Комбинаторные методы в теории случайных процессов - Такач Л.
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а с - положительная константа.
7. Найти функцию W (х), определенную формулой (10), где
1 - ае~х при х ^0,
(1 - а) ех при х < 0 и 0 < а < 1.
8. Положим
Н(х) | " ч ,
i/i - * />-*
(t, s) - J е sxdxW (t, x)
Q.
0
при Re (s) ^ 0, где функция W (t, x) задана соотношением (54) и
E {e~s^ = ecsu~^u [1_4> Wl
для Re (s) = 0, где ф (s) - преобразование Лапласа - Стильтьеса функции Н
(х). Доказать, что
Литература
171
для 0 < w < оо, где
A (s, w) = exp | - J e~sxdxM (x, w)
l о
и
оо ОО с
Af (jc, w) = ^ -^y- J е~^+а,)иип~1 [1 cu)] du = J - P{?
(u) > xj du
n= 1 0 0
при *^0. (См. Крамер [14].)
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172
Гл. 7. Процессы разорения
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