Modern Stochastics: Theory and Applications

We construct an estimator of the unknown drift parameter $\theta \in \mathbb{R}$ in the linear model where ${B}^{H_{1}}$ and ${B}^{H_{2}}$ are two independent fractional Brownian motions with Hurst indices $H_{1}$ and $H_{2}$ satisfying the condition $\frac{1}{2}\le H_{1}<H_{2}<1$. Actually, we reduce the problem to the solution of the integral Fredholm equation of the 2nd kind with a specific weakly singular kernel depending on two power exponents. It is proved that the kernel can be presented as the product of a bounded continuous multiplier and weak singular one, and this representation allows us to prove the compactness of the corresponding integral operator. This, in turn, allows us to establish an existence–uniqueness result for the sequence of the equations on the increasing intervals, to construct accordingly a sequence of statistical estimators, and to establish asymptotic consistency.

We establish the rate of convergence of distributions of sums of independent identically distributed random variables to the Gaussian distribution in terms of truncated pseudomoments by implementing the idea of Yu. Studnyev for getting estimates of the rate of convergence of the order higher than ${n}^{-1/2}$.

We deal with a generalization of the classical risk model when an insurance company gets additional funds whenever a claim arrives and consider some practical approaches to the estimation of the ruin probability. In particular, we get an upper exponential bound and construct an analogue to the De Vylder approximation for the ruin probability. We compare results of these approaches with statistical estimates obtained by the Monte Carlo method for selected distributions of claim sizes and additional funds.