## 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 $$u \geq 0$$.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 60J25 Continuous-time Markov processes on general state spaces
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### References:

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