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Continuous-time Random Walks for the Numerical Solution of Stochastic Differential Equations [article]

Nawaf Bou-Rabee, Eric Vanden-Eijnden (Courant Institute, NYU)
2015 arXiv   pre-print
This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics  ...  These schemes are obtained by spatially discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be  ...  The main tools used in this analysis are properties of the SDE solution and the following variation of constants formula for a mild solution of an inhomogeneous, linear differential equation on a Banach  ... 
arXiv:1502.05034v2 fatcat:2wxfuoplfzhcvbcg52lmr6zaue

Numerical approximation of random periodic solutions of stochastic differential equations

Chunrong Feng, Yu Liu, Huaizhong Zhao
2017 Zeitschrift für Angewandte Mathematik und Physik  
In this paper, we discuss the numerical approximation of random periodic solutions (r.p.s.) of stochastic differential equations (SDEs) with multiplicative noise.  ...  We prove that the latter is an approximated random periodic solution with an error to the exact one at the rate of √(Δ t) in the mean-square sense in Euler-Maruyama method and Δ t in the Milstein method  ...  reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.  ... 
doi:10.1007/s00033-017-0868-7 fatcat:wlz4ac4v3bbphmykpjgcg4ciwy

Numerical solution of random differential models

J.-C. Cortés, L. Jódar, L. Villafuerte, R. Company
2011 Mathematical and computer modelling  
This paper deals with the construction of a numerical solution of random initial value problems by means of a random improved Euler method.  ...  Conditions for the mean square convergence of the proposed method are established.  ...  a b s t r a c t This paper deals with the construction of a numerical solution of random initial value problems by means of a random improved Euler method.  ... 
doi:10.1016/j.mcm.2010.12.037 fatcat:zk2pfyyme5hatovm2kklswl2em

Numerical solution of systems of random differential equations with Gaussian statistics

Carl G Looney
1985 Journal of Mathematical Analysis and Applications  
A direct approach is used to compute a numerical solution for a system of (nonlinear) ordinary differential equations with Gaussian statistics.  ...  It is shown that for a large class of problems the accuracy of the solution can be made as great as desired by taking the step size h sufficiently small. B 1985 Academic Prerq Inc. 222  ...  Barry and Boyce [7] have a numerical method for certain differential equations with random boundaries.  ... 
doi:10.1016/0022-247x(85)90108-8 fatcat:zcgzgmtqyranrdfg5anz2idjby

Mean square numerical solution of random differential equations: Facts and possibilities

J.C. Cortés, L. Jódar, L. Villafuerte
2007 Computers and Mathematics with Applications  
This paper deals with the construction of numerical solutions of random initial value differential problems.  ...  The random Euler method is presented and the conditions for the mean square convergence are established.  ...  This motivates the main goal of this paper that is to introduce the random Euler method for constructing reliable numerical solutions of the problem (1.1) in the sense that approximations be mean square  ... 
doi:10.1016/j.camwa.2006.05.030 fatcat:ziixx5t5pnf5hdrcntjykr5pa4

Solving linear and quadratic random matrix differential equations: A mean square approach

M.-C. Casabán, J.-C. Cortés, L. Jódar
2016 Applied Mathematical Modelling  
Solving linear and quadratic random matrix differential equations: A mean square approach. Applied Mathematical Modelling. 40(21-22):9362-9377.  ...  Abstract In this paper linear and Riccati random matrix differential equations are solved taking advantage of the so called L p -random calculus.  ...  class of randomness, has been developed in recent years taking advantage of the mean square random calculus.  ... 
doi:10.1016/j.apm.2016.06.017 fatcat:65ylhcxhxba7rkqm7mefon3lxi

Symbolic and numeric scheme for solution of linear integro-differential equations with random parameter uncertainties and Gaussian stochastic process input

I.E. Poloskov, C. Soize
2018 Applied Mathematical Modelling  
The paper describes a theoretical apparatus and an algorithmic part of application of the Green matrix-valued functions for time-domain analysis of systems of linear stochastic integro-differential equations  ...  The first one explicitly defines closed relations for symbolic and numeric computations of the conditional mean and covariance functions, and the second one calculates unconditional characteristics by  ...  As it was said before, the main aim of numerical experiments performed by MCM is a numerical verification of the presented mathematical approach.  ... 
doi:10.1016/j.apm.2017.11.024 fatcat:qj32iyg2zzb65p2adx274uoqcm

Numerical Solution of Stiff ODEs with Physics-Informed RPNNs [article]

Evangelos Galaris, Gianluca Fabiani, Francesco Calabrò, Daniela di Serafino, Constantinos Siettos
2021 arXiv   pre-print
We propose a numerical method based on physics-informed Random Projection Neural Networks for the solution of Initial Value Problems (IVPs) of Ordinary Differential Equations (ODEs) with a focus on stiff  ...  The numerical solution of the IVPs is obtained by constructing a system of nonlinear algebraic equations, which is solved with respect to the output weights by the Gauss-Newton method, using a simple adaptive  ...  The estimation of the optimal output weights is achieved by solving a system of linear equations arising from a mean-squares minimization problem.  ... 
arXiv:2108.01584v2 fatcat:rtzek2qcara4df4mpq3xwmhojy

