Internal truncated distributions: applications to Wiener process range distribution when deleting a minimum stochastic volatility interval from its domain

Figures & data

Figure 1. Probability density function of $\tilde{R}\left(T\right)$.

Figure 2. Distribution function of $\tilde{R}\left(T\right)$.

Figure 3. Probability density function of R(T).

Figure 4. Cumulative distribution function of $R\left(T\right)$.

Figure 5. Survival function of R(T).

Figure 6. Hazard rate function of R(T).

Figure 7. Reversed hazard rate function of R(T).

Table 1. The cumulative distribution function and the mean values of $R\left(T\right)$.

			r		$G_{R\left(T\right)}\left(r\right)$
$T_v$	$\alpha$	$\beta$	$0<r<\alpha$	$\beta <r<\mathrm{\infty }$	$0<r<\alpha$	$\beta <r<\mathrm{\infty }$	$M_1\left(T\right)$
50	1	2	0.0001	2.0001	0	1.05967940010^−27	5.82394824710^110
			0.2	5	5.87305581910^−2673	0.0008782816312
			0.4	10	2.18921372410^−666	0.4090517634
			0.6	15	4.86469350710^−295	0.8658348072
			0.8	20	3.40404482510^−165	0.9823686576
			0.85	25	3.79149608610^−146	0.9993962512
75	5	10	0.0001	10.0001	0	0.00002271565807	5.97742326610^11
			1	15	1.92833638610^−158	0.6059423057
			2	20	1.30401169210^−38	0.9007829441
			3	25	1.16932798910^−16	0.9825565074
			4	30	4.24400555210^−9	0.9986745990
			4.5	35	4.30725204110^−7	1
150	50	60	0.0001	60.0001	0	1	4.25986873510^16
			10	65	0.007873975506	1
			20	70	0.6001910845	1
			30	75	0.9441013896	1
			40	80	0.9968448720	1
			45	85	1	1