##
**Ruin probability in the Cramér-Lundberg model with risky investments.**
*(English)*
Zbl 1236.91087

Let \(Y\) be a geometric Brownian motion. That is, the solution to the stochastic differential equation \(d Z_t = a Z_t \;d t + \sigma Z_t \;d W_t\), where \(W\) is a standard Brownian motion. By \(P\) we denote an independent compound Poisson process \(P_t = \sum_{k=1}^{N(t)} \xi_k\), where \(N\) is a Poisson process with rate \(\lambda\) and \(\{\xi_k\}\) are iid positive random variables. \(N\), \(\{\xi_k\}\) and \(W\) are assumed to be independent. \(Z\) models the price of a risky asset and \(P\) models the aggregate claim of an insurance portfolio. If all the surplus is invested in the risky asset, the surplus process of the insurance portfolio is
\[
X_t = X_0 Z_t + \int_0^t \frac{Z_t}{Z_s} c_s \;d s - \sum_{k=1}^{N_t} \xi_k \frac{Z_t}{Z_{T_k}}\;,
\]
where \(\{c_s\}\) is the premium rate and \(\{T_k\}\) are the occurrence times of the Poisson process. The premium rate is assumed to be bounded, \(0 \leq c_s \leq c < \infty\) for some \(c > 0\).

The case \(2 a > \sigma^2\) has been considered in several papers, see for instance [A. Frolova, Y. Kabanov and S. Pergamenshchikov, Finance Stoch. 6, No. 2, 227–235 (2002; Zbl 1002.91037); V. Kalashnikov and R. Norberg, Stochastic Processes Appl. 98, No. 2, 211–228 (2002; Zbl 1058.60095); J. Paulsen, Stochastic Processes Appl. 75, No. 1, 135–148 (1998; Zbl 0932.60044); H. Nyrhinen, Stochastic Processes Appl. 92, No. 2, 265–285 (2001; Zbl 1047.60040); S. Pergamenshchikov and O. Zeitouny, Stochastic Processes Appl. 116, No. 2, 267–278 (2006); erratum ibid. 119, No. 1, 305–306 (2009; Zbl 1088.60076)]. They show that, even for light-tailed claim size distributions, the ruin probability decays at a polynomial rate.

In [Pergamenshchikov and Zeitouny, loc.cit.] it was proved, under the assumption that the claim size distribution has unbounded support, that the ruin probability is \(\psi(u) = P[\inf_t X_t < 0\mid X_0 = u] = 1\) if \(2 a \leq \sigma^2\).

The present paper deals with the case of bounded claims, \(P[\xi_k \leq M] = 1\) for some \(M < \infty\). Since \(Z_t = \exp\{(a - \sigma^2/2) t + \sigma W_t\}\), we have that \(\liminf_{t \to \infty} Z_t = 0\). Let \(Y_t = 1/Z_t = \exp\{-(a-\sigma^2/2) t - \sigma W_t\}\), one can observe that ruin for \(X\) occurs if and only if ruin occurs for the process \[ X_0 + \int_0^t Y_s c_s \;d s - \sum_{k=1}^{N_t} \xi_k Y_{T_k}\;. \] It is concluded, that it suffices to show the result for \(c_s = c\), and that \(\psi(u) \leq \psi(v)\) for \(v \leq u\).

It is first shown that \(X\) will leave an interval \([0,n]\) in finite time for any \(n > 0\). This could be done in a simpler way. There is \(\varepsilon > 0\), such that \(P[\xi_k > \varepsilon] > 1/2\). We can assume that \(n > \varepsilon\). Let for \(t \in {\mathbb N}\) \[ A_t = \{N_{t+1}=N_t,\;\varepsilon Z_{t+1}Y_t > 2 n\} \] if \(2 X_t \geq \varepsilon\) and \[ A_t = \{N_{t+1}=N_t+1,\;\sup_{t \leq s \leq t+1} \{c s + (Z_s+\varepsilon) Y_t\} < \varepsilon, \xi_{N_t+1} > \varepsilon\} \] otherwise. Then \(A_t\) is independent of \(\mathcal{F}_t\) and \(P[A_n] \geq \alpha > 0\). By the generalised Borel-Cantelli lemma, there are infinitely many \(t\), where \(A_t\) holds. But then \(X_{t+1} > n\) or \(X_{t+1} < 0\). This gives the result.

