly more efficient than a tensorized approach. Expand
Nonparametric Adjoint-Based Inference for Stochastic Differential Equations
• Computer Science
• 2016 IEEE International Conference on Data Science and Advanced Analytics (DSAA)
• 2016
A nonparametric method to infer the drift and diffusion functions of a stochastic differential equation that can build predictive models starting with repeated time series and/or high-dimensional longitudinal data and applies to real data on hourly measurements of ground level ozone. Expand
Learning governing equations for stochastic dynamical systems
This dissertation presents work on automating discovery of governing equations for stochastic dynamical systems from noisy, vector-valued time series, and develops a numerical approximation for the likelihood of the SDE using an innovative density tracking by quadrature (DTQ) method. Expand
Bayesian Inference of Stochastic Pursuit Models from Basketball Tracking Data
• Computer Science
• 2016
A Metropolis algorithm to perform Bayesian inference for models given by coupled stochastic differential equations is developed and shown to be capable of efficient inference for an electrical oscillator model. Expand
Scalable SDE Filtering and Inference with Apache Spark
• Computer Science
• BigMine
• 2016
A Metropolis algorithm to sample from the high-dimensional joint posterior density of all SDE parameters and state time series, relying on an innovative density tracking by quadrature (DTQ) method to compute the likelihood of the SDE, the part of the posterior that requires the most computational effort to evaluate. Expand
Fast predictive multi-fidelity prediction with models of quantized fidelity levels
• Computer Science
• J. Comput. Phys.
• 2019
A novel approach for the construction of multi-fidelity surrogate models with “discrete” fidelity levels is introduced, and it is shown that the approach is applicable to competitive ecological systems with different numbers of species, discrete-state Markov chains with a different number of states, polymer networks with adifferent number of connections, and nano-particle plasmonic arrays with aDifferent number of scatterers. Expand

#### References

SHOWING 1-10 OF 43 REFERENCES
A weak trapezoidal method for a class of stochastic differential equations
• Mathematics
• 2009
We present a numerical method for the approximation of solutions for the class of stochastic differential equations driven by Brownian motions which induce stochastic variation in fixed directions.Expand
On numerical density approximations of solutions of SDEs with unbounded coefficients
• Mathematics, Computer Science
• 2018
A rigorous analysis of the numerical method to compute probability density functions of solutions of stochastic differential equations that covers systems of equations with unbounded coefficients is provided. Expand
Algorithms for Linear Stochastic Delay Differential Equations
This work derives a new algorithm to compute the density function of the solution with no sampling by discretizing the stochastic equation in time, and indicates that the algorithm is a fast, accurate alternative to existing methods. Expand
Computing the Density Function for a Nonlinear Stochastic Delay System
• Mathematics
• 2015
Abstract We develop a numerical method to compute the density of a specific nonlinear stochastic delay system, with no sampling. This system arises as a switch-type control model for human balance.Expand
The Law of the Euler Scheme for Stochastic Differential Equations: II. Convergence Rate of the Density
• Mathematics, Computer Science
• Monte Carlo Methods Appl.
• 1996
It is proven that the discretization error can be expanded in terms of powers of $\frac1n$ under a nondegeneracy condition of Hormander type for the infinitesimal generator of $(X_t)$. Expand
Transient Fokker-Planck-Kolmogorov equation solved with smoothed particle hydrodynamics method
• Mathematics
• 2013
SUMMARY Probabilistic theories aim at describing the properties of systems subjected to random excitations by means of statistical characteristics such as the probability density function ψ (pdf).Expand
Hermite Spectral Method to 1-D Forward Kolmogorov Equation and Its Application to Nonlinear Filtering Problems
• Mathematics, Computer Science
• IEEE Transactions on Automatic Control
• 2013
The Hermite spectral method (HSM) to numerically solve the forward Kolmogorov equation (FKE) is investigated and the HSM to FKE algorithm surpasses the particle filters as a real-time solver to NLF. Expand
Nonparametric Adjoint-Based Inference for Stochastic Differential Equations
• Computer Science
• 2016 IEEE International Conference on Data Science and Advanced Analytics (DSAA)
• 2016
A nonparametric method to infer the drift and diffusion functions of a stochastic differential equation that can build predictive models starting with repeated time series and/or high-dimensional longitudinal data and applies to real data on hourly measurements of ground level ozone. Expand
Spectral solution of delayed random walks.
• Mathematics, Medicine
• Physical review. E, Statistical, nonlinear, and soft matter physics
• 2012
A spectral method for computing the probability density function for delayed random walks is developed, which is exact to machine precision and faster than existing approaches to nonlinear stochastic delay differential equations. Expand
The Path Integral Approach to Financial Modeling and Options Pricing
In this paper we review some applications of the path integral methodology of quantum mechanics to financial modeling and options pricing. A path integral is defined as a limit of the sequence ofExpand