Second order weak Runge Kutta type methods for Itô equations
|
|
- Dana Reed
- 5 years ago
- Views:
Transcription
1 Mathematics and Computers in Simulation Second order weak Runge Kutta type methods for Itô equations Vigirdas Mackevičius, Jurgis Navikas Department of Mathematics and Informatics, Vilnius University, Naugarduko 4, 600 Vilnius, Lithuania Received 1 October 000; accepted 13 November 000 Abstract A standard second order weak Runge Kutta method for a stochastic differential equation can be applied only in the case where the equation is understood in the Stratonovich sense. To adapt Runge Kutta type methods for Itô equations, we propose to use a rather simple additional derivative-free term. 001 IMACS. Published by Elsevier Science B.V. All rights reserved. MSC: 60H35; 65C30; 68U0 Keywords: Stochastic differential equation; Runge Kutta method; Weak approximation 1. Introduction: Runge Kutta type methods for SDEs Consider a one-dimensional stochastic differential equation of the form dx t = bx t dt + σx t db t, X 0 = x, 1 with a driving Brownian motion B ={B t,t 0}. By standard four-stage Runge Kutta type approximations of its solution we mean approximations X h ={Xkh h,k = 0, 1,,...}, h>0, of the form where X h 0 = x, Xh k+1h = axh kh,h, B k, B k = B k+1h B kh, ax,s,y = x + 3 q i F i s + i=0 3 r i G i y, i=0 Corresponding author. address: vigirdas.mackevicius@maf.vu.lt V. Mackevičius /01/$ IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S
2 30 V. Mackevičius, J. Navikas / Mathematics and Computers in Simulation F 0 = bx, G 0 = σx + α 00 F 0 s, H 1 = x + α 10 F 0 s + β 10 G 0 y, F 1 = bh 1, G 1 = σh 1, H = x + α 0 F 0 + α 1 F 1 s + β 0 G 0 + β 1 G 1 y, F = bh, G = σh, H 3 = x + α 30 F 0 + α 31 F 1 + α 3 F s + β 30 G 0 + β 31 G 1 + β 3 G y, F 3 = bh 3, G 3 = σh 3, In [3], it is shown that if Eq. 1 is understood in the Stratonovich sense, then, under some boundedness and smoothness conditions, 1 unknown parameters α ij, β ij, q i, and r i can be chosen so that the approximation X h has the second order accuracy in the weak sense or in the sense of distributions. This means that see, e.g. [,4,5], for each fixed T>0 EfX h T EfX T = Oh, h 0, for a rather wide class of functions f : R R including polynomials. Unfortunately, this cannot be done when Eq. 1 is understood in the Itô sense. The reason is that, in the Itô case, the system of 15 equations to be satisfied by the parameters has no solutions. For example, one of the equations is q 0 + q 1 + q + q 3 bx = bx 1/σ σ x which, in Stratonovich case, becomes q 0 + q 1 + q + q 3 bx = bx, i.e. q 0 + q 1 + q + q 3 = 1. Our main idea is adding to the method-defining function a = ax,s,y a derivative-free two-stage correction term, which neutralizes bad terms in the equations for parameters such as 1/σ σ x in the example above. More precisely, we consider the approximations of the form with X h 0 = x, Xh k+1h = AXh kh,h, B k, 3 Ax,s,y= ax,s,y + Kx,s, where a is of the form and K is of the form Kx,s = γ 1 σ x + γ σxs σx + γ 3 σxs. In this paper, we show that, with this refinement, there exist second order weak Runge Kutta approximations for Itô equations and give a simulation example.. Second order conditions for weak Runge Kutta approximations In [3], we derived the conditions sufficient for the second order accuracy of a weak approximation of the general form 3 not necessarily Runge Kutta type of the solution of the Itô Eq. 1 A y x = σx, A s x = bx 1 σσ x, A yy x = σσ x, 4 A ys + A yyy x = bσ x + 1 σ σ x, A ss + A yys A yyyy x = bb x + 1 σ b x,
