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Georgiy Shevchenko

Publications and source records attributed to Georgiy Shevchenko.

At least 19 recordsLinked to original sources
arXiv2023

Stochastic selection problem for a Stratonovich SDE with power non-linearity

In our paper [Bernoulli 26(2), 2020, 1381-1409], we found all strong Markov solutions that spend zero time at $0$ of the Stratonovich stochastic differential equation $d X=|X|^{\alpha}\circ dB$, $\alpha\in (0,1)$. These solutions have the form $X_t^\theta=F(B^\theta_t)$, where $F(x)=\frac{1}{1-\alpha}|x|^{1/(1-\alpha)}\text{sign}\, x$ and $B^\theta$ is the skew Brownian motion with skewness parameter $\theta\in [-1,1]$ starting at $F^{-1}(X_0)$. In this paper we show how an addition of small external additive noise $\varepsilon W$ restores uniqueness. In the limit as $\varepsilon\to 0$, we recover heterogeneous diffusion corresponding to the physically symmetric case $\theta=0$.

math.PR↗
arXiv2022

Minimax identity with robust utility functional for a non-concave utility

We study the minimax identity for a non-decreasing upper-semicontinuous utility function satisfying mild growth assumption. In contrast to the classical setting, we do not impose the assumption that the utility function is concave. By considering the concave envelope of the utility function we obtain equalities and inequalities between the robust utility functionals of an initial utility function and its concavification. Furthermore, we prove similar equalities and inequalities in the case of implementing an upper bound on the final endowment of the initial model.

math.PR↗
arXiv2022

Tail Measures and Regular Variation

A general framework for the study of regular variation (RV) is that of Polish star-shaped metric spaces, while recent developments in [1] have discussed RV with respect to some properly localised boundedness $\mathcal{B}$ imposing weak assumptions on the structure of Polish space. Along the lines of the latter approach, we discuss the RV of Borel measures and random processes on general Polish metric spaces. Tail measures introduced in [2] appear naturally as limiting measures of regularly varying time series. We define tail measures on a measurable space indexed by $\mathcal{H}(D)$, a countable family of homogeneous coordinate maps, and show some tractable instances for the investigation of RV when $\mathcal{B}$ is determined by $\mathcal{H}(D)$. This allows us to study the regular variation of cadlag processes on $D(R^l, R^d)$ retrieving in particular results obtained in [1] for RV of stationary cadlag processes on the real line removing $l=1$ therein. Further, we discuss potential applications and open questions.

math.PR↗
arXiv2022

The harmonic mean formula for random processes

Motivated by the harmonic mean formula in [1], we investigate the relation between the sojourn time and supremum of a random process $X(t),t\in \mathbb{R}^d$ and extend the harmonic mean formula for general stochastically continuous $X$. We discuss two applications concerning the continuity of distribution of supremum of $X$ and representations of classical Pickands constants.

math.PR↗
arXiv2020

Gaussian processes with Volterra kernels

We study Volterra processes $X_t = \int_0^t K(t,s) dW_s$, where $W$ is a standard Wiener process, and the kernel has the form $K(t,s) = a(s) \int_s^t b(u) c(u-s) du$. This form generalizes the Volterra kernel for fractional Brownian motion (fBm) with Hurst index $H>1/2$. We establish smoothness properties of $X$, including continuity and Holder property. It happens that its Holder smoothness is close to well-known Holder smoothness of fBm but is a bit worse. We give a comparison with fBm for any smoothness theorem. Then we investigate the problem of inverse representation of $W$ via $X$ in the case where $c\in L^1[0,T]$ creates a Sonine pair, i.e. there exists $h\in L^1[0,T]$ such that $c * h = 1$. It is a natural extension of the respective property of fBm that generates the same filtration with the underlying Wiener process. Since the inverse representation of the Gaussian processes under consideration are based on the properties of Sonine pairs, we provide several examples of Sonine pairs, both well-known and new. Key words: Gaussian process, Volterra process, Sonine pair, continuity, Holder property, inverse representation.

