# Georgiy ShevchenkoKyiv School of Economics

Georgiy Shevchenko

Ph.D.

## About

129

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Introduction

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July 2016 - August 2022

January 2009 - June 2021

## Publications

Publications (129)

For a continuous-time lattice random walk $X^\Lambda=\set{X^\Lambda_t,t\ge 0}$ in a random environment $\Lambda$, we study the asymptotic behavior, as $t\rightarrow \infty$, of the normalized additive functional $c_t\int_0^{t} f(X^\Lambda_s)ds$, $t\ge 0$. We establish a limit theorem for it, which is similar to that in the non-lattice case, under l...

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^\thet...

We study Volterra processes Xt=∫0tK(t,s)dWs, where W is a standard Wiener process, and the kernel has the form K(t,s)=a(s)∫stb(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 Hölder property. It happens that its Hölde...

The minimax identity for a nondecreasing upper-semicontinuous utility function satisfying mild growth assumption is studied. In contrast to the classical setting, concavity of the utility function is not asumed. By considering the concave envelope of the utility function, equalities and inequalities between the robust utility functionals of an init...

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 u...

Motivated by the classical harmonic mean formula, estabished by Aldous in 1989, we investigate the relation between the sojourn time and supremum of a random process X(t),t∈Rd 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 repre...

We study the standard utility maximization problem 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, or concavification, of the utility function, we identify the...

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...

We consider a fractionally integrated Bessel process defined by Y s δ , H = ∫ 0 ∞ ( u H − ( 1 / 2 ) − ( u − s ) + H − ( 1 / 2 ) ) d X u δ , where X δ is the Bessel process of dimension δ > 2. We discuss the relation of this process to the fractional Brownian motion at its maximum, study the basic properties of the process and prove its Hölder conti...

We study boundary non-crossing probabilities $$\begin{aligned} P_{f,u} := \mathrm {P}\big (\forall t\in {\mathbb {T}}\ X_t + f(t)\le u(t)\big ) \end{aligned}$$for a 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...

A general framework for the study of regular variation is that of Polish star-shaped metric spaces, while recent developments in [1] have discussed regular variation in relation to a boundedness and weaker assumptions are imposed therein on the structure of Polish space. Along the lines of the latter approach, we discuss the regular variation of me...

We consider a version of the secretary problem where elements may vanish during the selection and become unchoosable. We construct a selection strategy and identify the probability to select the best element, which turns out to be asymptotically maximal as number of elements increases indefinitely. As an auxiliary result of independent interest we...

For a continuous-time random walk X = {Xt, t ⩾ 0} (in general non-Markov), we study the asymptotic behaviour, as t → ∞, of the normalized additive functional $c_t\int _0^{t} f(X_s)\,{\rm d}s$ , t⩾ 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 est...

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 pro...

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 $\alpha$-stable l...

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 arbitrary compact separable metrizable space $\mathbb T$. We obtain upper and lower bounds for the probabilities, which are matching in the sense that they lead to preci...

In this paper we determine all solutions of the Stratonovich SDE $dX=|X|^{\alpha}\circ d B$, $\alpha\in(-1,1)$, which are strong Markov processes spending zero time in 0. We show that for $\alpha\in (0,1)$ these solutions have the form $X_t^\theta=|(1-\alpha)B^\theta+|X_0|^{1-\alpha}\operatorname{sign} X_0|^{1/(1-\alpha)}\operatorname{sign} ((1-\al...

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 m...

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...

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 $H>1/2$, we establish a new result on existence and uniqueness of a mild solution. Compared to the existing results, we show uniq...

We study a stochastic differential equation, the diffusion coefficient of which is a function of some adapted stochastic process. The various conditions for the existence and uniqueness of weak and strong solutions are presented. The drift parameter estimation in this model is investigated, and the strong consistency of the least squares and maximu...

The three-dimensional wave equation is studied in the paper. The right hand side of the equation has a symmetric α-stable distribution. Two cases are considered, namely the cases where the perturbation is a (1) “white noise” and (2) “col-ored noise”. It is proved for both cases that a candidate for a solution (a function represented by the Kirchhof...

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 m...

This paper deals with stochastic differential heat equation which is the typical example of stochastic partial differential equations (SPDE). In particular, paper is devoted to the estimation of diffusion parameter $\sigma$ for the random field which is the solution of stochastic differential heat equation for R^d, d = 1, 2, 3. The estimtion of dif...

