随机微分方程数值解法1.pptVIP

Stability Issues for SDEsExplicit and Implicit MethodsK Burrage and P M BurrageDepartment of MathematicsUniversity of QueenslandBrisbane, Australiakb@maths.uq.edu.aupmb@maths.uq.edu.au  Contents Stability Concepts A-stability and mean-square stability T-stability Composite methods High order implicit methods Prediction-Correction Stability Concepts many SDEs cannot be solved analytically need accurate and stable numerical schemes stable if decays to zero, so should numerical solution if bounded, numerical solution should be bounded linear test equation deterministic Stratonovich Ito Stability of Ito if a) stochastically stable b) stochastically A-stable, if in addition, c) stochastically A-stable in the large, if in addition to the above 2, Theorem 1. If uniformly in is stochastically A-stable if d) Numerical A-stability A method is A-stable if, when applied to test equation, Theorem 2 (Higham): Real-valued, non-negative, iid ; Numerical MS-stability general stochastic Runge-Kutta method apply to linear SDE test equation Example - method R2: change of variables require Example - method E1: Stability function: T-stability (Saito and Mitsui) Measures asymptotic stability in weak formulation. T-stability (Burrage and Tian) Strong form Implicit Methods Class: For strong order 1: Minimum error constants: Stability Mean square stability = MS stability function For MS stability function does not exist. Some methods: Implicit Full Euler: Applied to (2): can become unbounded Composite: Combination of 2 methods, Avoids problem of unboundedness Composition: Compose methods (5) and (6) Stability function of (6) Prediction-Correction Deterministic case RK Method as an s-stage corrector written as block (l+1)s-stage RK: Every correct