math.PR↗
arXiv2020

Boundary non-crossing probabilities of Gaussian processes: sharp bounds and asymptotics

We study boundary non-crossing probabilities $$ P_{f,u} := \mathrm P\big(\forall t\in \mathbb T\ X_t + f(t)\le u(t)\big) $$ for continuous centered Gaussian process $X$ indexed by some arbitrary compact separable metric space $\mathbb T$. We obtain both upper and lower bounds for $P_{f,u}$. The bounds are matching in the sense that they lead to precise logarithmic asymptotics for the large-drift case $P_{y f,u}$, $y \to+\infty$, which are two-term approximations (up to $o(y)$). The asymptotics are formulated in terms of the solution $\tilde f$ to the constrained optimization problem $$ \|h\|_{\mathbb H_X}\to \min, \quad h\in \mathbb H_X, h\ge f $$ in the reproducing kernel Hilbert space $\mathbb H_X$ of $X$. Several applications of the results are further presented.

math.PR↗
arXiv2020

Existence and uniqueness of mild solution to fractional stochastic heat equation

For a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset \mathbb {R}^d$ and driven by an $L^2(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$, a new result on existence and uniqueness of a mild solution is established. Compared to the existing results, the uniqueness in a fully nonlinear case is shown, not assuming the coefficient in front of the noise to be affine. Additionally, the existence of moments for the solution is established.

math.PR↗
arXiv2019

Stratonovich SDE with irregular coefficients: Girsanov's example revisited

In this paper we study the Stratonovich stochastic differential equation $\mathrm{d} X=|X|^α\circ\mathrm{d} B$, $α\in(-1,1)$, which has been introduced by Cherstvy et al. [New Journal of Physics 15:083039 (2013)] in the context of analysis of anomalous diffusions in heterogeneous media. We determine its weak and strong solutions, which are homogeneous strong Markov processes \chng{spending zero time at $0$: for $α\in (0,1)$, these solutions have the form $$ X_t^θ=\bigl((1-α)B_t^θ\bigr)^{1/(1-α)}, $$ where $B^θ$ is the $θ$-skew Brownian motion driven by $B$ and starting at $\frac{1}{1-α}(X_0)^{1-α}$, $θ\in [-1,1]$,} and $(x)^γ=|x|^γ\operatorname{sign} x$; for $α\in(-1,0]$, only the case $θ=0$ is possible. The central part of the paper consists in the proof of the existence of a quadratic covariation $[f(B^θ),B]$ for a locally square integrable function $f$ and is based on the time-reversion technique for Markovian diffusions.

math.PR↗
arXiv2019

Nonparametric estimation of the kernel function of symmetric stable moving average random functions

We estimate the kernel function of a symmetric alpha stable ($SαS$) moving average random function which is observed on a regular grid of points. The proposed estimator relies on the empirical normalized (smoothed) periodogram. It is shown to be weakly consistent for positive definite kernel functions, when the grid mesh size tends to zero and at the same time the observation horizon tends to infinity (high frequency observations). A simulation study shows that the estimator performs well at finite sample sizes, when the integrator measure of the moving average random function is $SαS$ and for some other infinitely divisible integrators.

math.ST↗
arXiv2019

Limit theorems for additive functionals of continuous time random walks

For a continuous-time random walk $X=\{X_t,t\ge 0\}$ (in general non-Markov), we study the asymptotic behavior, as $t\rightarrow \infty$, of the normalized additive functional $c_t\int_0^{t} f(X_s)ds$, $t\ge 0$. Similarly to the Markov situation, assuming that the distribution of jumps of $X$ belongs to the domain of attraction to $α$-stable law with $α>1$, we establish the convergence to the local time at zero of an $α$-stable Lévy motion. We further study a situation where $X$ is delayed by a random environment given by the Poisson shot-noise potential: $Λ(x,γ)= e^{-\sum_{y\in γ} ϕ(x-y)},$ where $ϕ\colon\mathbb R\to [0,\infty)$ is a bounded function decaying sufficiently fast, and $γ$ is a homogeneous Poisson point process, independent of $X$. We find that in this case the weak limit has both "quenched" component depending on $Λ$, and a component, where $Λ$ is "averaged".