Extending the deterministic Riemann–Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations

C. Burgos, J.-C. Cortés, L. Villafuerte, R.-J. Villanueva
2017 Chaos, Solitons & Fractals  
Extending the deterministic Riemann-Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations.  ...  The approach is based on a mean square chain rule, recently established, together with the random Fröbenius method.  ...  the mean square derivative of a random power series in order to formally construct the solution 159 of the random fractional linear differential equation with a random initial condition.  ... 
doi:10.1016/j.chaos.2017.02.008 fatcat:2iuhoodokvap7nybx2swdoarhu

A Competitive Mean-Squared Error Approach to Beamforming

Yonina C. Eldar, Arye Nehorai, Patricio S. La Rosa
2007 IEEE Transactions on Signal Processing  
However this does not guarantee a small mean-squared error (MSE), hence on average their resulting signal estimateŝ(t) can be far from s(t).  ...  However, this approach does not guarantee a small MSE, so that on average, the resulting estimate of s(t) may be far from s(t).  ...  La Rosa for conducting the numerical examples in this paper.  ... 
doi:10.1109/tsp.2007.897883 fatcat:ggokauh6dfep3l22wxpnvvst4a

Analytic-Numerical Solution of Random Parabolic Models: A Mean Square Fourier Transform Approach

Maria-Consuelo Casaban, Juan-Carlos Cortes, Lucas Jodar
2018 Mathematical Modelling and Analysis  
This paper deals with the construction of mean square analytic-numerical solution of parabolic partial differential problems where both initial condition and coefficients are stochastic processes.  ...  By using a random Fourier transform, an inf- nite integral form of the solution stochastic process is firstly obtained.  ...  To achieve this goal, we use a mean square Fourier transform approach.  ... 
doi:10.3846/mma.2018.006 fatcat:lfmggz2u3zbwzbvhkqlnsplwdi

Mean-Squared Error Beamforming for Signal Estimation: A Competitive Approach [chapter]

Yonina C. Eldar, Arye Nehorai
2005 Robust Adaptive Beamforming  
1.30) is also the solution in the case in which we have both an upper and lower bound on the norm xviii MEAN-SQUARED ERROR BEAMFORMING FOR SIGNAL ESTIMATION of s.  ...  − mŝ) * R −1 (y − mŝ) +ŝ 2 E (a − m) * R −1 (a − m) = (y − mŝ) * R −1 (y − mŝ) +ŝ 2 Tr R −1 C . (1.84) Differentiating (1.84) with respect toŝ and equating to 0, we have that s = 1 Tr (R −1 C) + m * R  ... 
doi:10.1002/0471733482.ch5 fatcat:h7q5t3frmrcgtear7i3vkpd4pm

Solving the random diffusion model in an infinite medium: A mean square approach

M.-C. Casabán, R. Company, J.-C. Cortés, L. Jódar
2014 Applied Mathematical Modelling  
This paper deals with the construction of an analytic-numerical mean square solution of the random diffusion model in an infinite medium.  ...  Mean square operational rules to the Fourier transform of a stochastic process are developed and stated. The main statistical moments of the stochastic process solution are also computed.  ...  The theory and applications of random differential equations is a very active area of mathematical research and there are several different approaches, from the so-called stochastic differential approach  ... 
doi:10.1016/j.apm.2014.04.063 fatcat:dr3j5njuavacha2czj2nagpamy

Prediction of mean square radius of gyration of tree-like polymers by a general kinetic approach

Mário Rui P.F.N. Costa, Rolando C.S. Dias
2007 Polymer  
This paper describes a kinetic method to predict the z-average molecular mean square radius of gyration of tree-like polymers formed by irreversible reactions, assuming Gaussian chains.  ...  An automated method for the solution of those equations is valid both before as well as after gelation for complex kinetic schemes.  ...  A general procedure for solving the resulting partial differential equations by the method of characteristics is a key point in this approach.  ... 
doi:10.1016/j.polymer.2007.01.033 fatcat:7o5454satfc2dkcimvu7c72nim

Numerical Solution of Stochastic Differential Equations [chapter]

2013 Introduction to Stochastic Analysis  
This chapter is an introduction and survey of numerical solution methods for stochastic differential equations.  ...  We include a review of fundamental concepts, a description of elementary numerical methods and the concepts of convergence and order for stochastic differential equation solvers.  ...  Summary Numerical methods for the solution of stochastic differential equations are essential for the analysis of random phenomena.  ... 
doi:10.1002/9781118603338.ch13 fatcat:ku7bvcu65vcntkluw3cej4qmwq
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