The Cramer–Lundberg model is considered as a model of insurance company. Since it is impossible to obtain an explicit solution for the non-ruin probability function of insurance company for an arbitrary distribution of the values of insurance claims, the authors consider the problem of estimating the convergence of the original non-ruin probability...

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 t...

This chapter considers the problem of the construction of an optimal filter in the linear two-dimensional partially observed Gaussian model and reduced it to the solution of two equations, one of them being a Riccati differential equation and the other one being a linear stochastic differential equation. For technical simplicity, the chapter also c...

This chapter demonstrates the construction of an Ito integral for random integrands and investigates its properties. In order to define the class of admissible integrands, statisticians need to remember a notion of progressively measurable process. The Ito integral can be naturally extended to a larger class of integrands. In order to proceed, the...

The main reason for stochastic integration lies in the necessity of modeling dynamical systems with randomness. This is done through stochastic differential equations, which are the main object of this chapter. To keep things simpler, the chapter considers a finite interval [0, T]; in the case of whole half-line [0,+∞], only marginal changes are ne...

In this chapter, a Wiener process is constructed on the interval [0, 1]. The sequences of Haar and Schauder functions on this interval are also considered. There exists a Wiener process with continuous trajectories. However, the trajectories are irregular in the sense that almost all trajectories have no derivative at one specific point. It is hard...

A stochastic process is a function of two variables, one of them being a time variable and the other one a sample point (elementary event). This chapter considers some examples of random processes and draws their trajectories. There are two main approaches to characterizing a stochastic process: by the properties of its trajectories and by some num...

This chapter first considers real-valued stochastic process on some interval [0, T]. It then discusses modification of stochastic processes, including stochastically equivalent and indistinguishable processes. Any indistinguishable processes are stochastically equivalent. Under some mentioned assumptions, indistinguishability is deduced from stocha...

This chapter describes notion of (sub-, super-) martingale. A vector process is called (sub-, super-) martingale if the corresponding property has each of its components. Evidently, any martingale is a (sub-, super) martingale. The chapter concentrates on the discrete-time martingale processes. This specific field is much simpler than the correspon...

The term “Markov chain” is used for a homogeneous Markov chain, while a general one is called a time-inhomogeneous Markov chain. This chapter summarizes the behavior of the continuous-time Markov chain. Continuous-time Markov chain spends an exponential time at state i and then switches to another state according to probabilities qij . The sequence...

This chapter first considers real-valued or vector-valued stochastic processes with independent increments. It then discusses three different approaches on how to introduce the Poisson process. The first approach is based on the characterization theorem for the processes with independent increments and gives a general definition of the (possibly no...

This chapter begins with considering drift and diffusion parameter estimation in the linear regression models (autoregressive model, and homogeneous diffusion model) with discrete time, continuous time, fractional Brownian motion and the Wiener noise. In order to estimate drift parameter assuming diffusion parameter to be known, the properties of q...

This chapter first discusses Gaussian vectors. The coordinates of a Gaussian vector are independent (in standard sense, as the random variables), if and only if they are non-correlated. If some subsets of coordinates of a random vector are Gaussian, it does not mean that the vector itself is Gaussian. Next, the chapter considers Gaussian processes...

We define a random measure generated by a real anisotropic harmonizable fractional stable field $Z^H$ with stability parameter $\alpha\in(1,2)$ and Hurst index $H\in(1/2,1)$ and prove that the measure is $\sigma$-additive in probability. An integral with respect to this measure is constructed, which enables us to consider a wave equation in $\mathb...

We estimate the kernel function of a symmetric alpha stable (\(S\alpha S\)) 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...

Stochastic heat equation on [0, T] × ℝd, d ≥ 1, driven by a general stochastic measure µ(t), t ∊ [0, T], is studied in this paper. The existence, uniqueness, and Hölder regularity of a mild solution are proved.

This short note is devoted to the “Fractality and Fractionality” workshop, held on 17–20 May 2016 in Lorentz Center (Leiden, Netherlands).

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 $\alpha$-stable spatial white noise multiplied by some regular function $\sigma$. We define a candidate solution $U$...

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 $\alpha>1$. It is shown that the approximations for its solution, which are defined by truncating the LePage series for $Z$, converge to the solution.

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 H...