Using a martingale argument, it is shown that there is \(u^*\), such that \(P[\inf X_t < u^*] = 1\). Next it is shown that, if \(P[\inf X_t < u_0] = 1\) for some \(u_0\), then also \(P[\inf X_t < u_0-M/2] = 1\). From this, it can be concluded that \(\psi(u) = 1\) for all \(u \geq 0\).

The case \(2 a > \sigma^2\) has been considered in several papers, see for instance [A. Frolova, Y. Kabanov and S. Pergamenshchikov, Finance Stoch. 6, No. 2, 227–235 (2002; Zbl 1002.91037); V. Kalashnikov and R. Norberg, Stochastic Processes Appl. 98, No. 2, 211–228 (2002; Zbl 1058.60095); J. Paulsen, Stochastic Processes Appl. 75, No. 1, 135–148 (1998; Zbl 0932.60044); H. Nyrhinen, Stochastic Processes Appl. 92, No. 2, 265–285 (2001; Zbl 1047.60040); S. Pergamenshchikov and O. Zeitouny, Stochastic Processes Appl. 116, No. 2, 267–278 (2006); erratum ibid. 119, No. 1, 305–306 (2009; Zbl 1088.60076)]. They show that, even for light-tailed claim size distributions, the ruin probability decays at a polynomial rate.

In [Pergamenshchikov and Zeitouny, loc.cit.] it was proved, under the assumption that the claim size distribution has unbounded support, that the ruin probability is \(\psi(u) = P[\inf_t X_t < 0\mid X_0 = u] = 1\) if \(2 a \leq \sigma^2\).

The present paper deals with the case of bounded claims, \(P[\xi_k \leq M] = 1\) for some \(M < \infty\). Since \(Z_t = \exp\{(a - \sigma^2/2) t + \sigma W_t\}\), we have that \(\liminf_{t \to \infty} Z_t = 0\). Let \(Y_t = 1/Z_t = \exp\{-(a-\sigma^2/2) t - \sigma W_t\}\), one can observe that ruin for \(X\) occurs if and only if ruin occurs for the process \[ X_0 + \int_0^t Y_s c_s \;d s - \sum_{k=1}^{N_t} \xi_k Y_{T_k}\;. \] It is concluded, that it suffices to show the result for \(c_s = c\), and that \(\psi(u) \leq \psi(v)\) for \(v \leq u\).

It is first shown that \(X\) will leave an interval \([0,n]\) in finite time for any \(n > 0\). This could be done in a simpler way. There is \(\varepsilon > 0\), such that \(P[\xi_k > \varepsilon] > 1/2\). We can assume that \(n > \varepsilon\). Let for \(t \in {\mathbb N}\) \[ A_t = \{N_{t+1}=N_t,\;\varepsilon Z_{t+1}Y_t > 2 n\} \] if \(2 X_t \geq \varepsilon\) and \[ A_t = \{N_{t+1}=N_t+1,\;\sup_{t \leq s \leq t+1} \{c s + (Z_s+\varepsilon) Y_t\} < \varepsilon, \xi_{N_t+1} > \varepsilon\} \] otherwise. Then \(A_t\) is independent of \(\mathcal{F}_t\) and \(P[A_n] \geq \alpha > 0\). By the generalised Borel-Cantelli lemma, there are infinitely many \(t\), where \(A_t\) holds. But then \(X_{t+1} > n\) or \(X_{t+1} < 0\). This gives the result.

Using a martingale argument, it is shown that there is \(u^*\), such that \(P[\inf X_t < u^*] = 1\). Next it is shown that, if \(P[\inf X_t < u_0] = 1\) for some \(u_0\), then also \(P[\inf X_t < u_0-M/2] = 1\). From this, it can be concluded that \(\psi(u) = 1\) for all \(u \geq 0\).

Reviewer: Hanspeter Schmidli (Köln)

### MSC:

91B30 | Risk theory, insurance (MSC2010) |

60J25 | Continuous-time Markov processes on general state spaces |

### Keywords:

Cramér–Lundberg model; geometric Brownian motion; ruin probability; risky investments; martingale methods
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\textit{S. Xiong} and \textit{W.-S. Yang}, Stochastic Processes Appl. 121, No. 5, 1125--1137 (2011; Zbl 1236.91087)

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### References:

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