arXiv subjects

Alexander Kalinin

Publications and source records attributed to Alexander Kalinin.

12 recordsLinked to original sources

arXiv2026

On McKean-Vlasov SDEs with polynomial drifts for SIS epidemic models

We present a tractable class of one-dimensional McKean-Vlasov equations that allow for unique strong solutions and extend the dynamics of various SIS epidemic models that are well-established in the literature. While the distribution-dependent drift coefficients are of polynomial type, the diffusion coefficients may involve sums of power functions. Our analysis includes various scenarios of extinction and persistence of the disease and an effective Euler-Maruyama scheme, for which we derive an explicit strong error estimate in $p$th moment for $p\geq 2$.

math.PR↗

arXiv2026

Stochastic Volterra equations with random functional coefficients in Banach spaces

We derive unique Banach-valued solutions to stochastic Volterra equations with random coefficients that may depend on pure chance and involve singular kernels. In particular, for controlled and distribution-dependent coefficients these solutions become strong, as a measurability analysis of the Wasserstein metric confirms. The presented novel approach is based on the proof that a stochastic Volterra integral admits a progressively measurable modification in a weak sense and on sharp moment estimates for non-negative product measurable processes.

math.PR↗

arXiv2024

Resolvent and Gronwall inequalities and fixed points of evolution operators

We introduce kernels and resolvents on preordered sets and derive sharp resolvent inequalities that entail Gronwall inequalities for functions of several variables. In this way, we can prove a fixed point result for operators on topological spaces that extends Banach's fixed point theorem and allows for a wide range of applications.

math.FA↗

arXiv2023

The oriented derivative

We show that the derivatives in the sense of Fréchet and Gâteaux can be viewed as derivatives oriented towards a star convex set with the origin as center. The resulting oriented differential calculus extends the mean value theorem, the chain rule and the Taylor formula in Banach spaces. Moreover, the oriented derivative decomposes additively along countably infinite orthogonal sums in Hilbert spaces.

math.FA↗

arXiv2022

Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs in arbitrary moments

We deduce stability and pathwise uniqueness for a McKean-Vlasov equation with random coefficients and a multidimensional Brownian motion as driver. Our analysis focuses on a non-Lipschitz drift coefficient and includes moment estimates for random It\^o processes that are of independent interest. For deterministic coefficients we provide unique strong solutions, even if the drift fails to be of affine growth. The theory that we develop rests on It\^o's formula and leads to $p$-th moment and pathwise $\alpha$-exponential stability for $p\geq 2$ and $\alpha > 0$ with explicit Lyapunov exponents, regardless of whether a Lyapunov function exists.

math.PR↗

arXiv2021

Mild to classical solutions for XVA equations under stochastic volatility

We extend the valuation of contingent claims in presence of default, collateral and funding to a random functional setting and characterise pre-default value processes by martingales. Pre-default value semimartingales can also be described by BSDEs with random path-dependent coefficients and martingales as drivers. En route, we generalise previous settings by relaxing conditions on the available market information, allowing for an arbitrary default-free filtration and constructing a broad class of default times. Moreover, under stochastic volatility, we characterise pre-default value processes via mild solutions to parabolic semilinear PDEs and give sufficient conditions for mild solutions to exist uniquely and to be classical.

math.PR↗

arXiv2021

Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs: a multidimensional Yamada-Watanabe approach

We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean-Vlasov equations with random non-Lipschitz continuous coefficients. In the deterministic case, we also obtain the existence of unique strong solutions. By using our approach, which is based on an extension of the Yamada-Watanabe ansatz to the multidimensional setting and which does not rely on the construction of Lyapunov functions, we prove first moment and pathwise exponential stability. Furthermore, Lyapunov exponents are computed explicitly.

math.PR↗

arXiv2019

On the support of solutions of stochastic differential equations with path-dependent coefficients

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron-Martin space under the flow of mild solutions to a system of path-dependent ordinary differential equations. Our result extends the Stroock-Varadhan support theorem for diffusion processes to the case of stochastic differential equations with path-dependent coefficients. The proof is based on functional Ito calculus.

math.PR↗

arXiv2019

Support characterization for regular path-dependent stochastic Volterra integral equations

We consider a stochastic Volterra integral equation with regular path-dependent coefficients and a Brownian motion as integrator in a multidimensional setting. Under an imposed absolute continuity condition, the unique solution is a semimartingale that admits almost surely Hölder continuous paths. Based on functional Itô calculus, we prove that the support of its law in the Hölder norm can be described by a flow of mild solutions to ordinary integro-differential equations that are constructed by means of the vertical derivative of the diffusion coefficient.

math.PR↗

arXiv2018

Mild and viscosity solutions to semilinear parabolic path-dependent PDEs

We study and compare two concepts for weak solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). The first is that of mild solutions as it appears, e.g., in the log-Laplace functionals of historical superprocesses. The aim of this paper is to show that mild solutions are also solutions in a viscosity sense. This result is motivated by the fact that mild solutions can provide value functions and optimal strategies for problems of stochastic optimal control. Since unique mild solutions exist under weak conditions, we obtain as a corollary a general existence result for viscosity solutions to semiilinear parabolic PPDEs.

math.PR↗

arXiv2017

Markovian Integral Equations

We analyze multidimensional Markovian integral equations that are formulated with a time-inhomogeneous progressive Markov process that has Borel measurable transition probabilities. In the case of a path-dependent diffusion process, the solutions to these integral equations lead to the concept of mild solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). Our goal is to establish uniqueness, stability, existence, and non-extendibility of solutions among a certain class of maps. By requiring the Feller property of the Markov process, we give weak conditions under which solutions become continuous. Moreover, we provide a multidimensional Feynman-Kac formula and a one-dimensional global existence- and uniqueness result.

math.PR↗

arXiv2014

Using crowdsourcing system for creating site-specific statistical machine translation engine

A crowdsourcing translation approach is an effective tool for globalization of site content, but it is also an important source of parallel linguistic data. For the given site, processed with a crowdsourcing system, a sentence-aligned corpus can be fetched, which covers a very narrow domain of terminology and language patterns - a site-specific domain. These data can be used for training and estimation of site-specific statistical machine translation engine

cs.CL↗