Approximations for stochastic differential equations with reflecting convex boundaries
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.
scientific article; zbMATH DE number 828756