Sample Path Analysis and Distributions of Boundary Crossing Times by Shelemyahu Zacks

Author:Shelemyahu Zacks
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(4.94)

and

(4.95)

In addition let T S = min{T L , T U }. We wish to find the distribution of T S and of T L ∗ = T S I{T L = T S } and that of T U ∗ = T S I{T U = T S }. Let g(y; t, β 1, β 2) be the defective density . It is clear that

(4.96)

We have to derive the formula for g(y; t, β 1, β 2). Stadje and Zacks (2003) derived the restricted density

(4.97)

Let β = β 1 + β 2 and

(4.98)

Notice that Y (t) cannot cross the lower boundary after crossing the upper boundary within the intervals (n β, (n + 1)β). Define g 0(y; t, β 1, β 2) = I{0 < t ≤ β}g(y; t, β 1, β 2). This function is given in the following lemma.

Lemma 4.2.

0 < t ≤ β

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