Untitled

Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients

“…Two results motivated us to try to prove convergence of the stochastic Euler approximation Y N N to the exact solution X T of the SDE (1.1) in the strong meansquare sense as the number of time steps N goes to infinity. Gyöngy (1998) established pathwise convergence for SDEs with locally Lipschitz continuous coefficients. More precisely, theorem 1 in Gyöngy (1998)…”

“…Gyöngy (1998) established pathwise convergence for SDEs with locally Lipschitz continuous coefficients. More precisely, theorem 1 in Gyöngy (1998)…”

“…Bernard & Fleury (2001) establish convergence in probability under weak hypotheses such as local Lipschitz continuity. Pathwise convergence results can be found in Gyöngy (1998), Fleury (2005), or in . A number of authors obtain convergence for modified Euler schemes.…”

On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions

“…The main contribution of this work is to reveal that a slow convergence phenomenon of the form (1.2) also arises in two (d = 2) and three (d = 3) space dimensions. Upper error bounds and numerical approximation schemes for SDEs with non-globally Lipschitz continuous coefficients can, for example, be found in [7][8][9][10][11][12][13][14][15] and the references mentioned therein. Lower error bounds for strong approximation schemes for SDEs with globally Lipschitz continuous coefficients can, for example, be found in the overview article by Müller-Gronbach & Ritter [16] and the references mentioned therein.…”

“…In particular, it is of fundamental importance in this research area to reveal explicit conditions on the coefficients of the SDE which are both necessary and sufficient for numerical approximations to converge with positive strong/weak convergence rates. There are a number of articles in the literature which provide sufficient conditions for strong convergence rates for numerical approximations (cf., for example, [7][8][9][10][11][12][13][14][15] and the references mentioned therein). These conditions are far from being necessary for strong convergence rates.…”