European call option issued on a bond governed by a modified geometric Ornstein-Uhlenbeck process, is investigated. Objective price of such option as a function of the mean and the variance of a geometric Ornstein-Uhlenbeck process is studied. It is proved that the "Ornstein-Uhlenbeck" market is arbitrage-free and complete. We obtain risk-neutral measure and calculate the fair price of a call option. We consider also the bond price, governed by a modified fractional geometric Ornstein-Uhlenbeck process with Hurst index ∈ (1/2, 1). Limit behaviour of the variance of the process as → 1/2 and → 1 is studied, the monotonicity of the variance and the objective price of the option as a function of Hurst index is established.

Abstract. The existence and uniqueness of a solution of a stochastic differential equation with a non-Lipschitz diffusion for cases of both centered and non-centered Poisson measures is proved. We prove that the pathwise uniqueness of a solution and the existence of a weak solution imply the existence of a strong solution for such equations.

Abstract. The existence and uniqueness of a solution of a stochastic differential equation with random coefficients, non-Lipschitzian diffusion, and with centered as well as with non-centered Poisson measures are proved. We estimate the probability that a solution eventually becomes negative. We find conditions for the existence of a nonnegative solution.

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In the paper we investigate the dynamics of a life insurance company depending on changes in macro indicators. For the sensitivity study, profitability testing is performed for the average life insurance contract. Profitability testing is based on an estimate of the expected present value of future cash flows. Various approaches to the choice of discount rates are investigated. The choice of strategy for using the rates of the zero-coupon yield curve built on the UAH domestic government bonds is substantiated. The dependence of the zero-coupon yield curve on the change in the key rate is shown. The dynamics of the model at different levels of the interest rates, namely guaranteed investment income, average profitability of investment activity, key rate, is considered. The scenario of the predicted decrease in the key rate at the stage of post-war reconstruction of Ukraine is considered. The profitability of the insurance company was tested based on the macro parameters of the specified scenarios. The expediency of using the chosen approach for researching the dynamics of a life insurance company in the conditions of the current high volatility of the Ukrainian economy is shown.