3 V. Mackevičius, J. Navikas / Mathematics and Computers in Simulation where x = x, 0, 0. Using this equations system, let us try to derive the relations for the parameters of the Runge Kutta type approximation. We first need to calculate the derivatives A y, A s, A yy, A ys, A yyy, A ss, A yys, and A yyyy of the function A that appear in Eq The expressions of derivatives are rather long, but we are actually interested on their values at the points x only. Denoting α i = j α ij and β i = j β ij, we have A y x = r 0 + r 1 + r + r 3 σ x, A yy x = r 1β 1 + r β + r 3 β 3 σ σ x, A yyy x = 3r 1β1 + r β + r 3β3 σ σ x + 6r β 1 β 1 + r 3 β 31 β 1 + r 3 β 3 β σ σ x, A yyyy x = 4r 1β1 3 + r β 3 + r 3β3 3 σ 3 σ x + [1r β 1 β1 + r 3β 31 β1 + r 3β 3 β +4r β 1 β 1 β + r 3 β 31 β 1 β 3 + r 3 β 3 β β 3 ]σ σ σ x + 4r 3 β 3 β 1 β 1 σ σ 3 x, A s x = q 0 + q 1 + q + q 3 bx, A ss x = q 1α 1 + q α + r 3 α 3 bb x, A ys x = r 0α 0 + r 1 α 1 + r α + r 3 α 3 bσ x + q 1 β 1 + q β + q 3 β 3 b σx, A yys x = [r 1β 1 α 0 + r β 0 α 0 + β 1 α 1 + r 3 β 30 α 0 + β 31 α 1 + β 3 α ]bσ x +r 1 α 1 β 1 + r α β + r 3 α 3 β 3 bσ σ x + q β 1 β 1 + q 3 β 31 β 1 + q 3 β 3 β +r α 1 β 1 + r 3 α 31 β 1 + r 3 α 3 β b σσ x + q 1 β 1 + q β + q 3β 3 b σ x. Note that the partial derivatives of A with respect to y contain only the diffusion coefficient σ, the derivatives with respect to s contain the shift coefficient b, and, finally, the mixed partial derivatives with respect to both y and s contain both coefficients. Substituting these values into Eq. 4, we get the following 17 equations for comparison, in the parentheses, we indicate the corresponding values of the right sides in the Stratonovich case r 0 + r 1 + r + r 3 = 1, r 1 β 1 + r β + r 3 β 3 = 1, 1 5. r 1 β1 + r β + r 3β3 = 1 6, r β 1 β 1 + r 3 β 31 β 1 + r 3 β 3 β = 0, r 1 β1 3 + r β 3 + r 3β3 3 = 0, r β 1 β1 + r 3β 31 β1 + r 3β 3 β + r β 1 β 1 β + r 3 β 31 β 1 β 3 + r 3 β 3 β β 3 = 0, r 3 β 3 β 1 β 1 = 0, q 0 + q 1 + q + q 3 = 1, q 1 α 1 + q α + r 3 α 3 = 1, r 0 α 0 + r 1 α 1 + r α + r 3 α 3 = 1, q 1 β 1 + q β + q 3 β 3 = 1, For calculations, we used MAPLE.
4 3 V. Mackevičius, J. Navikas / Mathematics and Computers in Simulation r 1 β 1 α 0 + r β 0 α 0 + β 1 α 1 + r 3 β 30 α 0 + β 31 α 1 + β 3 α = 0, 1 4 q β 1 β 1 + q 3 β 31 β 1 + q 3 β 3 β + r α 1 β 1 + r 3 α 31 β 1 + r 3 α 3 β = 0, 1 4 r 1 α 1 β 1 + r α β + r 3 α 3 β 3 = 0, 1 4 q 1 β1 + q β + q 3β3 = 1, 1 γ 1 γ γ 3 = 1, γ 1 γ γ 3 = Here we have again grouped the equations into groups according to the participation of the coefficients. The first stochastic or diffusion group contains the parameters r i and β ij related to the diffusion coefficient σ ; the second deterministic or drift one 5.8 and 5.9 contains the parameters q i and α ij related to the drift coefficient b; the third mixed group contains all the parameters just mentioned; finally, the last two equations connects the additional parameters γ i. 3. Example We first easily choose the values of the parameters γ i, i = 1,, 3, satisfying Eqs and 5.17 γ 1 = 1 4, γ = 1, γ 3 = 1. It is convenient to put the remaining parameters into the Butcher-type array of the form cf. [1] From several solutions of Eqs that we succeeded to find using MAPLE, we have chosen the following rather nice one having many zeros
5 V. Mackevičius, J. Navikas / Mathematics and Computers in Simulation Thus, our approximation is given by the following method-defining function Ax,s,y= x + 1 bxs bx + bxs + σxys 1 σ x 4 5 bxs y σxy σ x bxs + 1 σxy y 1 σx + bxs + σxyy 6 1 σσx + σxs σx σxs. 4 It is interesting to compare it with the classical Milstein [4] approximation defined by the function ax,s,y= x + σxy + bx 1 σxσ x s + 1 σxσ xy + 1 bσ x σ xσ x ys + 1 bxb x σ xb x s Fig. 1. A comparison of Runge Kutta type and Euler approximations. Solid line: exact values of EfX t. Solid polygonal line: the Runge Kutta approximation. Dashed polygonal line: the Euler approximation. Number of simulated trajectories of approximations n =