We consider a mixed stochastic differential equation \(\mathop{\mathrm{d}}\nolimits \!X_{t} = a(t,X_{t})\mathop{\mathrm{d}}\nolimits \!t + b(t,X_{t})\mathop{\mathrm{d}}\nolimits \!W_{t} + c(t,X_{t})\mathop{\mathrm{d}}\nolimits \!B_{t}^{H}\) driven by independent multidimensional Wiener process and fractional Brownian motion. Under Hörmander type co...

We show that if a random variable is the final value of an adapted log-Holder
continuous process, then it can be represented as a stochastic integral with
respect to fractional Brownian motion with adapted integrand. In order to
establish this representation result, we extend the definition of fractional
integral.

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, as well as of the moments when the solutions hit certain
sets.

We show that small ball estimates together with Holder continuity assumption
allow to obtain new representation results in models with long memory. In order
to apply these results, we establish small ball probability estimates for
Gaussian processes whose incremental variance admits two-sided estimates and
the incremental covariance preserves sign....

For a mixed stochastic differential equation containing both Wiener process
and a H\"older continuous process with exponent $\gamma>1/2$, we prove a
stochastic viability theorem. As a consequence, we get a result about
positivity of solution and a pathwise comparison theorem. An application to
option price estimation is given.

In this paper we study asymptotic behaviour of power variations of a linear combination of independent Wiener process and fractional Brownian motion. This results are applied to construct consistence parameter estimators and approximate confidence intervals in mixed models.

The paper is concerned with a mixed stochastic delay differential equation
involving both a Wiener process and a $\gamma$-H\"older continuous process with
$\gamma>1/2$ (e.g. a fractional Brownian motion with Hurst parameter greater
than $1/2$). It is shown that its solution depends continuously on the
coefficients and the initial data. Two applicat...

This is an extended version of the lecture notes to a mini-course devoted to
fractional Brownian motion and delivered to the participants of 7th Jagna
International Workshop.

We consider a mixed stochastic differential equation
$d{X_t}=a(t,X_t)d{t}+b(t,X_t) d{W_t}+c(t,X_t)d{B^H_t}$ driven by independent
multidimensional Wiener process and fractional Brownian motion. Under Hormander
type conditions we show that the distribution of $X_t$ possesses a density with
respect to the Lebesgue measure.

We study integral representations of random variables with respect to general
H\"older continuous processes and with respect to two particular cases;
fractional Brownian motion and mixed fractional Brownian motion. We prove that
arbitrary random variable can be represented as an improper integral, and that
the stochastic integral can have any distr...

A lower estimate is given for the accuracy of approximation of random variables by functionals of the increments of a fractional Brownian motion with Hurst index H>1/2.

We show that if a random variable is a final value of an adapted Holder
continuous process, then it can be represented as a stochastic integral with
respect to fractional Brownian motion, and the integrand is an adapted process,
continuous up to the final point.

We prove that the standard conditions that provide unique solvability of a
mixed stochastic differential equations also guarantee that its solution
possesses finite moments. We also present conditions supplying existence of
exponential moments. For a special equation whose coefficients do not satisfy
the linear growth condition, we find conditions...

We consider a problem of statistical estimation of an unknown drift parameter
for a stochastic differential equation driven by fractional Brownian motion.
Two estimators based on discrete observations of solution to the stochastic
differential equations are constructed. It is proved that the estimators
converge almost surely to the parameter value...

We consider multifractional process given by double Ito--Wiener integrals,
which generalize the multifractional Rosenblatt process. We prove that this
process is continuous and has a square integrable local time.

In this article we study a homogeneous transient diffusion process $X$. We
combine the theories of differential equations and of stochastic processes to
obtain new results for homogeneous diffusion processes, generalizing the
results of Salminen and Yor. The distribution of local time of $X$ is found in
a closed form. To this end, a second order di...

We consider a stochastic delay differential equation driven by a Holder
continuous process and a Wiener process. Under fairly general assumptions on
its coefficients, we prove that this equation is uniquely solvable. We also
give sufficient conditions for finiteness of its moments and establish a limit
theorem.

For a mixed stochastic differential driven by independent fractional Brownian
motions and Wiener processes, the existence and integrability of the Malliavin
derivative of its solution are established. It is also proved that the solution
possesses exponential moments.

We obtain results on both weak and almost sure asymptotic behaviour of power
variations of a linear combination of independent Wiener process and fractional
Brownian motion. These results are used to construct strongly consistent
parameter estimators in mixed models.

