We give explicit necessary and sufficient conditions for the viability of polyhedrons with respect to Itô equations. Using the viability criterion we obtain a comparison theorem for multi-dimensional Itô processes.

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We give necessary and sufficient conditions for the invariance property of closed subsets of IRm by It6 and Stratonovich equations. The conditions are expressed in terms of properly defined contingent cones and in a form suitable for applications.

SummaryIn this paper, we propose some derivative designed for small stock investors. Using the Black-Scholes model we derive an explicit formula for the price of the derivative, computing its discounted expected payoff. The payoff is modelled on the payoff of the catastrophe bonds, random occurrence of a natural disaster is replaced by a random stock price falling. Different variants of the proposed derivative are obtained by introducing a parameter to the payoff of the derivative. By Monte Carlo method, to reduce the risk of large losses associated with the investment, indicated the variant of this instrument, appropriate to selected typical values of volatility of considered stock .