6 34 V. Mackevičius, J. Navikas / Mathematics and Computers in Simulation containing the derivatives of b and σ up to the second order. For a simulation example, we have chosen the equation dx t = 1 X t + Xt + 1 dt + Xt + 1dB t, X t = 0, having the solution X t = sin ht + B t, t 0. To compare approximations with the exact solution, we have to chose the test function f such that EfX t could be explicitly found. To produce, for example, a third order polynomial of t, we take fx = Parcsinh x, where Py = y 3 6y + 8y. Then ft:= EfX t = E[t + B t 3 6t + B t + 8t + B t ] = t 3 3t + t = tt 1t. In Fig. 1, we see the simulation results for the constructed Runge Kutta type approximation. The approximation steps were taken h = 0.3, 0., 0.1. As usual, approximations of the function f were obtained by averaging the values fx h,i kh, where Xh,i, i = 1,,...,n, are n independent simulated trajectories of the approximation. The number of trajectories was taken n = For comparison, we have simultaneously generated trajectories of the Euler approximation defined by ax,s,y = x + bxs + σxy. We see that the Runge Kutta type approximation behaves significantly better, especially, for large values of h. We have also compared the Runge Kutta type approximation with the Milstein one. It is, however, difficult to show this graphically, since the corresponding graphs appeared to be visually indistinguishable! References [1] J. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge Kutta and General Linear Methods, Wiley, New York, [] P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 199. [3] V. Mackevičius, Second order weak approximations for Stratonovich stochastic differential equations, Lith. Math. J [4] G.N. Milshtein, A method of second-order accuracy for the integration of stochastic differential equations, Theor. Probab. Appl [5] D. Talay, Discrétization d une équation différentielle stochastique et calcul approché d espérences de fonctionnelles de la solution, Modélisation Mathématique et Analyse Numérique
c 2002 Society for Industrial and Applied Mathematics
SIAM J. SCI. COMPUT. Vol. 4 No. pp. 507 5 c 00 Society for Industrial and Applied Mathematics WEAK SECOND ORDER CONDITIONS FOR STOCHASTIC RUNGE KUTTA METHODS A. TOCINO AND J. VIGO-AGUIAR Abstract. A general
More informationLecture 4: Numerical Solution of SDEs, Itô Taylor Series, Gaussian Process Approximations
Lecture 4: Numerical Solution of SDEs, Itô Taylor Series, Gaussian Process Approximations Simo Särkkä Aalto University Tampere University of Technology Lappeenranta University of Technology Finland November
More informationMean-square Stability Analysis of an Extended Euler-Maruyama Method for a System of Stochastic Differential Equations
Mean-square Stability Analysis of an Extended Euler-Maruyama Method for a System of Stochastic Differential Equations Ram Sharan Adhikari Assistant Professor Of Mathematics Rogers State University Mathematical
More informationNumerical Integration of SDEs: A Short Tutorial
Numerical Integration of SDEs: A Short Tutorial Thomas Schaffter January 19, 010 1 Introduction 1.1 Itô and Stratonovich SDEs 1-dimensional stochastic differentiable equation (SDE) is given by [6, 7] dx
More informationIntroduction to the Numerical Solution of SDEs Selected topics
0 / 23 Introduction to the Numerical Solution of SDEs Selected topics Andreas Rößler Summer School on Numerical Methods for Stochastic Differential Equations at Vienna University of Technology, Austria
More informationLecture 4: Numerical Solution of SDEs, Itô Taylor Series, Gaussian Approximations
Lecture 4: Numerical Solution of SDEs, Itô Taylor Series, Gaussian Approximations Simo Särkkä Aalto University, Finland November 18, 2014 Simo Särkkä (Aalto) Lecture 4: Numerical Solution of SDEs November
More informationarxiv: v2 [math.na] 16 Jan 2017
arxiv:171.551v2 [math.na] 16 Jan 217 The weak rate of convergence for the Euler-Maruyama approximation of one-dimensional stochastic differential equations involving the local times of the unknown process
More informationDiscretization of SDEs: Euler Methods and Beyond
Discretization of SDEs: Euler Methods and Beyond 09-26-2006 / PRisMa 2006 Workshop Outline Introduction 1 Introduction Motivation Stochastic Differential Equations 2 The Time Discretization of SDEs Monte-Carlo
More informationNumerical methods for solving stochastic differential equations
Mathematical Communications 4(1999), 251-256 251 Numerical methods for solving stochastic differential equations Rózsa Horváth Bokor Abstract. This paper provides an introduction to stochastic calculus
More informationStochastic Differential Equations.
