Related papers: An extension of Wiener integration with the use of…
Approximation theory is a substantial field of mathematical analysis that emerged in the 19th century and has been developed by mathematicians across the globe ever since. Its importance has increased over time, as it provides solutions to…
Along the ideas of Curtain and Glover, we extend the balanced truncation method for infinite-dimensional linear systems to bilinear and stochastic systems. Specifically , we apply Hilbert space techniques used in many-body quantum mechanics…
Given a bounded linear operator $T$ on separable Hilbert space, we develop an approach allowing one to construct a matrix representation for $T$ having certain specified algebraic or asymptotic structure. We obtain matrix representations…
The integration-by-parts formula discovered by Malliavin for the Ito map on Wiener space is proved using the two-parameter stochastic calculus. It is also shown that the solution of a one-parameter stochastic differential equation driven by…
The Lie-Trotter formula, together with its higher-order generalizations, provides a direct approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly…
We develop an operator-theoretic formulation of stochastic calculus for fractional Brownian motion with Hurst parameter H in (0, 1/2). The approach is based on adjointness between stochastic integration and differentiation in the…
This paper introduces a factorization for the inverse of discrete Fourier integral operators that can be applied in quasi-linear time. The factorization starts by approximating the operator with the butterfly factorization. Next, a…
In this paper, by using a Taylor development type formula, we show how it is possible to associate differential operators with stochastic differential equations driven by a fractional Brownian motion. As an application, we deduce that…
By combining the Malliavin calculus with Fourier techniques, we develop a high-order asymptotic expansion theory for a sequence of vector-valued random variables. Our asymptotic expansion formulas give the development of the characteristic…
By means of the Ito-Nisio theorem, we introduce and discuss a general approach to series representations of path integrals. We then argue that the optimal basis for both ``primitive'' and partial averaged approaches is the Wiener…
We study the existence of a unique solution to semilinear fractional backward doubly stochastic differential equation driven by a Brownian motion and a fractional Brownian motion with Hurst parameter less than 1/2. Here the stochastic…
In this work, we generalise the stochastic local time space integration introduced in \cite{Ei00} to the case of Brownian sheet. %We develop a stochastic local time-space calculus with respect to the Brownian sheet. This allows us to prove…
Gaussian ODE filtering is a probabilistic numerical method to solve ordinary differential equations (ODEs). It computes a Bayesian posterior over the solution from evaluations of the vector field defining the ODE. Its most popular version,…
In this work, we investigate a theory of stochastic integration for operator-valued processes with respect to semimartingales taking values in the dual of a nuclear space. Our construction of this particular stochastic integral relies on…
Using the Euler--Maruyama technique, we show that a class of Wiener processes exist that are obtained by computing an arbitrary positive power of them. This can be accomplished with a proper set of definitions that makes meaningful the…
We consider some versions and generalizations of an approach to the expansion of iterated Ito stochastic integrals of arbitrary multiplicity $k$ $(k\in\mathbb{N})$ based on generalized multiple Fourier series. Expansions of iterated…
We propose a stochastic extension of deformation quantization on a Hilbert space. The Moyal product is defined in this context on the space of functionals belonging to all of the Sobolev spaces of the Malliavin calculus.
Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for…
We introduce an adaptation of integral approximation operators to set-valued functions (SVFs, multifunctions), mapping a compact interval $[a,b]$ into the space of compact non-empty subsets of ${\mathbb R}^d$. All operators are adapted by…
Finite Cartesian products of operators play a central role in monotone operator theory and its applications. Extending such products to arbitrary families of operators acting on different Hilbert spaces is an open problem, which we address…