Table 2.

Comparison of the estimation of the expected first hitting time and the theoretical upper bounds.

Algorithm	$\overrightarrow{\mathit{z}}$	n	10	40	70	100	130	160	190	220	250	280
BSO-I	$N(0,1)$	$1+\sqrt{2\pi n}\frac{a}{k}-\sqrt{\frac{2\pi }{n}}\frac{\epsilon }{k}$	80.27	159.53	210.72	251.66	286.80	318.07	346.51	372.79	397.33	420.44
		$\widehat{E\left(T_{\epsilon }\right|X_0)}$	79.65	159.57	212.51	252.09	287.14	316.84	345.61	373.66	398.08	418.75
	$U\left(-\frac{1}{2},\frac{1}{2}\right)$	$1+2\sqrt{6\pi n}\frac{a}{k}-\frac{2\sqrt{6\pi }}{\sqrt{n}}\frac{\epsilon }{k}$	275.59	550.17	727.49	869.32	991.04	1099.35	1197.90	1288.93	1373.94	1453.98
		$\widehat{E\left(T_{\epsilon }\right|X_0)}$	275.81	548.63	727.97	868.67	990.57	1102.96	1198.00	1288.83	1373.06	1455.95
	$U(-1,1)$	$1+\sqrt{6\pi n}\frac{a}{k}-\sqrt{\frac{6\pi }{n}}\frac{\epsilon }{k}$	138.29	275.59	364.24	435.16	496.02	550.17	599.45	644.96	687.47	727.49
		$\widehat{E\left(T_{\epsilon }\right|X_0)}$	137.27	275.45	363.95	435.81	498.34	553.30	598.41	641.36	685.46	723.91
BSO-II	$N(0,1)$	$1+\frac{5\sqrt{2\pi n}}{4+\sqrt{2}}\frac{a}{k}-\frac{5\sqrt{2\pi }}{4\sqrt{n}+\sqrt{2n}}\frac{\epsilon }{k}$	74.20	147.40	194.68	232.49	264.93	293.81	320.08	344.35	367.01	388.35
		$\widehat{E\left(T_{\epsilon }\right|X_0)}$	72.97	145.55	193.03	232.77	262.05	294.26	318.51	347.10	365.51	387.87
	$U\left(-\frac{1}{2},\frac{1}{2}\right)$	$1+\frac{10\sqrt{6\pi n}}{4+\sqrt{2}}\frac{a}{k}-\frac{10\sqrt{6\pi }}{4\sqrt{n}+\sqrt{2n}}\frac{\epsilon }{k}$	254.58	508.16	671.91	802.89	915.30	1015.32	1106.33	1190.40	1268.90	1342.82
		$\widehat{E\left(T_{\epsilon }\right|X_0)}$	251.80	505.49	674.53	804.58	915.61	1014.22	1103.21	1190.82	1271.88	1342.12
	$U(-1,1)$	$1+\frac{5\sqrt{6\pi n}}{4+\sqrt{2}}\frac{a}{k}-\frac{5\sqrt{6\pi }}{4\sqrt{n}+\sqrt{2n}}\frac{\epsilon }{k}$	127.79	254.58	336.45	401.95	458.15	508.16	553.67	595.70	634.95	671.91
		$\widehat{E\left(T_{\epsilon }\right|X_0)}$	128.10	253.61	334.30	400.20	458.07	507.69	554.34	594.77	634.45	669.75

The points larger than the theoretical upper bounds are highlighted in boldface.