Chapter 3 Stochastic Differential Equations. 3.1 Existence and Uniqueness. One of the ways of constructing a Diffusion process is to solve the stochastic differential equation dx(t) = σ(t, x(t)) dβ(t)
More informationStochastic perspective and parameter estimation for RC and RLC electrical circuits
Int. J. Nonlinear Anal. Appl. 6 (5 No., 53-6 ISSN: 8-68 (electronic http://www.ijnaa.semnan.ac.ir Stochastic perspective and parameter estimation for C and LC electrical circuits P. Nabati a,,. Farnoosh
More informationStrong Predictor-Corrector Euler Methods for Stochastic Differential Equations
Strong Predictor-Corrector Euler Methods for Stochastic Differential Equations Nicola Bruti-Liberati 1 and Eckhard Platen July 14, 8 Dedicated to the 7th Birthday of Ludwig Arnold. Abstract. This paper
More informationWEAK VERSIONS OF STOCHASTIC ADAMS-BASHFORTH AND SEMI-IMPLICIT LEAPFROG SCHEMES FOR SDES. 1. Introduction
WEAK VERSIONS OF STOCHASTIC ADAMS-BASHFORTH AND SEMI-IMPLICIT LEAPFROG SCHEMES FOR SDES BRIAN D. EWALD 1 Abstract. We consider the weak analogues of certain strong stochastic numerical schemes considered
More informationJournal of Computational and Applied Mathematics. Higher-order semi-implicit Taylor schemes for Itô stochastic differential equations
Journal of Computational and Applied Mathematics 6 (0) 009 0 Contents lists available at SciVerse ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More informationStochastic Differential Equations
Chapter 5 Stochastic Differential Equations We would like to introduce stochastic ODE s without going first through the machinery of stochastic integrals. 5.1 Itô Integrals and Itô Differential Equations
More informationStochastic differential equation models in biology Susanne Ditlevsen
Stochastic differential equation models in biology Susanne Ditlevsen Introduction This chapter is concerned with continuous time processes, which are often modeled as a system of ordinary differential
More informationApproximating diffusions by piecewise constant parameters
Approximating diffusions by piecewise constant parameters Lothar Breuer Institute of Mathematics Statistics, University of Kent, Canterbury CT2 7NF, UK Abstract We approximate the resolvent of a one-dimensional
More informationI forgot to mention last time: in the Ito formula for two standard processes, putting
I forgot to mention last time: in the Ito formula for two standard processes, putting dx t = a t dt + b t db t dy t = α t dt + β t db t, and taking f(x, y = xy, one has f x = y, f y = x, and f xx = f yy
More informationIntroduction to standard and non-standard Numerical Methods
Introduction to standard and non-standard Numerical Methods Dr. Mountaga LAM AMS : African Mathematic School 2018 May 23, 2018 One-step methods Runge-Kutta Methods Nonstandard Finite Difference Scheme
More informationA nonparametric method of multi-step ahead forecasting in diffusion processes
A nonparametric method of multi-step ahead forecasting in diffusion processes Mariko Yamamura a, Isao Shoji b a School of Pharmacy, Kitasato University, Minato-ku, Tokyo, 108-8641, Japan. b Graduate School
More informationEULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS
Qiao, H. Osaka J. Math. 51 (14), 47 66 EULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS HUIJIE QIAO (Received May 6, 11, revised May 1, 1) Abstract In this paper we show
More informationON THE FIRST TIME THAT AN ITO PROCESS HITS A BARRIER
ON THE FIRST TIME THAT AN ITO PROCESS HITS A BARRIER GERARDO HERNANDEZ-DEL-VALLE arxiv:1209.2411v1 [math.pr] 10 Sep 2012 Abstract. This work deals with first hitting time densities of Ito processes whose
More informationMath 216 Final Exam 14 December, 2012
Math 216 Final Exam 14 December, 2012 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationA numerical method for solving uncertain differential equations
Journal of Intelligent & Fuzzy Systems 25 (213 825 832 DOI:1.3233/IFS-12688 IOS Press 825 A numerical method for solving uncertain differential equations Kai Yao a and Xiaowei Chen b, a Department of Mathematical
More informationFourth Order RK-Method
Fourth Order RK-Method The most commonly used method is Runge-Kutta fourth order method. The fourth order RK-method is y i+1 = y i + 1 6 (k 1 + 2k 2 + 2k 3 + k 4 ), Ordinary Differential Equations (ODE)
More informationIntroduction to numerical simulations for Stochastic ODEs
Introduction to numerical simulations for Stochastic ODEs Xingye Kan Illinois Institute of Technology Department of Applied Mathematics Chicago, IL 60616 August 9, 2010 Outline 1 Preliminaries 2 Numerical
More informationStochastic Integration and Stochastic Differential Equations: a gentle introduction
Stochastic Integration and Stochastic Differential Equations: a gentle introduction Oleg Makhnin New Mexico Tech Dept. of Mathematics October 26, 27 Intro: why Stochastic? Brownian Motion/ Wiener process
More information1 Brownian Local Time
1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) =
More informationON LIE GROUP CLASSIFICATION OF A SCALAR STOCHASTIC DIFFERENTIAL EQUATION