In this paper, we consider a stochastic differential equation driven by a
fractional Brownian motion (fBm) and a Wiener process and having jumps. We
prove that this equation has a unique solution and show that all its moments
are finite.

We study the problem of optimal approximation of a fractional Brownian motion
by martingales. We prove that there exist a unique martingale closest to
fractional Brownian motion in a specific sense. It shown that this martingale
has a specific form. Numerical results concerning the approximation problem are
given.

For a mixed stochastic differential equation involving standard Brownian
motion and an almost surely H\"older continuous process $Z$ with H\"older
exponent $\gamma>1/2$, we establish a new result on its unique solvability. We
also establish an estimate for difference of solutions to such equations with
different processes $Z$ and deduce a correspon...

We consider a mixed stochastic differential equation involving both standard
Brownian motion and fractional Brownian motion with Hurst parameter $H>1/2$.
The mean-square rate of convergence of Euler approximations of solution to this
equation is obtained.

We show that a pathwise stochastic integral with respect to fractional
Brownian motion with an adapted integrand $g$ can have any prescribed
distribution, moreover, we give both necessary and sufficient conditions when
random variables can be represented in this form. We also prove that any random
variable is a value of such integral in some improp...

We consider a mixed stochastic differential equation driven by possibly dependent fractional Brownian motion and Brownian motion. Under mild regularity assumptions on the coefficients, it is proved that the equation has a unique solution.

We construct absolute continuous stochastic processes that converge to
anisotropic fractional and multifractional Brownian sheets in Besov-type
spaces.

An anisotropic harmonizable multifractional stable field is defined. Its continuity is proved. Existence and square integrability of local time are established. It is proved that the local time is jointly continuous in the Gaussian case.

The optimal stopping problem in a Lévy model is investigated. We show that the stopping region is nonempty for a wide class of models and payoff functions. In the general case, we establish sufficient conditions on the payoff function that provide nonemptiness of the stopping region. For a zero discounting rate we also give conditions for the stopp...

We discuss a constant which arises in several problems related to optimal exercise of American derivative securities.

A real harmonizable multifractional stable process is defined, its H\"older
continuity and localizability are proved. The existence of local time is shown
and its regularity is established.

We construct a multifractional process which has prescribed local versions at infinity and at finite points. Applications to financial mathematics are discussed.

We discuss a constant which arises in several problems related to optimal exercise of American derivative securities.

A Rosenblatt process and its multifractional counterpart are considered. For a multifractional Rosenblatt process, we investigate the local properties of its trajectories, namely the continuity and localizability. We prove the existence of square integrable local times for both processes.

We study suﬃcient conditions for the weak convergence of stochastic integrals with respect to processes of bounded variation, martingales, or semimartin- gales. A semimartingale theorem is extended to the multidimensional case. We apply a limit procedure and pass from processes of bounded variation to risk processes. An “inverse” problem for the we...

A new generalization of fractional Brownian motion (called multifractal Brownian motion) is considered for the case where the Hürst index H is a function of time t. The pathwise continuity of multifractal Brownian motion is proved. Global and local Hölder properties are also studied.

We introduce the notion of Vε-arbitrage (in other words, an arbitrage under the taxation proportional to the portfolio size) for a multiperiod discrete time model of a financial market. For a Vε-arbitrage, we prove a result analogous to the classical fundamental asset pricing theorem. Differences between a Vε-arbitrage and some other notions of arb...

Let \(S^{1}_{t}\), \(S_{t}^{2}\) be correlated geometric Brownian motions. We consider the following problem: find the stopping time τ
*≤T such that
$$\sup_{\tau\in[0,T]}\mathsf{E}[S_\tau^1-S_\tau^2]=\mathsf{E}[S_{\tau^*}^1-S_{\tau^*}^2]$$
where the supremum is taken over all stopping times from [0,T]. A similar problem, but on infinite interval, w...

Properties of solutions of stochastic diﬀerential equations with nonhomo- geneous coeﬃcients and non-Lipschitz diﬀusion are studied in the paper. Conditions on the coeﬃcients of an equation are obtained ensuring that a solution does not vanish over a finite time interval in the case of the diﬀusion (formula presented). We prove a limit theorem that...

For anticipative stochastic differential equations with Skorohod and white noise integrals the existence of approximate (in the sense that the difference between the left- and the right-hand sides of the equation is small in appropriate norm) solutions is proved under fairly mild conditions.