17.3 Ruin Probability in the Limit Risk Model of Good and Bad Periods

Let us define

$\displaystyle R_H(t)=u+ct-\lambda^H B_H(t),$ (17.10)

where $ u$, $ c$, and $ \lambda $ are positive constants and the ruin time:

$\displaystyle T(R_H)=\inf\{t> 0 : R_H(t)< 0\},$ (17.11)

if the set is non-empty and $ T(R_H)=\infty$ otherwise.

The ruin probability of the process of (17.10) is given by (Michna; 1998):

$\displaystyle \textrm{P}\{T(R_H)\leq t\}\leq 1-\Phi\left\{\frac{u+ct}{\sigma(\l...
...H}}\right\} \left[1-\Phi\left\{\frac{u-ct}{\sigma(\lambda t)^H}\right\}\right],$ (17.12)

where the functional $ T$ is given in (17.11) and $ \sigma^2=E\{B_H^2(1)\}$.

The next result enables us to approximate the ruin probability of the process $ R_H(t)$ for a sufficiently large initial capital. For every $ t>0$:

$\displaystyle \lim_{u\rightarrow\infty}\frac{P\{T(R_H)\leq t\}}{P\{\lambda^H B_H(t)>u+ct\}}=1,$ (17.13)

where the functional $ T$ is given in (17.11).

Now, let us consider the infinite time ruin probability. The lower and upper bounds for the ruin probability are given by:

$\displaystyle \textrm{P}\{T(R_H)<\infty\}\geq 1-\Phi\left\{\frac{u^{1-H}c^H} {\sigma(\lambda H)^H(1-H)^{1-H}}\right\},$ (17.14)

and

$\displaystyle \textrm{P}\{T(R_H)<\infty\}\leq\frac{2c}{\sqrt{8\pi}(1-H)}\int_0^...
...{-\frac{1}{2}\lambda^{-2H}\sigma^{-2}(ux^{-\frac{H}{1-H}}+cx)^2\right\}\,dx \,.$ (17.15)

See Norros (1994) for the lower bound and Debicki, Michna, and Rolski (1998) for the upper bound analysis.

The next property will show the asymptotic behavior of the infinite time ruin probability. Let the Hurst parameter satisfy $ 0<H<1$. 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
·

$\displaystyle \hspace{28mm} \cdot\left[1-\Phi\left\{\left(\frac{1-H}{H}\right)^{H}\frac{u^{1-H}c^H}{(1-H)\lambda^{H}\sigma}\right\}\right] \{1+o(1)\},$ (17.16)

as $ u\rightarrow \infty$ where $ P_H$ is the Pickands constant, Piterbarg (1996). The value of the Pickands constant is known only for $ H=0.5$ and $ H=1$. Some approximations of its value can be found in Burnecki and Michna (2002) and Debicki, Michna, and Rolski (2003). The above result permits to approximate the infinite time ruin probability in the model of good and bad periods for large values of the initial capital.

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

$\displaystyle \tau_a(u)=\inf\{t>0: B_H(t)+at>u\}\,,$ (17.17)

where $ a\geq 0$ and the function

$\displaystyle w(t,s) = \left\{\begin{array}{ll} c_1s^{\frac{1}{2}-H}{(t-s)}^{\frac{1}{2}-H} & s\in (0,t)\\ 0, & s\not\in (0,t), \end{array} \right.$ (17.18)

where $ \frac{1}{2}<H<1$,

$\displaystyle c_1={\left\{H(2H-1){{\mathcal{B}}}\left(\frac{3}{2}-H, H-\frac{1}{2}\right)\right\}}^{-1}\,,$ (17.19)

and $ {{\mathcal{B}}}$ denotes the beta function. Note that $ \tau_a<\infty$ almost surely for $ a\geq 0$. According to Norros, Valkeila, and Virtamo (1999) the following centered Gaussian process

$\displaystyle M(t)=\int_0^t w(t,s)~dB_H(s),$ (17.20)

possesses independent increments and its variance is

$\displaystyle \textrm{E}M^2(t)=c_2^2t^{2-2H}\, ,$ (17.21)

where

$\displaystyle c_2={\left\{H(2H-1)(2-2H){\mathcal{B}}\left(H-\frac{1}{2}, 2-2H\right)\right\}}
^{-\frac{1}{2}}.
$

In particular $ M(t)$ is a martingale. For all $ a>0$ we have


P{T(R_H)<&infin#infty;}=

$\displaystyle \textrm{E}\exp\left\{-\frac{c+a}{\lambda^H\sigma}\int_0^{\tau_{a}...
...)~dB_H(s)-\frac{1}{2\lambda^{2H}\sigma^2}c_2^2(c+a)^2
{\tau_a}^{2-2H}\right\}.
$

The above formula enables us to simulate the infinite time ruin probability for an arbitrary value of the initial capital. Using the structure of the common distribution of $ (M(t), B_H(t))$ we get the following estimator of the ruin probability valid for $ 0<H<1$:

$\displaystyle \textrm{P}\{T(R_H)<\infty\}=\textrm{E}\exp\left\{-\frac{(c+a)}{\l...
...u^{1-2H}_{a}u +\frac{(a^2-c^2)}{2\lambda^{2H}\sigma^2} {\tau_a}^{2-2H}\right\}.$ (17.22)

Let us note that putting $ a=c$ in (17.22) we obtain a simple formula

$\displaystyle \textrm{P}\{T(R_H)<\infty\}= \textrm{E}\exp\left\{-\frac{2c\tau^{1-2H}_{c}u}{\lambda^{2H}\sigma^2}\right\}.$ (17.23)

For similar methods of simulation based on the change of measure technique applied to fluid models see Debicki, Michna, and Rolski (2003).


Table 17.1: Ruin probabilities for $ H=0.7$ and fixed $ \mu =20, \sigma =10$, and $ t=10$.
$ u$ $ {c}$ $ \lambda $ $ \Psi(t)$ $ \Psi$ . 
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
29187 STFgood01.xpl


Table 17.2: Ruin probabilities for $ H=0.8$ and fixed $ \mu =20, \sigma =10$, and $ t=10$.
$ u$ $ {c}$ $ \lambda $ $ \Psi(t)$ $ \Psi$ . 
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
29193 STFgood02.xpl