Journal of Nonlinear Mathematical Physics ISSN: 1402-9251 (Print) 1776-0852 (Online) Journal homepage: http://www.tandfonline.com/loi/tnmp20 ON LIE GROUP CLASSIFICATION OF A SCALAR STOCHASTIC DIFFERENTIAL
More informationStochastic Differential Equations
CHAPTER 1 Stochastic Differential Equations Consider a stochastic process X t satisfying dx t = bt, X t,w t dt + σt, X t,w t dw t. 1.1 Question. 1 Can we obtain the existence and uniqueness theorem for
More informationDownloaded 12/15/15 to Redistribution subject to SIAM license or copyright; see
SIAM J. NUMER. ANAL. Vol. 40, No. 4, pp. 56 537 c 2002 Society for Industrial and Applied Mathematics PREDICTOR-CORRECTOR METHODS OF RUNGE KUTTA TYPE FOR STOCHASTIC DIFFERENTIAL EQUATIONS KEVIN BURRAGE
More informationWORD SERIES FOR THE ANALYSIS OF SPLITTING SDE INTEGRATORS. Alfonso Álamo/J. M. Sanz-Serna Universidad de Valladolid/Universidad Carlos III de Madrid
WORD SERIES FOR THE ANALYSIS OF SPLITTING SDE INTEGRATORS Alfonso Álamo/J. M. Sanz-Serna Universidad de Valladolid/Universidad Carlos III de Madrid 1 I. OVERVIEW 2 The importance of splitting integrators
More informationStochastic Integration (Simple Version)
Stochastic Integration (Simple Version) Tuesday, March 17, 2015 2:03 PM Reading: Gardiner Secs. 4.1-4.3, 4.4.4 But there are boundary issues when (if ) so we can't apply the standard delta function integration
More informationCoupled differential equations
Coupled differential equations Example: dy 1 dy 11 1 1 1 1 1 a y a y b x a y a y b x Consider the case with b1 b d y1 a11 a1 y1 y a1 ay dy y y e y 4/1/18 One way to address this sort of problem, is to
More informationLogistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations
Logistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations S. Y. Ha and J. Park Department of Mathematical Sciences Seoul National University Sep 23, 2013 Contents 1 Logistic Map 2 Euler and
More informationGaussian processes for inference in stochastic differential equations
Gaussian processes for inference in stochastic differential equations Manfred Opper, AI group, TU Berlin November 6, 2017 Manfred Opper, AI group, TU Berlin (TU Berlin) inference in SDE November 6, 2017
More informationSIMULATION OF STOCHASTIC DIFFERENTIAL EQUATIONS
Ann. Inst. Statist. Math. Vol. 45, No. 3, 419-432 (1993) SIMULATION OF STOCHASTIC DIFFERENTIAL EQUATIONS YOSHIHIRO SAITO 1 AND TAKETOMO MITSUI 2 1Shotoku Gakuen Women's Junior College, 1-38 Nakauzura,
More informationNested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model
Nested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model Xiaowei Chen International Business School, Nankai University, Tianjin 371, China School of Finance, Nankai
More informationApplied Math for Engineers
Applied Math for Engineers Ming Zhong Lecture 15 March 28, 2018 Ming Zhong (JHU) AMS Spring 2018 1 / 28 Recap Table of Contents 1 Recap 2 Numerical ODEs: Single Step Methods 3 Multistep Methods 4 Method
More informationAn adaptive numerical scheme for fractional differential equations with explosions
An adaptive numerical scheme for fractional differential equations with explosions Johanna Garzón Departamento de Matemáticas, Universidad Nacional de Colombia Seminario de procesos estocásticos Jointly
More informationSimulation Method for Solving Stochastic Differential Equations with Constant Diffusion Coefficients
Journal of mathematics and computer Science 8 (2014) 28-32 Simulation Method for Solving Stochastic Differential Equations with Constant Diffusion Coefficients Behrouz Fathi Vajargah Department of statistics,
More informationSDE Coefficients. March 4, 2008
SDE Coefficients March 4, 2008 The following is a summary of GARD sections 3.3 and 6., mainly as an overview of the two main approaches to creating a SDE model. Stochastic Differential Equations (SDE)
More informationSolving Ordinary Differential equations
Solving Ordinary Differential equations Taylor methods can be used to build explicit methods with higher order of convergence than Euler s method. The main difficult of these methods is the computation
More informationIntroduction to Random Diffusions
Introduction to Random Diffusions The main reason to study random diffusions is that this class of processes combines two key features of modern probability theory. On the one hand they are semi-martingales
More informationA posteriori error estimates applied to flow in a channel with corners
Mathematics and Computers in Simulation 61 (2003) 375 383 A posteriori error estimates applied to flow in a channel with corners Pavel Burda a,, Jaroslav Novotný b, Bedřich Sousedík a a Department of Mathematics,
More informationAn Approximation to the Solution of the Brusselator System by Adomian Decomposition Method and Comparing the Results with Runge-Kutta Method
Int. J. Contemp. Mat. Sciences, Vol. 2, 27, no. 2, 983-989 An Approximation to te Solution of te Brusselator System by Adomian Decomposition Metod and Comparing te Results wit Runge-Kutta Metod J. Biazar
More informationRunge Kutta methods for numerical solution of stochastic dierential equations
Journal of Computational and Applied Mathematics 18 00 19 41 www.elsevier.com/locate/cam Runge Kutta methods for numerical solution of stochastic dierential equations A.Tocino a; ; 1, R.Ardanuy b a Departamento
