arXiv subjects

Elena Boguslavskaya

Publications and source records attributed to Elena Boguslavskaya.

8 recordsLinked to original sources

arXiv2026

Fractional Wiener chaos: Part 2. Interface spectral chaos

We construct a product spectral decomposition from power-normalised parabolic-cylinder functions. At noninteger real orders, these functions are not square integrable with respect to the corresponding full-line Gaussian measure. Bringing the admissible half-line branches into the same Gaussian space and matching them at the origin gives a corrected orthonormal eigenbasis. A unitary transformation to a weighted probability space makes the eigenfunction corresponding to the lowest eigenvalue constant and permits consistent countable products. The resulting law is non-Gaussian under deformation, and the decomposition recovers classical Wiener chaos when the deformation is removed. We identify the associated closed gradient, adjoint divergence and deformed number operator. The Rodrigues formula covers all orders occurring in the spectral branches, including negative orders. Under deformation, however, changing both branch orders by the same nonzero amount breaks the required relation between them. Fractional evolution defined through spectral calculus preserves the basis. After unitary identification, the one-coordinate fractional evolution operators converge in operator norm to Gaussian and three-dimensional radial Ornstein--Uhlenbeck fractional evolutions at the respective parameter endpoints. Convergence is uniform on compact time intervals away from zero. An application example with inverse stable clock is provided in the accompanying github repository.

math.PR↗

arXiv2026

Entire Logarithmic Signatures of Bounded-Variation Paths in Finite Dimensions

We classify signatures of bounded-variation paths in finite-dimensional real normed spaces whose logarithms are entire, in the sense of superexponential homogeneous decay. The zero-first-level fibre is trivial; if the first level is $v\neq0$, the signature is $a\e^v a^{-1}$ with $a$ a bounded-variation signature, and every such conjugate is entire. For a given path, $a$ may be chosen from a prefix. For a tree-reduced representative, a gate-selected prefix gives weak path conjugacy to a line. The proof uses a finite-dimensional spectral-growth statement: if $q\geq1$, $A:\C\to M_q(\C)$ is entire, and $\norm{\e^{A(z)}}\leq C\e^{τ\abs{z}}$, then the $k$th characteristic-polynomial coefficient of $A(z)$ has degree at most $k$ for $1\leq k\leq q$. Universal matrix isospectrality, resonant developments, metric-tree fixed points and compactness, and Stieltjes--Fourier reconstruction establish the bounded-variation modified Lyons--Sidorova conjecture under a hypothesis imposed only on the whole path.

math.PR↗

arXiv2024

Fractional Wiener Chaos

The area of fractional calculus has made its way into various pure and applied scientific fields, as evidenced by its integration into numerous disciplines. An increasing number of researchers are exploring various approaches to incorporating fractional calculus into stochastic analysis. In this paper, we generalise the Wiener chaos expansion by constructing a fractional Wiener chaos expansion based on the parabolic cylinder function with an exponential factor. In the process, we demonstrate that this parabolic cylinder function with an exponential factor, which we call "a power normalised parabolic cylinder function" acts as an extension of a Hermite polynomial and retains the same martingale properties inherent in the Hermite polynomial, as well as other basic properties of Hermite polynomials. Hence, it is accurate to state that power normalised cylindrical functions can be viewed as fractional versions of Hermite polynomials.

math.PR↗

arXiv2020

Revisiting integral functionals of geometric Brownian motion

In this paper we revisit the integral functional of geometric Brownian motion $I_t= \int_0^t e^{-(μs +σW_s)}ds$, where $μ\in\mathbb{R}$, $σ> 0$, and $(W_s )_s>0$ is a standard Brownian motion. Specifically, we calculate the Laplace transform in $t$ of the cumulative distribution function and of the probability density function of this functional.

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↗

arXiv2015

Utility maximization in Wiener-transformable markets

We consider a utility maximization problem in a broad class of markets. Apart from traditional semimartingale markets, our class of markets includes processes with long memory, 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. We introduce the notion of a Wiener-transformable Gaussian process, and give examples of such processes, and their representations. The representation for the solution of the utility maximization problem in our specific setting is presented for various utility functions.

math.PR↗

arXiv2015

Solving optimal stopping problems for Lévy processes in infinite horizon via $A$-transform

We present a method to solve optimal stopping problems in infinite horizon for a Lévy process when the reward function can be non-monotone. To solve the problem we introduce two new objects. Firstly, we define a random variable $η(x)$ which corresponds to the argmax of the reward function. Secondly, we propose a certain integral transform which can be built on any suitable random variable. It turns out that this integral transform constructed from $η(x)$ and applied to the reward function produces an easy and straightforward description of the optimal stopping rule. We check the consistency of our method with the existing literature, and further illustrate our results with a new example. The method we propose allows to avoid complicated differential or integro-differential equations which arise if the standard methodology is used.

math.PR↗

arXiv2015

An explicit solution for optimal investment in Heston model

In this paper we consider a variation of the Merton's problem with added stochastic volatility and finite time horizon. It is known that the corresponding optimal control problem may be reduced to a linear parabolic boundary problem under some assumptions on the underlying process and the utility function. The resulting parabolic PDE is often quite difficult to solve, even when it is linear. The present paper contributes to the pool of explicit solutions for stochastic optimal control problems. Our main result is the exact solution for optimal investment in Heston model.

q-fin.MF↗