Let us define
The ruin probability of the process of (17.10) is given by (Michna; 1998):
The next result enables us to approximate the ruin probability of
the process for a sufficiently large initial
capital. For every
:
Now, let us consider the infinite time ruin probability. The lower and upper bounds for the ruin probability are given by:
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(17.14) |
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(17.15) |
The next property will show the asymptotic behavior of the infinite
time ruin probability. Let the Hurst parameter satisfy . Then
(Hüsler and Piterbarg; 1999):
P{T(R_H)<&infin#infty;}=
P_H&pi#pi;c^1-HH^H-32
u^(1-H)(1H-1)2^12H-
12(1-H)^H+1H-32&lambda#lambda;^1-H
&sigma#sigma;^1H-1
·
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(17.16) |
For an arbitrary value of the initial capital there exists a simulation method of the infinite time ruin probability based on the Girsanov-type theorem. To present this method we introduce the stopping time
P{T(R_H)<&infin#infty;}=
Let us note that putting in (17.22) we obtain a simple formula
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. | |||
25 | 50 | 2 | 8 | . | 1257e-2 | 0 | . | 28307 |
25 | 60 | 2 | 1 | . | 3516e-2 | 0 | . | 03932 |
30 | 60 | 2 | 6 | . | 6638e-3 | 0 | . | 02685 |
35 | 60 | 2 | 3 | . | 6826e-3 | 0 | . | 01889 |
40 | 60 | 2 | 2 | . | 2994e-3 | 0 | . | 01363 |
40 | 70 | 3 | 1 | . | 0363e-1 | 0 | . | 38016 |
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. | |||
25 | 50 | 2 | 0 | . | 22240 | 0 | . | 40728 |
25 | 60 | 2 | 0 | . | 09890 | 0 | . | 08029 |
30 | 60 | 2 | 0 | . | 06570 | 0 | . | 06583 |
35 | 60 | 2 | 0 | . | 04496 | 0 | . | 05471 |
40 | 60 | 2 | 0 | . | 03183 | 0 | . | 04646 |
40 | 70 | 3 | 0 | . | 23622 | 0 | . | 55505 |