More informationExact Linearization Of Stochastic Dynamical Systems By State Space Coordinate Transformation And Feedback I g-linearization
Applied Mathematics E-Notes, 3(2003), 99-106 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Exact Linearization Of Stochastic Dynamical Systems By State Space Coordinate
More informationCHAPTER 80 NUMERICAL METHODS FOR FIRST-ORDER DIFFERENTIAL EQUATIONS
CHAPTER 8 NUMERICAL METHODS FOR FIRST-ORDER DIFFERENTIAL EQUATIONS EXERCISE 33 Page 834. Use Euler s method to obtain a numerical solution of the differential equation d d 3, with the initial conditions
More informationMathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )
Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical
More informationOn Mean-Square and Asymptotic Stability for Numerical Approximations of Stochastic Ordinary Differential Equations
On Mean-Square and Asymptotic Stability for Numerical Approximations of Stochastic Ordinary Differential Equations Rózsa Horváth Bokor and Taketomo Mitsui Abstract This note tries to connect the stochastic
More informationfor all f satisfying E[ f(x) ] <.
. Let (Ω, F, P ) be a probability space and D be a sub-σ-algebra of F. An (H, H)-valued random variable X is independent of D if and only if P ({X Γ} D) = P {X Γ}P (D) for all Γ H and D D. Prove that if
More information[Ahmed*, 5(3): March, 2016] ISSN: (I2OR), Publication Impact Factor: 3.785
IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY DENSITIES OF DISTRIBUTIONS OF SOLUTIONS TO DELAY STOCHASTIC DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS INITIAL DATA ( PART II)
More informationDivergence theorems in path space II: degenerate diffusions
Divergence theorems in path space II: degenerate diffusions Denis Bell 1 Department of Mathematics, University of North Florida, 4567 St. Johns Bluff Road South, Jacksonville, FL 32224, U. S. A. email:
More informationLecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C.
Lecture 9 Approximations of Laplace s Equation, Finite Element Method Mathématiques appliquées (MATH54-1) B. Dewals, C. Geuzaine V1.2 23/11/218 1 Learning objectives of this lecture Apply the finite difference
More informationHigher order weak approximations of stochastic differential equations with and without jumps
Higher order weak approximations of stochastic differential equations with and without jumps Hideyuki TANAKA Graduate School of Science and Engineering, Ritsumeikan University Rough Path Analysis and Related
More informationAn estimate for the probability of dependent events
Statistics and Probability Letters 78 (2008) 2839 2843 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro An estimate for the
More information1 Partial differentiation and the chain rule
1 Partial differentiation and the chain rule In this section we review and discuss certain notations and relations involving partial derivatives. The more general case can be illustrated by considering
More informationComputational Physics (6810): Session 8
Computational Physics (6810): Session 8 Dick Furnstahl Nuclear Theory Group OSU Physics Department February 24, 2014 Differential equation solving Session 7 Preview Session 8 Stuff Solving differential
More informationc 2003 Society for Industrial and Applied Mathematics
SIAM J. SCI. COMPUT. Vol. 24, No. 5, pp. 1809 1822 c 2003 Society for Industrial and Applied Mathematics EXPONENTIAL TIMESTEPPING WITH BOUNDARY TEST FOR STOCHASTIC DIFFERENTIAL EQUATIONS KALVIS M. JANSONS
More informationPeter E. Kloeden Eckhard Platen. Numerical Solution of Stochastic Differential Equations
Peter E. Kloeden Eckhard Platen Numerical Solution of Stochastic Differential Equations Peter E. Kloeden School of Computing and Mathematics, Deakin Universit y Geelong 3217, Victoria, Australia Eckhard
More informationMATH 251 Examination II April 7, 2014 FORM A. Name: Student Number: Section:
MATH 251 Examination II April 7, 2014 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must
More informationNumerical Solution of Differential Equations
1 Numerical Solution of Differential Equations A differential equation (or "DE") contains derivatives or differentials. In a differential equation the unknown is a function, and the differential equation
More informationOn A General Formula of Fourth Order Runge-Kutta Method
On A General Formula of Fourth Order Runge-Kutta Method Delin Tan Zheng Chen Abstract In this paper we obtain a general formula of Runge-Kutta method in order with a free parameter t By picking the value
More informationMSc Dissertation topics:
.... MSc Dissertation topics: Omar Lakkis Mathematics University of Sussex Brighton, England November 6, 2013 Office Location: Pevensey 3 5C2 Office hours: Autumn: Tue & Fri 11:30 12:30; Spring: TBA. O
More information25. Chain Rule. Now, f is a function of t only. Expand by multiplication:
25. Chain Rule The Chain Rule is present in all differentiation. If z = f(x, y) represents a two-variable function, then it is plausible to consider the cases when x and y may be functions of other variable(s).
