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2 2015

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Ruin probability in a risk model with variable premium intensity and risky investments

Mishura Y., Perestyuk M., Ragulina O.

Opuscula Mathematica

2015

Vol. 35, no. 3

333--352

We consider a generalization of the classical risk model when the premium intensity depends on the current surplus of an insurance company. All surplus is invested in the risky asset, the price of which follows a geometric Brownian motion. We get an exponential bound for the infinite-horizon ruin probability. To this end, we allow the surplus process to explode and investigate the question concerning the probability of explosion of the surplus process between claim arrivals.

Diffusion approximation of recurrent schemes for financial markets, with application to the Ornstein-Uhlenbeck process

Mishura Y.

Opuscula Mathematica

2015

Vol. 35, no. 1

99--116

We adapt the general conditions of the weak convergence for the sequence of processes with discrete time to the diffusion process towards the weak convergence for the discrete-time models of a financial market to the continuous-time diffusion model. These results generalize a classical scheme of the weak convergence for discrete-time markets to the Black-Scholes model. We give an explicit and direct method of approximation by a recurrent scheme. As an example, an Ornstein-Uhlenbeck process is considered as a limit model.