arXiv subjects

Yuliia Mishura

Publications and source records attributed to Yuliia Mishura.

3 recordsLinked to original sources

arXiv2025

Functional limit theorems for a time-changed multidimensional Wiener process

We study the asymptotic behaviour of a properly normalized time-changed multidimensional Wiener process; the time change is given by an additive functional of the Wiener process itself. At the level of generators, the time change means that we consider the Laplace operator -- which generates a multidimensional Wiener process -- and multiply it by a (possibly degenerate) state-space dependent intensity. We assume that the intensity admits limits at infinity in each octant of the state space, but the values of these limits may be different. Applying a functional limit theorem for the superposition of stochastic processes, we prove functional limit theorems for the normalized time-changed multidimensional Wiener process. Among the possible limits there is a multidimensional analogue of skew Brownian motion.

math.PR↗

arXiv2016

Asymptotic behavior of homogeneous additive functionals of the solutions of Itô stochastic differential equations with nonregular dependence on parameter

We study the asymptotic behavior of mixed functionals of the form $I_T(t)=F_T(ξ_T(t))+\int_0^tg_T(ξ_T(s))\,dξ_T(s)$, $t\ge0$, as $T\to\infty$. Here $ξ_T(t)$ is a strong solution of the stochastic differential equation $dξ_T(t)=a_T(ξ_T(t))\,dt+dW_T(t)$, $T>0$ is a parameter, $a_T=a_T(x)$ are measurable functions such that $\left|a_T(x)\right|\leq C_T$ for all $x\in \mathbb {R}$, $W_T(t)$ are standard Wiener processes, $F_T=F_T(x)$, $x\in \mathbb {R}$, are continuous functions, $g_T=g_T(x)$, $x\in \mathbb {R}$, are locally bounded functions, and everything is real-valued. The explicit form of the limiting processes for $I_T(t)$ is established under very nonregular dependence of $g_T$ and $a_T$ on the parameter $T$.

math.PR↗

arXiv2016

Functional limit theorems for additive and multiplicative schemes in the Cox--Ingersoll--Ross model

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.

math.PR↗