More informationExact Simulation of Diffusions and Jump Diffusions
Exact Simulation of Diffusions and Jump Diffusions A work by: Prof. Gareth O. Roberts Dr. Alexandros Beskos Dr. Omiros Papaspiliopoulos Dr. Bruno Casella 28 th May, 2008 Content 1 Exact Algorithm Construction
More informationOn the classification of certain curves up to projective tranformations
On the classification of certain curves up to projective tranformations Mehdi Nadjafikhah Abstract The purpose of this paper is to classify the curves in the form y 3 = c 3 x 3 +c 2 x 2 + c 1x + c 0, with
More informationKolmogorov Equations and Markov Processes
Kolmogorov Equations and Markov Processes May 3, 013 1 Transition measures and functions Consider a stochastic process {X(t)} t 0 whose state space is a product of intervals contained in R n. We define
More informationBrownian Motion. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Brownian Motion
Brownian Motion An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Background We have already seen that the limiting behavior of a discrete random walk yields a derivation of
More informationGeometric projection of stochastic differential equations
Geometric projection of stochastic differential equations John Armstrong (King s College London) Damiano Brigo (Imperial) August 9, 2018 Idea: Projection Idea: Projection Projection gives a method of systematically
More informationNUMERICAL METHODS IN THE WEAK SENSE FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH SMALL NOISE
SIAM J. NUMER. ANAL. c 1997 Society for Industrial and Applied Mathematics Vol. 34, No. 6, pp. 2142 2167, December 1997 004 NUMERICAL METHODS IN THE WEAK SENSE FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH
More informationQualitative behaviour of numerical methods for SDEs and application to homogenization
Qualitative behaviour of numerical methods for SDEs and application to homogenization K. C. Zygalakis Oxford Centre For Collaborative Applied Mathematics, University of Oxford. Center for Nonlinear Analysis,
More informationLangevin Methods. Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 10 D Mainz Germany
Langevin Methods Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 1 D 55128 Mainz Germany Motivation Original idea: Fast and slow degrees of freedom Example: Brownian motion Replace
More informationOrdinary Differential Equations (ODEs)
Ordinary Differential Equations (ODEs) 1 Computer Simulations Why is computation becoming so important in physics? One reason is that most of our analytical tools such as differential calculus are best
More informationLesson 4: Population, Taylor and Runge-Kutta Methods
Lesson 4: Population, Taylor and Runge-Kutta Methods 4.1 Applied Problem. Consider a single fish population whose size is given by x(t). The change in the size of the fish population over a given time
More informationCOMPUTER-AIDED MODELING AND SIMULATION OF ELECTRICAL CIRCUITS WITH α-stable NOISE
APPLICATIONES MATHEMATICAE 23,1(1995), pp. 83 93 A. WERON(Wroc law) COMPUTER-AIDED MODELING AND SIMULATION OF ELECTRICAL CIRCUITS WITH α-stable NOISE Abstract. The aim of this paper is to demonstrate how
More informationMini project ODE, TANA22
Mini project ODE, TANA22 Filip Berglund (filbe882) Linh Nguyen (linng299) Amanda Åkesson (amaak531) October 2018 1 1 Introduction Carl David Tohmé Runge (1856 1927) was a German mathematician and a prominent
More informationRunge-Kutta Method for Solving Uncertain Differential Equations
Yang and Shen Journal of Uncertainty Analysis and Applications 215) 3:17 DOI 1.1186/s4467-15-38-4 RESEARCH Runge-Kutta Method for Solving Uncertain Differential Equations Xiangfeng Yang * and Yuanyuan
More informationBernardo D Auria Stochastic Processes /12. Notes. March 29 th, 2012
1 Stochastic Calculus Notes March 9 th, 1 In 19, Bachelier proposed for the Paris stock exchange a model for the fluctuations affecting the price X(t) of an asset that was given by the Brownian motion.
