Modern Stochastics: Theory and Applications

In this paper, we consider the Cox–Ingersoll–Ross (CIR) process in the regime where the process does not hit zero. We construct additive and multiplicative discrete approximation schemes for the price of asset that is modeled by the CIR process and geometric CIR process. In order to construct these schemes, we take the Euler approximations of the CIR process itself but replace the increments of the Wiener process with iid bounded vanishing symmetric random variables. We introduce a “truncated” CIR process and apply it to prove the weak convergence of asset prices. We establish the fact that this “truncated” process does not hit zero under the same condition considered for the original nontruncated process.

We consider the two-line fitting problem. True points lie on two straight lines and are observed with Gaussian perturbations. For each observed point, it is not known on which line the corresponding true point lies. The parameters of the lines are estimated.

We consider a multivariate functional measurement error model $AX\approx B$. The errors in $[A,B]$ are uncorrelated, row-wise independent, and have equal (unknown) variances. We study the total least squares estimator of X, which, in the case of normal errors, coincides with the maximum likelihood one. We give conditions for asymptotic normality of the estimator when the number of rows in A is increasing. Under mild assumptions, the covariance structure of the limit Gaussian random matrix is nonsingular. For normal errors, the results can be used to construct an asymptotic confidence interval for a linear functional of X.

We consider the problem of optimal estimation of the linear functional $A_{N}\xi ={\sum _{k=0}^{N}}a(k)\xi (k)$ depending on the unknown values of a stochastic sequence $\xi (m)$ with stationary increments from observations of the sequence $\xi (m)+\eta (m)$ at points of the set $\mathbb{Z}\setminus \{0,1,2,\dots ,N\}$, where $\eta (m)$ is a stationary sequence uncorrelated with $\xi (m)$. We propose formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional in the case of spectral certainty, where spectral densities of the sequences are exactly known. We also consider the problem for a class of cointegrated sequences. We propose relations that determine the least favorable spectral densities and the minimax spectral characteristics in the case of spectral uncertainty, where spectral densities are not exactly known while a set of admissible spectral densities is specified.

Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables (not necessarily identically distributed), and η be a counting random variable independent of this sequence. We obtain sufficient conditions on $\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution function of the random sum $S_{\eta }=\xi _{1}+\xi _{2}+\cdots +\xi _{\eta }$ belongs to the class of $\mathcal{O}$-exponential distributions.

We prove a large deviation type estimate for the asymptotic behavior of a weighted local time of $\varepsilon W$ as $\varepsilon \to 0$.