2016/09/26 by Larisa Yaroslavtseva, Yaroslavtseva, Larisa
Economics, Econometrics and Finance · Social Sciences · Decision Sciences · #Stochastic processes and financial applications #Insurance, Mortality, Demography, Risk Management #Risk and Portfolio Optimization
paper · pdf · doi:10.48550/arxiv.1609.08073
Recently, it has been shown in [Hairer, M., Hutzenthaler, M., Jentzen, A.,\nLoss of regularity for Kolmogorov equations, Ann. Probab. 43, 2 (2015),\n468--527] that there exists a system of stochastic differential equations (SDE)\non the time interval [0,T] with infinitely often differentiable and bounded\ncoefficients such that the Euler scheme with equidistant time steps converges\nto the solution of this SDE at the final time in the strong sense but with no\npolynomial rate. Even worse, in [Jentzen, A., M "uller-Gronbach, T., and\nYaroslavtseva, L. On stochastic differential equations with arbitrary slow\nconvergence rates for strong approximation, Commun. Math. Sci. 14, 7 (2016),\n1477-1500] it has been shown that for any sequence (an)n\∈ mathbb\nN\⊂ (0,\∞), which may converge to zero arbitrary slowly, there\nexists an SDE on [0,T] with infinitely often differentiable and bounded\ncoefficients such that no approximation of the solution of this SDE at the\nfinal time based on n evaluations of the driving Brownian motion at fixed\ntime points can achieve a smaller absolute mean error than the given number\nan. In the present article we generalize the latter result to the case when\nthe approximations may choose the location as well as the number of the\nevaluation sites of the driving Brownian motion in an adaptive way dependent on\nthe values of the Brownian motion observed so far.\n