More informationDifferential Equations
Differential Equations Overview of differential equation! Initial value problem! Explicit numeric methods! Implicit numeric methods! Modular implementation Physics-based simulation An algorithm that
More informationConsistency and Convergence
Jim Lambers MAT 77 Fall Semester 010-11 Lecture 0 Notes These notes correspond to Sections 1.3, 1.4 and 1.5 in the text. Consistency and Convergence We have learned that the numerical solution obtained
More informationNumerical method for approximating the solution of an IVP. Euler Algorithm (the simplest approximation method)
Section 2.7 Euler s Method (Computer Approximation) Key Terms/ Ideas: Numerical method for approximating the solution of an IVP Linear Approximation; Tangent Line Euler Algorithm (the simplest approximation
More informationDifferential Equations
Differential Equations Definitions Finite Differences Taylor Series based Methods: Euler Method Runge-Kutta Methods Improved Euler, Midpoint methods Runge Kutta (2nd, 4th order) methods Predictor-Corrector
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 216 17 INTRODUCTION TO NUMERICAL ANALYSIS MTHE612B Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.
More informationControlled Diffusions and Hamilton-Jacobi Bellman Equations
Controlled Diffusions and Hamilton-Jacobi Bellman Equations Emo Todorov Applied Mathematics and Computer Science & Engineering University of Washington Winter 2014 Emo Todorov (UW) AMATH/CSE 579, Winter
More informationON THE COMPLEXITY OF STOCHASTIC INTEGRATION
MATHEMATICS OF COMPUTATION Volume 7, Number 34, Pages 685 698 S 5-5718)114-X Article electronically published on March, ON THE COMPLEXITY OF STOCHASTIC INTEGRATION G. W. WASILKOWSKI AND H. WOŹNIAKOWSKI
More informationMath 215/255 Final Exam (Dec 2005)
Exam (Dec 2005) Last Student #: First name: Signature: Circle your section #: Burggraf=0, Peterson=02, Khadra=03, Burghelea=04, Li=05 I have read and understood the instructions below: Please sign: Instructions:.
More informationNumerical Methods for Ordinary Differential Equations
CHAPTER 1 Numerical Methods for Ordinary Differential Equations In this chapter we discuss numerical method for ODE. We will discuss the two basic methods, Euler s Method and Runge-Kutta Method. 1. Numerical
More informationarxiv: v1 [math.na] 31 Oct 2016
RKFD Methods - a short review Maciej Jaromin November, 206 arxiv:60.09739v [math.na] 3 Oct 206 Abstract In this paper, a recently published method [Hussain, Ismail, Senua, Solving directly special fourthorder
More informationQuasi-invariant measures on the path space of a diffusion
Quasi-invariant measures on the path space of a diffusion Denis Bell 1 Department of Mathematics, University of North Florida, 4567 St. Johns Bluff Road South, Jacksonville, FL 32224, U. S. A. email: dbell@unf.edu,
More information5 December 2016 MAA136 Researcher presentation. Anatoliy Malyarenko. Topics for Bachelor and Master Theses. Anatoliy Malyarenko
5 December 216 MAA136 Researcher presentation 1 schemes The main problem of financial engineering: calculate E [f(x t (x))], where {X t (x): t T } is the solution of the system of X t (x) = x + Ṽ (X s
More informationOrdinary Differential Equations
Ordinary Differential Equations In this lecture, we will look at different options for coding simple differential equations. Start by considering bicycle riding as an example. Why does a bicycle move forward?
More informationLecture 1: Pragmatic Introduction to Stochastic Differential Equations
Lecture 1: Pragmatic Introduction to Stochastic Differential Equations Simo Särkkä Aalto University, Finland (visiting at Oxford University, UK) November 13, 2013 Simo Särkkä (Aalto) Lecture 1: Pragmatic
More informationSolution of Stochastic Optimal Control Problems and Financial Applications
Journal of Mathematical Extension Vol. 11, No. 4, (2017), 27-44 ISSN: 1735-8299 URL: http://www.ijmex.com Solution of Stochastic Optimal Control Problems and Financial Applications 2 Mat B. Kafash 1 Faculty
More informationApproximation of random dynamical systems with discrete time by stochastic differential equations: I. Theory
Random Operators / Stochastic Eqs. 15 7, 5 c de Gruyter 7 DOI 1.1515 / ROSE.7.13 Approximation of random dynamical systems with discrete time by stochastic differential equations: I. Theory Yuri A. Godin
More informationMalliavin Calculus in Finance
Malliavin Calculus in Finance Peter K. Friz 1 Greeks and the logarithmic derivative trick Model an underlying assent by a Markov process with values in R m with dynamics described by the SDE dx t = b(x
More informationPhysics 584 Computational Methods
Physics 584 Computational Methods Introduction to Matlab and Numerical Solutions to Ordinary Differential Equations Ryan Ogliore April 18 th, 2016 Lecture Outline Introduction to Matlab Numerical Solutions
More informationChecking the Radioactive Decay Euler Algorithm
Lecture 2: Checking Numerical Results Review of the first example: radioactive decay The radioactive decay equation dn/dt = N τ has a well known solution in terms of the initial number of nuclei present
More information