Modern Stochastics: Theory and Applications

In this paper the fractional Cox–Ingersoll–Ross process on ${\mathbb{R}_{+}}$ for $H<1/2$ is defined as a square of a pointwise limit of the processes ${Y_{\varepsilon }}$, satisfying the SDE of the form $d{Y_{\varepsilon }}(t)=(\frac{k}{{Y_{\varepsilon }}(t){1_{\{{Y_{\varepsilon }}(t)>0\}}}+\varepsilon }-a{Y_{\varepsilon }}(t))dt+\sigma d{B^{H}}(t)$, as $\varepsilon \downarrow 0$. Properties of such limit process are considered. SDE for both the limit process and the fractional Cox–Ingersoll–Ross process are obtained.

In this paper we define the fractional Cox–Ingersoll–Ross process as $X_{t}:={Y_{t}^{2}}\mathbf{1}_{\{t<\inf \{s>0:Y_{s}=0\}\}}$, where the process $Y=\{Y_{t},t\ge 0\}$ satisfies the SDE of the form $dY_{t}=\frac{1}{2}(\frac{k}{Y_{t}}-aY_{t})dt+\frac{\sigma }{2}d{B_{t}^{H}}$, $\{{B_{t}^{H}},t\ge 0\}$ is a fractional Brownian motion with an arbitrary Hurst parameter $H\in (0,1)$. We prove that $X_{t}$ satisfies the stochastic differential equation of the form $dX_{t}=(k-aX_{t})dt+\sigma \sqrt{X_{t}}\circ d{B_{t}^{H}}$, where the integral with respect to fractional Brownian motion is considered as the pathwise Stratonovich integral. We also show that for $k>0$, $H>1/2$ the process is strictly positive and never hits zero, so that actually $X_{t}={Y_{t}^{2}}$. Finally, we prove that in the case of $H<1/2$ the probability of not hitting zero on any fixed finite interval by the fractional Cox–Ingersoll–Ross process tends to 1 as $k\to \infty $.

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 investigate the convergence of hitting times for jump-diffusion processes. Specifically, we study a sequence of stochastic differential equations with jumps. Under reasonable assumptions, we establish the convergence of solutions to the equations and of the moments when the solutions hit certain sets.

We present large sample properties and conditions for asymptotic normality of linear functionals of powers of the periodogram constructed with the use of tapered data.