2017/07/27 by Thomas Müller-Gronbach, Müller-Gronbach, Thomas, Larisa Yaroslavtseva +1
Economics, Econometrics and Finance · Social Sciences · #FOS: Mathematics #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1707.08818
openalex publication_date 2017/07/27 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
Recently, it has been shown in [Jentzen, A., M "uller-Gronbach, T., and\nYaroslavtseva, L., Commun. Math. Sci., 14, 2016] that there exists a system of\nautonomous stochastic differential equations (SDE) on the time interval [0,T]\nwith infinitely differentiable and bounded coefficients such that no strong\napproximation method based on evaluation of the driving Brownian motion at\nfinitely many fixed times in [0,T], e.g. on an equidistant grid, can converge\nin absolute mean to the solution at the final time with a polynomial rate in\nterms of the number of Brownian motion values that are used. In the literature\non strong approximation of SDEs, polynomial error rate results are typically\nachieved under the assumption that the first order derivatives of the\ncoefficients of the equation satisfy a polynomial growth condition. This\nassumption is violated for the pathological SDEs from the above mentioned\nnegative result. However, in the present article we construct an SDE with\nsmooth coefficients that have first order derivatives of at most linear growth\nsuch that the solution at the final time can not be approximated with a\npolynomial rate, whatever method based on observations of the driving Brownian\nmotion at finitely many fixed times is used. Most interestingly, it turns out\nthat using a method that adjusts the number of evaluations of the driving\nBrownian motion to its actual path, the latter SDE can be approximated with\nrate 1 in terms of the average number of evaluations that are used. To the best\nof our knowledge, this is only the second example in the literature of an SDE\nfor which there exist adaptive methods that perform superior to non-adaptive\nones with respect to the convergence rate.\n