An analytic approximation of solutions of stochastic differential equations
type: Item
instance of: scholarly article
title: An analytic approximation of solutions of stochastic differential equations
zbMATH Open document ID: 1075.60067
DOI: 10.1016/S0898-1221(04)90074-0
publication date: 2005-03-08T00:00:00Z
review text: In general, it is not possible to determine explicitly the solution of the equation \[ dX(t)=a(t,X(t)) dt +b(t,X(t)) dW(t), \;t \in [0,1], \quad X(0)=\eta. \] To obtain an approximate solution, some applicable analytic or numerical methods are usually used. An analytic approximation for the solution of the equation above is given, namely the drift and diffusion coefficients are Taylor approximations of the functions \(a\) and \(b\). If \(0=t_0<t_1<t_2<\ldots<t_n=1,\) \(\delta_n=\max_{0\leq k \leq n-1}(t_{k+1}-t_k)\) is an arbitrary partition of the interval \([0,1]\), then the approximation is the form \[ \begin{multlined} X_t^n =X_{t_k}^n+\int_{t_k}^t \sum_{i=1}^{m_1} \frac{a_x^{(i)}(s, X_{t_k}^n)}{i!} (X_s^n-X_{t_k}^n)^{i} \,ds+\\ \int_{t_k}^t \sum_{i=1}^{m_2} \frac{b_x^{(i)}(s, X_{t_k}^n)}{i!} (X_s^n-X_{t_k}^n)^i \,dw_s, \quad 0 \leq k\leq{n-1},\end{multlined} \] where \(X_{0}^n=\eta\) a.s. The authors compare in the \(L^p\)-norm, \(p\geq 2\), the solution of the first equation to its approximation. They prove that the sequence of the approximate solutions tends with probability one to the solution of first equation.
reviewed by: Rózsa Horváth-Bokor
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 2142355
zbMATH Keywords: Taylor approximations
zbMATH Keywords: \(L^p\) -convergence, convergence with probability one
author: Svetlana Janković
author: Dejan Ilić
MaRDI profile type: Publication
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published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 2142355