math.PR↗
arXiv2018

Replication of Wiener-transformable stochastic processes with application to financial markets with memory

We investigate Wiener-transformable markets, where the driving process is given by an adapted transformation of a Wiener process. This includes processes with long memory, like fractional Brownian motion and related processes, and, in general, Gaussian processes satisfying certain regularity conditions on their covariance functions. Our choice of markets is motivated by the well-known phenomena of the so-called `constant' and `variable depth' memory observed in real world price processes, for which fractional and multifractional models are the most adequate descriptions. Motivated by integral representation results in general Gaussian setting, we study the conditions under which random variables can be represented as pathwise integrals with respect to the driving process. From financial point of view, it means that we give the conditions of replication of contingent claims on such markets. As an application of our results, we consider the utility maximization problem in our specific setting. Note that the markets under consideration can be both arbitrage and arbitrage-free, and moreover, we give the representation results in terms of bounded strategies.

math.PR↗
arXiv2018

Existence and uniqueness of mild solution to stochastic heat equation with white and fractional noises

We prove the existence and uniqueness of a mild solution for a class of non-autonomous parabolic mixed stochastic partial differential equations defined on a bounded open subset $D \subset \mathbb{R}^d$ and involving standard and fractional $L^2(D)$-valued Brownian motions. We assume that the coefficients are homogeneous, Lipschitz continuous and the coefficient at the fractional Brownian motion is an affine function.

math.PR↗
arXiv2017

Wave equation with a coloured stable noise

We define a random measure generated by a real anisotropic harmonizable fractional stable field $Z^H$ with stability parameter $α\in(1,2)$ and Hurst index $H\in(1/2,1)$ and prove that the measure is $σ$-additive in probability. An integral with respect to this measure is constructed, which enables us to consider a wave equation in $\mathbb R^3$ with a random source generated by $Z^H$. We show that the solution to this equation, given by Kirchhoff's formula, has a modification, which is Hölder continuous of any order up to $(3H-1)\wedge 1$. In the case where $H\in(2/3,1)$, we show further that the modification is absolutely continuous.

math.PR↗
arXiv2016

Stochastic wave equation in a plane driven by spatial stable noise

The main object of this paper is the planar wave equation \[\bigg(\frac{\partial^2}{\partial t^2}-a^2\varDelta\bigg)U(x,t)=f(x,t),\quad t\ge0, x\in \mathbb {R}^2,\] with random source $f$. The latter is, in certain sense, a symmetric $α$-stable spatial white noise multiplied by some regular function $σ$. We define a candidate solution $U$ to the equation via Poisson's formula and prove that the corresponding expression is well defined at each point almost surely, although the exceptional set may depend on the particular point $(x,t)$. We further show that $U$ is Hölder continuous in time but with probability 1 is unbounded in any neighborhood of each point where $σ$ does not vanish. Finally, we prove that $U$ is a generalized solution to the equation.

math.PR↗
arXiv2016

Approximations for a solution to stochastic heat equation with stable noise

We consider a Cauchy problem for stochastic heat equation driven by a real harmonizable fractional stable process $Z$ with Hurst parameter $H>1/2$ and stability index $α>1$. It is shown that the approximations for its solution, which are defined by truncating the LePage series for $Z$, converge to the solution.

math.PR↗
arXiv2016

Fractionally integrated inverse stable subordinators

A fractionally integrated inverse stable subordinator (FIISS) is the convolution of a power function and an inverse stable subordinator. We show that the FIISS is a scaling limit in the Skorokhod space of a renewal shot noise process with heavy-tailed, infinite mean `inter-shot' distribution and regularly varying response function. We prove local Hölder continuity of FIISS and a law of iterated logarithm for both small and large times.

math.PR↗
arXiv2015

Convergence of hitting times for jump-diffusion processes

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.

math.PR↗