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In particular, the methods are applicable to ODEs of Carath\u00e9odory type, whose coefficient functions are only integrable with respect to the time variable but are not assumed to be continuous. A further field of application are ODEs with coefficient functions that contain weak singularities with respect to the time variable. The main result consists of precise bounds for the discretization error with respect to the<jats:inline-formula id=\"j_cmam-2016-0048_ineq_9999_w2aab3b7b1b1b6b1aab1c14b1b1Aa\"><jats:alternatives><m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><m:mrow><m:msup><m:mi>L<\/m:mi><m:mi>p<\/m:mi><\/m:msup><m:mo>\u2062<\/m:mo><m:mrow><m:mo>(<\/m:mo><m:mi>\u03a9<\/m:mi><m:mo>;<\/m:mo><m:msup><m:mi>\u211d<\/m:mi><m:mi>d<\/m:mi><\/m:msup><m:mo>)<\/m:mo><\/m:mrow><\/m:mrow><\/m:math><jats:tex-math>{L^{p}(\\Omega;{\\mathbb{R}}^{d})}<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-norm. In addition, convergence rates are also derived in the almost sure sense. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. The theoretical results are illustrated through a few numerical experiments.<\/jats:p>","DOI":"10.1515\/cmam-2016-0048","type":"journal-article","created":{"date-parts":[[2017,2,16]],"date-time":"2017-02-16T10:01:28Z","timestamp":1487239288000},"page":"479-498","source":"Crossref","is-referenced-by-count":23,"title":["Error Analysis of Randomized Runge\u2013Kutta Methods for Differential Equations with Time-Irregular Coefficients"],"prefix":"10.1515","volume":"17","author":[{"given":"Raphael","family":"Kruse","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Mathematik , Technische Universit\u00e4t Berlin , Sekr. MA 5-3, Stra\u00dfe des 17. 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