An estimate for the ruin probability in a model with variable premiums and with investments in a bond and several stocks. (Ukrainian, English) Zbl 1194.91090

Androshchuk, M. O.

Teor. Jmovirn. Mat. Stat. 76, 1-13 (2007); translation in Theory Probab. Math. Stat. 76, 1-13 (2008).

The author deals with the risk process generalizing the classical Cramér-Lundberg process. The risk process is described by the equation \[ R_t(u,\overline{K})=u-\sum_{k=1}^{N_t}U_k+\int_0^tp(R_s)ds+ \sum_{i=1}^{n}\int_0^t\frac{K_s^i}{S_s^i}dS_s^i+ \int_0^t\frac{R_s-\sum_{i=1}^{n}K_s^i}{e^{\delta s}}de^{\delta s},t\geq0, \] where \(u>0\) is the initial capital of an insurance company, \(\{N_t,t\geq0\}\) is the Poisson process with a constant intensity \(\beta>0\) which models the flow of insurance claims, \(U_k,k\geq1\) is a sequence of nonnegative identically distributed random variables (the insurance payments according to claims), \(p(R_s)\) is the process of the insurance premiums (price function) depending on the current reserve of the company at a moment \(s\), \(K_s\) is the amount of money invested by the insurer in a stock \(i,i=1,\dots,n\), the rest of the money is invested in a riskless bond (deposited in a bank account) with a constant rate \(\delta\). The evolution of the price of stocks is described by the geometrical Brownian motions \[ dS_t^i=S_t^i(a_idt+b_tdW_t^i), \] where \(a_i>0\) and \(b_i\) are fixed constants and where \(W_t^i,t\geq0\) are standard Brownian motions being independent of the compound Poisson process and such that \(\text{Cov}(W_t^i,W_t^j)=\rho_{ij}t,i=1,\dots,n,j=1,\dots,n\). The feature of the process is that its price function depends on the current reserve of an insurance company as well as on its portfolio consisting of a riskless bond and a finite number of risky assets, modelled by geometric Brownian motions. An analogue of the classical exponential estimate for the ruin probability is obtained in this case. It turns out that the estimate for the model with investments is better than the corresponding estimate for the classical model without investments.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

91B30	Risk theory, insurance (MSC2010)
60G44	Martingales with continuous parameter
62P05	Applications of statistics to actuarial sciences and financial mathematics
91G50	Corporate finance (dividends, real options, etc.)

Keywords:

semimartingale; local martingale; ruin probability; investment strategy