Related papers: Quasi-shuffle algebras in non-commutative stochast…
For stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the Ito flow map is given. A similar formula is also obtained for solutions of linear matrix-valued SDEs driven by arbitrary…
We develop a stochastic calculus that makes it easy to capture a variety of predictable transformations of semimartingales such as changes of variables, stochastic integrals, and their compositions. The framework offers a unified treatment…
We give an infinitesimal meaning to the symbol $dX_t$ for a continuous semimartingale $X$ at an instant in time $t$. We define a vector space structure on the space of differentials at time $t$ and deduce key properties consistent with the…
A peculiar feature of It\^o's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative…
Branched rough paths, defined as paths with values in the character group of the Connes-Kreimer Hopf algebra $\mathcal{H}_\mathrm{CK}$, constitute integration theories that may fail to satisfy the usual integration by parts identity. Using…
We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfill the minimality criterion, first, the family must be a minimal…
Given a matrix of distribution functions and a quasi-stochastic matrix, i.e. an irreducible nonnegative matrix with maximal eigenvalue one and associated unique positive left and right eigenvectors, the article studies the properties of an…
For non-anticipative functionals, differentiable in Chitashvili's sense, the It\^o formula for cadlag semimartingales is proved. Relations between different notions of functional derivatives are established.
We develop a calculus of space-time controlled fields for rough stochastic systems. This approach provides a unified composition rule for evaluating random fields along rough semimartingales and yields a rough stochastic It\^o-Wentzell…
Based on an extension of the martingale comparison method some comparison results for path-dependent functions of semimartingales are established. The proof makes essential use of the functional It\^o calculus. A main tool is an extension…
This paper first summarizes the foundations of stochastic calculus via regularization and constructs through this procedure It\^o and Stratonovich integrals. In the second part, a survey and new results are presented in relation with finite…
An estimation method is proposed for a wide variety of discrete time stochastic processes that have an intractable likelihood function but are otherwise conveniently specified by an integral transform such as the characteristic function,…
Stochastic exponentials are defined for semimartingales on stochastic intervals, and stochastic logarithms are defined for semimartingales, up to the first time the semimartingale hits zero continuously. In the case of (nonnegative) local…
This paper does not suppose a priori that the evolution of the price of a financial asset is a semimartingale. Since possible strategies of investors are self-financing, previous prices are forced to be finite quadratic variation processes.…
We exhibit conditions under which the flow of marginal distributions of a discontinuous semimartingale $\xi$ can be matched by a Markov process, whose infinitesimal generator is expressed in terms of the local characteristics of $\xi$. Our…
The purpose of these notes is to distribute, mostly without proofs, fundamental definitions and results concerning the theory of semimartingales and stochastic integration. The material serves as a foundational guide for those interested in…
In this paper we study the path-regularity and martingale properties of the set-valued stochastic integrals defined in our previous work Ararat et al. (2023). Such integrals have some fundamental differences from the well-known…
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…
We derive a functional change of variable formula for {\it non-anticipative} functionals defined on the space of right continuous paths with left limits. The functional is only required to possess certain directional derivatives, which may…
We extend the Ito -to- Stratonovich analysis or quantum stochastic differential equations, introduced by Gardiner and Collett for emission (creation), absorption (annihilation) processes, to include scattering (conservation) processes.…