Related papers: A Generalized Ito Formula
The work considers a system of fractional order partial differential equations. The existence and uniqueness theorems for the classical solution of initial-boundary value problems are proved in two cases: 1) the right-hand side of the…
By means of Ito calculus it is possible to find, in a straight-forward way, the analytical solution to some equations related to the passive tracer transport problem in a velocity field that obeys the multidimensional Burgers equation and…
There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of…
An innovative theoretical framework for stochastic dynamics based on a decomposition of a stochastic differential equation (SDE) has been developed with an evident advantage in connecting deterministic and stochastic dynamics, as well as…
Given strong uniqueness for an It\^o's stochastic equation, we prove that its solution can beconstructed on "any" probability space by using, for example, Euler's polygonal approximations. Stochastic equations in $\mathbb{R}^{d}$ and in…
In this article we study a class of generalised linear systems of difference equations with given non-consistent initial conditions and infinite many solutions. We take into consideration the case that the coefficients are square constant…
In earlier papers it was shown that the generic tropical variety of an ideal can contain information on algebraic invariants as for example the depth in a direct way. The existence of generic tropical varieties has so far been proved in the…
We generalize several classical theorems in extremal combinatorics by replacing a global constraint with an inequality which holds for all objects in a given class. In particular we obtain generalizations of Tur\'an's theorem, the…
Relative orbifold Gromov-Witten theory is set-up and the degeneration formula is given.
The calculation of the decay rate of a metastable state in the path-integral formulation of stochastic processes is revisited. Previous derivations of this rate were achieved at the cost of a step that is difficult to justify…
The article is devoted to the expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on the generalized iterated Fourier series converging pointwise. The case of Fourier-Legendre…
This note examines the safety verification of the solution of Ito stochastic differential equations using the notion of stochastic zeroing barrier function. The main tools in the proposed method include Ito calculus and the concept of…
A formulation of singular classical theories (determined by degenerate Lagrangians) without constraints is presented. A partial Hamiltonian formalism in the phase space having an initially arbitrary number of momenta (which can be smaller…
We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique,…
We establish It\^o's formula along flows of probability measures associated with general semimartingales; this generalizes existing results for flows of measures on It\^o processes. Our approach is to first establish It\^o's formula for…
The present article is devoted to well-posedness by noise for the continuity equation. Namely, we consider the continuity equation with non-linear and partially degenerate stochastic perturbations in divergence form. We prove the existence…
We consider idealized financial markets in which price paths of the traded securities are cadlag functions, imposing mild restrictions on the allowed size of jumps. We prove the existence of quadratic variation for typical price paths,…
Point transformations of the 3-rd order ordinary differential equations are considered. Special classes of ordinary differential equations that are invariant under the general point transformations are constructed.
This paper is devoted to existence and uniqueness results for classes of nonlinear diffusion equations (or systems) which may be viewed as regular perturbations of Wasserstein gradient flows. First, in the case. where the drift is a…
Our goal is to prove existence results for classical solutions to some general nondegenerate Cauchy problems which are natural generalizations of Isaacs equations. For the latter we are able to extend our results by admitting local…