Related papers: Stochastic Calculus for Markov Processes Associate…

Motivated by applications to SPDEs we extend the It\^o formula for the square of the norm of a semimartingale $y(t)$ from Gy\"ongy and Krylov (Stochastics 6(3):153-173, 1982) to the case \begin{equation*} \sum_{i=1}^m \int_{(0,t]}…

The article is devoted to the expansions of iterated Stratonovich stochastic integrals on the basis of the method of generalized multiple Fourier series that converge in the sense of norm in Hilbert space $L_2([t, T]^k),$ $k\in\mathbb{N}.$…

We consider the stochastic CGL equation $$ \dot u- \nu\Delta u+(i+a) |u|^2u =\eta(t,x),\;\;\; \text {dim} \,x=n, $$ where $\nu>0$ and $a\ge 0$, in a cube (or in a smooth bounded domain) with Dirichlet boundary condition. The force $\eta$ is…

Quantum chaotic states over a noncommutative monoid, a unitalization of a noncommutative Ito algebra parametrizing a quantum stochastic Levy process, are described in terms of their infinitely divisible generating functionals over the…

Langevin equation with a multiplicative stochastic force is considered. That force is uncorrelated, it has the L\'evy distribution and the power-law intensity. The Fokker-Planck equations, which correspond both to the It\^o and Stratonovich…

We study the zeros sets of functions in the Dirichlet space. Using Carleson formula for Dirichlet integral, we obtain some new families of zero sets. We also show that any closed subset of $E \subset \TT$ with logarithmic capacity zero is…

We present an extension of Nelson's stochastic quantum mechanics to finite temperature. Utilizing the formulation of Thermo Field Dynamics (TFD), we can show that Ito's stochastic equations for tilde and non-tilde particle positions…

We show that a differential version of the classical Chebyshev-Markov-Stieltjes inequalities holds for a broad family of weight functions. Such a differential version appears to be new. Our results apply to weight functions which are…

The article is devoted to the mean-square approximation of iterated Ito and Stratonovich stochastic integrals in the context of the numerical integration of Ito stochastic differential equations. The expansion of iterated Ito stochastic…

This article introduces a certain class of stochastic processes, which we suggest to call mild Ito processes, and a new - somehow mild - Ito type formula for such processes. Examples of mild Ito processes are mild solutions of SPDEs and…

We consider the following quasi-linear parabolic system of backward partial differential equations: $(\partial_t+L)u+f(\cdot,\cdot,u, \nabla u\sigma)=0$ on $[0,T]\times \mathbb{R}^d\qquad u_T=\phi$, where $L$ is a possibly degenerate second…

From the path integral formalism for price fluctuations with non-Gaussian distributions I derive the appropriate stochastic calculus replacing Ito's calculus for stochastic fluctuations.

We establish an integral representation for the Dirichlet generating function of the coefficients of Euler's pentagonal number theorem. The Bromwich-type integral enables analytic continuation to the entire complex plane, filling a gap in…

This paper considers the problem of constructing finite-dimensional state space realizations for stochastic processes that can be represented as the outputs of a certain type of a causal system driven by a continuous semimartingale input…

We extend the It\=o formula \cite{MR1837298}*{Theorem 2.3} for semimartingales with rcll paths. We also comment on Local time process of such semimartingales. We apply the It\=o formula to L\'evy processes to obtain existence of solutions…

These notes rigorously construct the stochastic integral of a Hilbert Space valued process driven by a Cylindrical Brownian Motion. We expand upon this stochastic calculus to present an introduction to stochastic differential equations in…

The article is devoted to the construction of explicit one-step strong numerical methods with the orders 2.0 and 2.5 of convergence for Ito stochastic differential equations with multidimensional non-commutative noise. We consider the…

K. It\^{o} characterised in \cite{ito} zero-mean stationary Gauss Markov-processes evolving on a class of infinite-dimensional spaces. In this work we extend the work of It\^{o} in the case of Hilbert spaces: Gauss-Markov families that are…

We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming…

Using the recently developed covariant Ito-Langevin dynamics, we develop a non-equilibrium thermodynamic theory for small systems coupled to multiplicative noises. The theory is based on Ito-calculus, and is fully covariant under…