Approximations for stochastic differential equations with reflecting convex boundaries
type: Item
instance of: scholarly article
title: Approximations for stochastic differential equations with reflecting convex boundaries
zbMATH Open document ID: 0841.60042
DOI: 10.1016/0304-4149(95)00040-E
author: Roger Pettersson
published in: Stochastic Processes and their Applications
publication date: 1996-07-31T00:00:00Z
review text: The paper studies a numerical method for the solution on \([0,T]\) of a stochastic differential equation (SDE) reflecting on the boundary of a closed convex domain \(G \subset R^d\) (Skorokhod problem). The SDE is Lipschitzian, either autonomous or satisfying suitable conditions of linear growth type. \(G\) satisfies the Tanaka condition: there are \(\theta > 0\) and \(\varepsilon > 0\) such that, for every \(x \in G\), there is a point \(y_x\) for which \(x \in B_\theta(y_x)\) and \(B_\varepsilon(y_x) \subset G\). Here \(B_a(c)\) is the open ball centered in \(c\) with radius \(a\). The numerical method is an iterative projection scheme (Euler's method combined with projections on \(G\)) using a partition of \([0,T]\) with mesh \(\delta\). For bounded diffusion matrices, the paper proves mean square pointwise convergence of the approximation to the true solution with rate \(O((\delta \log 1/\delta)^{1/2})\) when \(\delta \to 0\). For particular forms of \(G\), stronger and/or faster convergence is shown. These results improve previous ones by other authors. One of the lemmas used in the proof is an interesting result on Brownian motion.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H20
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 60F25
zbMATH DE Number: 828756
zbMATH Keywords: Skorokhod problem
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: reflections
zbMATH Keywords: numerical methods
reviewed by: Carlos A. dos Santos Braumann
MaRDI profile type: Publication
cites work: Strong Solutions of Stochastic Differential Equations with Boundary Conditions
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cites work: Asymptotics via empirical processes. With comments and a rejoinder by the author
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cites work: On approximation of solutions of multidimensional SDE's with reflecting boundary conditions
cites work: Stochastic differential equations with reflecting boundary condition in convex regions
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description: scientific article; zbMATH DE number 828756